NAVAL POSTGRADUATE SCHOOL Monterey, California · Approved bN . -/Od # Aprvdb. _ Lawrence ijo ......

AD-A245 476

NAVAL POSTGRADUATE SCHOOLMonterey, California

SDTiC

'FI ACT

't ES0~ 1992

THESIS

SYMBOLIC SOLUTION OF A MULTILAYER OCEAN WAVEGUIDEPROBLEM WITH ARBITRARY DEPTH DEPENDENT AMBIENT

DENSITY AND SOUND SPEED PROFILES

by

LCDR Charles Joseph Young, Jr., USN

December 1991

Thesis Advisor: Lawrence J. Ziomek

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11 TITLE (Include Security Classification)SYMBOLIC SOLUTION OF' A MULTILAYER OCEAN WAVEGUIDE PROBLEM WITH ARBITRARY DEPTH DEPENDENTAMBIENT DENSITY AND SOUND SPEED PROFILES

12 PERSONAJ.AUTJOR(S) roung, unar e s J. , Jr.

13a TYPE OF REPORT T13b ThM[ COVERED 114 DATE OF REPORT (Year, Month, Day) 15 PAGE COUNT

Master's Thesis FPOM - TO December 1991 23216 SUPPLEMENTARY NOTATION The views expressed in this thesis are those of the author and donot reflect the official policy or position of the Department of Defense or theU.S. government.

17 COSA',

CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

FIELD GROUP SUBGROUP Multilayer waveguide model; arbitrary depth-dependentambient density and sound speed profiles; symbolic

solutions; weighted least squares estimation.19 ABSTRACT (Continue on reverse if necessary and identify by block number)

The main purpose of this thesis was to obtain the symbolic solution of a multilayer(four fluid media) ocean waveguiu= problem. The waveguide was assumed to have depth-dependent ambient density and sound speed profiles in all fluid media, and arbitrarilyshaped boundaries between all fluid media. A system of 28 equations in 17 unknownswas generated by satisfying all of the boundary conditions (including the boundarycondition at the source) in cylindrical coordinates. The problem was set up as aweighted least squares problem for symbolic solution by the computer program MathematicaDue to software and hardware constraints, a symbolic solution for the most general casewas not obtained. However, by making all of the boundaries plane, parallel boundaries,two cases were successfully programmed, yielding symbolic solutions which were verified

by comparison to previously known results.

20 D,STR;BbT',:0% AvAI'LABILT

Y OF:A %i''C 21 ABSTRACT SECJRIT" CASSIiCATiON

5- JNCLASSIF ED UNLiM'TED EJ SAVE AS RPT u DTIC 1'SEPS Unclassified12a %AME OF RES0ON4S!B E :D'V1DJAL 22b TELEPHONE (Include Area Code) 22L O"1ICE SYMBOL

Lawrence J. Ziomek (408 646-3206 1 EC/IDD Form 1473, JUN 86 Previous editions are obsolete SECL:PITV CLASS4 CA, OT% <l Ti S PACE

S/N 0102-LF-O14-6603 Unclassified

i

Approved for public release; distribution is unlimited

Symbolic Solution of a Multilayer Ocean Waveguide Problem With ArbitraryDepth Dependent Ambient Density and Sound Speed Profiles

by

Charles Joseph. Young, Jr.Lieutenant Commander, United States Navy

B.S., New Jersey Institute of Technology, 1978

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN ENGINEERING ACOUSTICS

from the

NAVAL POSTGRADUATE SCHOOLDecember 1991

Author!Charl ,o h Youg, Jr

Approved bN . -/Od #Aprvdb. _ Lawrence ijo 6k, Thesis Advisor

Hu fg-Mo ee ec d Reader

anthny A. At ley," d~irma .E ineering Acousticsr Academic Coi ittee

ABSTRACT

The main purpose of this thesis was to obtain the symbolic solution of a

multilayer (four fluid media) ocean waveguide problem. The waveguide was

assumed to have depth-dependent ambient density and sound-speed

profiles in all fluid media, and arbitrarily shaped boundaries between all

fluid media. A system of 28 equations in 17 unknowns was generated by

satisfying all of the boundary conditions (including the boundary condition

at the source) in cylindrical coordinates. The problem was set up as a

weighted least squares problem for symbolic solution by the computer

program Mathematica. Due to software and hardware constraints, a symbolic

solution for the most general case was not obtained. However, by making all

of the boundaries plane, parallel boundaries, two cases were successfully

programmed, yielding symbolic solutions which were verified by comparison

to previously known results.

Aceession For

DTI T:P 0(7,. d []

11/orit

TABLE OF CONTENTS

I. A MORE GENERAL SOLUTION TO THE LINEAR, THREE-DIMENSIONAL,LOSSLESS, HOMOGENEOUS WAVE EQUATION ................................................... I

II. DESCRIPTION OF THE GENERALIZED OCEAN WAVEGUIDE MODEL ....... 10

111. DEVELOPMENT OF THE GENERAL WAVEGUIDE MODEL SOLUTION ....... 13

A. VELOCITY POTENTIAL IN MEDIUM I .......................................................... 14

B. VELOCITY POTENTIAL IN MEDIUM II .................................................. 16

C. VELOCITY POTENTIAL IN MEDIUM III ............................................... 18

D. VELOCITY POTENTIAL IN MEDIUM IV ................................................. 19

E. BOUNDARY CONDITIONS .............................................................................. 2 1

F. SUMMARY OF BOUNDARY CONDITION EQUATIONS AND THEIRV A L ID IT Y ................................................................................................................ 6 7

G. DIFFERENCES NOTED DUE TO ARBITRARY BOUNDARY SHAPE ........... 75

H. VERIFICATION OF INITIAL RESULTS .......................... 79

IV. SOLUTION FOR THE UNKNOWN ARBITRARY CONSTANTS USINGSYMBOLIC ALGEBRA CAPABILITIES OF Maathematica ....................... 87

V. CONCLUSIONS AND RECOMMENDATIONS .......................... 223

L IST OF REFER EN CES ............................................................................................................. 226

IN IT IA L D ISTR IBUT ION L IST ......................................................................................... 227

iv

i. A MORE GENERAL SOLUTION TO THE LINEAR. THREE-DIMENSIONAL. LOSSLESS. HOMOGENEOUS WAVE EQUATION

The primary objective of this section is to derive the solution of the

linear, three-dimensional, lossless, homogeneous wave equation (1) in the

cylindrical coordinate system defined in Figure 1 for an arbitrary sound

speed profile (a function of the depth coordinate, y, only).

I a2q(t,r) (1)

t2(t,r) -c 2(y) A2 -

The Laplacian expressed in the defined cylindrical coordinate system is

given by (2).

2 a2 l a I a 2 a2;r2 r ar - = 2 a2_ (2)

The remaining quantities of (1) are defined as follows:

* q(t,r) is the velocity potential at tiwge t at a position r - (r,),y) expressed

in units of square meters per second, and

* c(y) is the arbitrary, depth-dependent speed of sound expressed in units

of meters per second.

The first assumption we will make in this derivation is that the source

of acoustic energy in the waveguide has a time-harmonic dependence. This

assumption may be justified by the facts that any arbitrary time dependence

can be expressed as a summation of time-harmonic terms using Fourier

U1

analysis. In addition, one of the basic tenets of linear acoustics allows the

velocity potential to be expressed as a linear combination (using the

superposition principle) of such terms.

Z!

(r,, 0)

xr y

(r, ,y)

y

Figure 1. The Cylindrical Coordinate System

Therefore, the resulting velocity potential will also have a time-

harmonic dependence given by

op(t,r) - q(r) e•Ix", (3)

where, f is the source frequency in Hertz.

Substituting (3) into (1) gives

V2[98r)ej2xft - 82[q(r) e12 l t) W (4)c2(y) at 2

2

Taking the time derivatives first yields

a[g(r) OURs f ] _ ( j nf 9r)) iat = () dt =j ()e~ r

Ap2[(r) eift] - a(4 (r) e 22fN) - a[j2xf qr) ejI xft]

at2 at at at

= (j2Xf) 2 oq(r) eI3ft

82 q(r) 2eift) - C2 ip(r) ei2 t, (5)at2

where co- 2f is the angular source frequency in radians per second.

Substituting (5) into (4) and observing that the Laplacian does not

operate on the time-harmonic term ei2xft yields

eIj xft V2qp(r) + c2 y 9(r) ei2x' - 0

Dividing out the complex exponential terms reveals the time-

independent lossless Helmholtz equation

V29(r) + k2(y) 9(r) - 0. (6)

where,

0- 2af 2x (7)(y ~) "cRy) "-.My)(7

is the wave number expressed in units of radians per meter and k is the

wavelength expressed in meters.

3

The next step is to find the solution to the lossless Helmholtz equation

(6). This will be accomplished using the method of separation of variables.

We will assume that the solution for 9(r) - I(r*,y) has the form

9(r)- 9(~y)- R~) V) Y~) .(8)

Substituting (8) into (6) yields

Performing the Laplacian operation reveals

82[R(r) 0(+) Y(y)1 I A[R(r) 0'(+) Y(y)1 I a2[R(r) V+$) Y(y)1ar2 Ir ar Ir

+ A2 RW(r) Y(y)1 + k2(y) IR(r) 044+) Y(y)I - 0 . (9)

The partial derivatives of (9) may be replaced with total derivatives

since the functions R, 4', and Y are each simply functions of the single

variables r, +, and y respectively. Continuing, (9) may be written as

dR Yr) 1 A)Yy dR(r) 1 Rr d2 '() Adr 2 r4$Y~) dr +r Yy d#2

+R(r) () dy2 +k2(y) IR(r) -t(+) Y(y)I - 0. (10)

Dividing (10) by the product IR(r) 4C+) Y(y)] yields

1 d2R(r) 1 ORr) 1 d24)(+) 1 2dy) _)R(r) dr2 + R(r) jr 4 rk()~ Y(y) dy2 +k(y-0

4

Separating out the depth dependence reveals

1 y k(y) - 1r d2R(r) I dR(r) 1 d24(*)

Y(y) dy 2 + R(r) dr 2 - rR(r) dr -r 24(() d+2 (11)

Since the left-hand side of ( 1) is a function of a single variable (y) and

the right-hand side is a function of two variables (r, +), the equality can only2be true if each side is equal to the same constant, say k2. Thus, the right-

hand side of ( 1) becomes

1 d2R(r) I dR(r) 1 d24() 2R(r) dr 2 rR(r) dr - r24,(+) d+2 kr "

Multiplying through by r2 and rearranging reveals

r 2 d2R(r) r dR(r) k2 r2 - I d24)()R(r) dr2 R(r) dr - 44+) d*2 (12)

Since the !eft-hand side (LHS) of (12) is a function of a single variable

(r) and the right-hand side (RHS) is a function of a different single variable

(#), the equality can hold only if each side is equal to the same constant, say

n2. Thus, the right-hand side of (12) becomes

I d24(4) 2

0+) d#2 -n. (13)

Multiplying (13) through by -4(#) and rearranging yields

d24'(*) +n2 0+)- 0 (14)

5

Equation (14) is a second order ordinary differential equation with the

following exact solution:

44() -A+ cos(nf) + B, sin(n*), (15)

where A, and B+ are in general complex constants whose values are

determined by satisfying boundary conditions.

Now, the left-hand side of (12) must be evaluated. Since the left-hand

side of (12) must also be equal to n2, we have

r2 d2R(r) r dR(r) 2 n2

R(r) dr 2 R(r) dr + kr _n (16)

R(r)Multiplying ( 6) by R- and rearranging reveals

d2R(r) I dR(r) [2 n(l

dr 2 'r dr + kr R(r)-0. (17)

Let

R(r) = g(kr r). (18)

Then

dR(r) d[g(k r r)1 d[g(k. r)1 d(kr r)dr dr d(k. r) dr

or,

6

dR(r) d[g(k r r)]dr - kr d(kr r) (19)

Additionally,

d2R(r) d dR(r)] dik d(g(k, r))] d [dgk, rfljd dr d- drr d(k, r) kr d(kr) dr

or,d2R(r) k 2 d2[g(kr ]

dr 2 r d(kr r) 2 (20)

Substituting (18) through (20) into (17) yields

k2 d2[g(k r+ kr d[g(k2_ r)] - 2n2 1

dk 2 r d(kr r) kr g(kr)0. (21)

2

Dividing (21) by kr yields

d 2[g(k r)1 1 d[g(kr r) I gkr)- 0 (22)

d(kr r)2 + kr r d(kr r) + ( r)2- g 2, -0.

Equation (22) is known as Bessel's differential equation, which has the

following exact solution:

g(kr r) - Ar Jn(kr r) + Br Nn(kr r), (23)

where J. and Nn are the Bessel functions of the first and second kind

respectively, and Ar and Br are in general complex constants whose values

are determined by satisfying boundary conditions.

7

Since we let R(r) - g(k. r), the exact solution for R(r) can be written as

follows:

R(r) - A, Jn (k r) + B, Nn(kr r). (24)2

Now, the left-hand side of ( 1) must be set equal to k , revealing

I d-("y) + k2(y) k (25)Y(y) dy 2 r (

Multiplying (25) by Y(y) and rearranging yields

d2y(y) 2] y2~ 2dy- ) Y(y) =0. (26)dy2

Let(27

ky(y) = k2(y) - k . (27)

Using (27) to rewrite (26) yields

d +Y(Y) ( (y)g(y)- 0 (28)dy2 k+

Since the coefficient k1 (y) in (28) is an arbitrary function of depth, y, an

exact solution to this differential equation cannot be determined.

8

In order to continue this derivation of a generalized solution to the

wave equation, we will assume that the solution to (28) can be determined

by other means, and is given simply by

Y(y) - Y(y). (29)

In certain cases, the sound speed profile, c(y), may be a function such

that an exact solution to (28) may be found (say by consulting tables). In

other specific cases, approximations can be made in order to put (28) in a

form whereby a known solution may be found. However, in the most general

case, (28) will have to be evaluated numerically.

Recall that (8) specifies the velocity potential as

qi(rO,y) - R(r) 0(+) Y(y). (30)

Substituting (15). (24), and (29) into (30) yields the following general

solution for the velocity potential:

qp(r,o,y) - [ArJn(krr) + BrNn(krr) [A, cos(no) + B, sin(n$)] Y(y). (31)

Finally, recall that (3) described the velocity potential as a function of

time and position as

9(t,r) - 9(r) ef xf , (32)

Therefore, the complete general solution for the velocity potential is

given by

p(t,r,+,y) - [Ar Jn(kr r)+ Br Nn(kr r)l [A, cos(n+) + B, sin(no)] Y(y) ei't1. (33)

9

II. DESCRIPTION OF THE GENERALIZED OCEAN WAVEGUIDE MODEL

The next logical step in this analysis is to apply the solution to the

linear, three-dimensional, lossless, homogeneous wave equation developed in

the previous section to the generalized ocean waveguide model. Before

continuing with the mathematical derivation, a short description of the

generalized ocean waveguide model chosen for analysis will be presented.

As shown in Figure 2. space has been separated into four distinct

media. Medium I (which may represent the air) is completely characterized

by its density (pl(y)) and sound speed profile (cl(y)), which may both be, in

general, arbitrary functions of depth coordinate, y, only. Medium II (which

may represent the ocean water) is completely characterized by its density

(P2(Y)) and sound speed profile (c2(y)), which may both be, in general,

arbitrary functions of depth coordinate, y, only. Medium III (which may

represent the upper layer (sediment) of the ocean bottom) is completely

characterized by its density (p3(y)) and sound speed profile (c3(y), which

may both be, in general, arbitrary functions of depth coordinate, y, only.

Medium IV (which may represent a second fluid layer of the ocean bottom)

is completely characterized by its density (p4(y)) and sound speed profile

(c4(y)), which may both be, in general, arbitrary functions of depth

coordinate, y, only.

Media I and II are separated by a boundary (which may represent the

ocean surface) at a depth of ys(r,+) meters, where ys may in general be an

10

arbitrary function of horizontal range, r, and azimuthal angle +. Media II and

III are separated by a boundary (which may represent the ocean bottom) at

a depth of yB1(r,+) meters, where yB1 may in general be an arbitrary function

of horizontal range, r, and azimuthal angle f. Media III and IV are separated

by a boundary (which may represent the interface between two different

layers in the bottom) at a depth of yB2(r,#) meters, where yB2 may in general

be an arbitrary function of horizontal range, r, and azimuthal angle +.

Medium II requires some additional examination. First, medium II

contains the omnidirectional, time-harmonic point source located at a fixed

depth of yo meters and at a fixed horizontal range of zero meters. Medium II

is also further separated into two distinct subregions, labelled medium Ila

and medium Ilb, by an artificial boundary located at the depth of the source.

This additional boundary is required to satisfy the source boundary

conditions. The density and sound speed profiles for these subregions are

completely defined by the generic medium II descriptions previously

discussed.

11

Medium I91 r,+y) Pj(y) cj(y)

Surfacey. -yS(r*+)

(Shown as

/X planar)

Medium IlaOmnnidirectional ,r+,) P( c( SueTime-Harmonic I -- -- -a~,y -2Y - - -2 y SoePoint Source Y Dept

/(A constant)

Medium hIb

%(r,4..y) p2(Y) c2(y)Bottom

y =Y r+(Shown as

7 planar)

Medium III

qp3(,+,y -P3Y) C(y)BottomLayerInterface

ya B r+(Shown asplanar)

Medium IV

Y y9j(r,+.y) P4(y) c4(Y)

Figure 2. Generalied Ocean Waveguide Model(Based on Ziomek (199 1, Figure 3.9-1)

12

III. DEVELOPMENT OF THE GENERAL WAVEGUIDE MODEL SOLUTION

The primary objective of this section is to derive general expressions

for the velocity potentials in media Ila and lib based on the wave equation

solution previously derived and the appropriate general boundary

conditions.

The next assumption required in this derivation is that the source of

acoustic energy in the waveguide is an omnidirectional point source which is

surrounded by the medium it is to excite acoustically. This assumption may

be justified by the fact that the velocity potential field generated by an

arbitrary acoustic source array may be expressed as the summation of the

velocity potential fields generated by the individual omnidirectional sources

which make up the array. Assuming that the source is surrounded by the

medium it is to drive acoustically is reasonable for many practical acoustic

systems.

Recall that the velocity potential is given by

(tr.,y) - [A, Jn(kr r) + Br Nn(kr r)] [A cos (nf) + B, sin (n+)] Y(y) ei2 ". (34)

The omnidirectional point source assumption implies that the source

radiates acoustic energy equally well in all directions. However, since we

have assumed an arbitrarily varying surface and an arbitrarily varying

bottom for this waveguide, this assumption doesnot imply that the velocity

potential field lacks an angular dependence (that is, in general, the velocity

13

potential at a given range and depth will depend on the azimuthal angle

because of the differences in the interactions of the sound rays with the

arbitrary surface and bottom). Assuming that the source is surrounded by

the medium (and since the source is located on the y-axis, this assumption

implies that the y-axis is surrounded by the medium) implies that the

arbitrary constant n must be an integer in order to ensure that the velocity

potential is single-valued for azimuthal angles * in excess of 2X radians (i.e.,

the velocity potential must be a periodic function with period 2x radians).

Since the velocity potential must be evaluated at the source (i.e., at r -

0), the arbitrary constant Br must be set equal to zero in order to eliminate

the Neumann function solution (since all of the Neumann functions tend to

infinity as r tends to zero) (see Boas (1983, p. 513 and p. 525) for further

proof). Thus, the general solution is reduced to

q(t,r,+,y) - Ar Jn(kr r) [A, cos (n+) + B, sin (n+)I Y(y) e' 2x'. (35)

A. VELOCITY POTENTIAL IN MEDIUM I

Based on (35), the velocity potential in medium I can be expressed as

T1(t,r,+,y) - Arl Jn(kr Ir) [A., cos (n+) + B+, sin (n+)I Y (y) ei 2 1L. (36)

where YI(y) represents the solution to (28) for the traveling wave in the

negative y direction.

14

Carrying out the indicated multiplications, and defining the following

new constants:

ArI A4I E B,

Ar, BI w Al,

reveals the general form for the velocity potential in medium I

q1(t,r,+,y) - [B! cos (n+) + Al sin (n*)] J(kr! r) Y1(y) eixft, (37)

where A, and B, are arbitrary complex constants whose values will be


Additionally,

y 2 2 (2Xf) 2

k1(y) - kr + kyl(y)- 2 . (38)c!(y)

where k1(y) is the wave number in medium I, kr is the constant radial

component of the propagation vector in medium I, kyl(y) is the depth

component of the propagation vector in medium I, and cl(y) is the depth-

dependent sound speed in medium I

15

It is worth noting that there is not a solution of the form Y*(y) in (37).

This is due to the fact that energy is considered to propagate in the negative

y direction out to negative infinity without reflection.

B. VELOCITY POTENTIAL IN MEDIUM II

It is clear from the configuration that the velocity potentials in media

Ila and lib will be combinations of "incident" waves traveling toward the

respective boundaries and "reflected" waves traveling away from these

boundaries. With this in mind, the velocity potential in medium Ia may be

written as (see equation (35))

",(t,r,+,y) - A2 a Jn(kr2 r) (A42a cos n+ + B,2a sin n+)

x [Ay2a Ya(y) + By2a Y;3(y)I e! 2 '2, (39)

where Y2a(y) represents the solution to (28) for the traveling wave in the

negative y direction (incident on the boundary at y - ys) and Y21 (y)

represents the solution to (28) for the traveling wave in the positive y

direction (reflected from the boundary at y = Ys).

Carrying out the indicated multiplications and defining the following

new constants:

Ar2a A02a Ay2a 2 A2&

16

..... . ...

Ar2a A,29 BY2& N B2a

Ar2& B+2a A C2a

Ar2a B2a By2a a D2a

reveals the general form for the velocity potential in medium Ila

q2,(tr,fy) - (A2a cos nf Y2 (y) + B2 acos nOY 2a(Y)

+C2a sin n+Y* (Y) + D sin nY 2a(y)) JD(k r) eiZxit, (40)

where A2., B2a' C2., and D2a are arbitrary complex constants whose values

will be determined by satisfying boundary conditions.

Similarly, the velocity potential in medium I lb is given by

9'b(t,r,+.y) (A2 b cos n+Yb(y)+B 2 b cos n+ Y2b(y)

+¢2b sin n$Y2b(y) + D2b sin n+Y2b(y)) J(kr r) el2xft (41)

where Y2b(y) represents the solution to (28) for the traveling wave in the

positive y direction (incident on the boundary at y - YB), Y2b(y) represents

the solution to (28) for the traveling wave in the negative y direction

17

(reflected from the boundary at y - y,,), and A2b, B2b, c2b' and D2b are

arbitrary complex constants whose values will be determined by satisfying

boundary conditions.

Additionally,

2( r2 + (y-2 + 2 y) 2 (42)c2(y)

where k2(y) is the wave number in medium II, kr2 is the constant radial

omponent of the propagation vetor in medium 11, k Y2(y) is the depth

component of the propagation vector in medium II, and c2(y) is the depth-

dependent sound speed in medium I.

C. VELOCITY POTENTIAL IN MEDIUM III

It is again clear from the configuration of our waveguide that the

velocity potential in medium III will be a combination of a traveling wave in

the positive y direction and a traveling wave in the negative y direction.

Thus, the velocity potential in medium III can be derived using the

techniques presented previously (for medium II). Performing this analysis

yields

T(t,r,#,y) - (A 3 cos n# Y;(y) + B3 cos n+ Y;(y)

+ C sin n+Y+(y) + D sin n+Y-(y)) J.(k r) ei2aft, (43)

18

where Y(y) represents the solution to (28) for the traveling wave in the

positive y direction (incident on the boundary at y - YB2), Y3(y) represents

the solution to (28) for the traveling wave in the negative y direction

(reflected from the boundary at y = YB2), and A3 , B , C, and D, are

arbitrary complex constants whose values will be determined by satisfying

boundary conditions.

Additionally,

2 2 2 (2nf) 2

k3(y) - kr3+ kY3 (y) (2 )2 (44)

where k3(y) is the wave number in medium Ill, k' 3 is the constant radial

component of the propagation vector in medium Ill, ky3(y) is the depth

component of the propagation vector in medium III, and c3(y) is the depth-

dependent sound speed in medium Il1.

D. VELOCITY POTENTIAL IN MEDIUM IV

Finally, in medium IV, the velocity potential is given by (see (35))

qp4(t,r,+,y) - Ar, Jn(kr4 r)(A,4 cos n# + B,4 sin n+) Y(y) eIxft , (45)

19

where Y*(y) represents the solution to (28) for the traveling wave in the

positive y direction.

Carrying out the indicated multiplications and defining the following

new constants:

Ar A44 A4

Ar4 B 4 . B4 ,

reveals the general form for the velocity potential in medium IV

qf(t,r*,y) - (A 4 cos n * B4 sin n4) Jn(kr4 r) Y(y) eIzft, (46)

where A4 and B4 are arbitrary complex constants whose values will be


Additionally,2 2 2 (Zmf) 2

k4(y) - kr4 + kY4 (y) - 2 (47)

c4(y)

where k4(Y) is the wave number in medium IV, kr4 is the constant radial

component of the propagation vector in medium IV, ky4(Y) is the depth

20

component of the propagation vector in medium IV, and c4(y) is the depth-

dependent sound speed in medium IV.

It is worth noting that there is not a solution of the form Y4(y) in (46).

This is due to the fact that energy is considered to propagate in the positive

y direction out to positive infinity without reflection.

E. BOUNDARY CONDITIONS

There are three different types of boundary conditions that must be

applied to the solution of our problem. The first is the condition of"continuity of acoustic pressure" across a boundary. This condition requires

that the acoustic pressure evaluated at a particular spatial location and time

on one side of the boundary be identically equal to the pressure evaluated at

the same spatial location and time on the other side of the boundary Kinsler

(1982, p. 125) describes this condition as meaning that there is no net force

acting on the massless boundary separating the two media.

The second type of boundary condition is that of continuity of the

normal component of the acoustic particle velocity across a boundary. This

condition requires that the normal component of the acoustic particle

velocity evaluated at a particular spatial location and time on one side of the

boundary be identically equal to the normal component of the acoustic

particle velocity evaluated at the same spatial location and time on the other

side of the boundary. Kinsler (1982, p. 126) describes this condition as

meaning that the media remain in contact with each other.

21

The final type of boundary condition is that of discontinuity of the

normal component of the acoustic particle velocity across the boundary at

the depth of the source. Officer (1958, p. 124 and following) and Ziomek

(1991, discussion following equation (3.9-37)) describe this condition as

being required to ensure that the solution to the wave equation reduces to

that of an omnidirectional point source when the boundaries at y - ys,

y - yB1, and y = yB2 are removed.

In developing the required boundary condition expressions, we will

consider the various types of conditions in the order previously discussed.

Thus, the first boundary condition to be applied to this problem is that of

continuity of acoustic pressure across the boundary at y - Ys. This implies:

pj(t.r,*,ys) - p2.(t,r,0,ys).

In general (from the fluid dynamics derivation), the acoustic pressure

can be related to the velocity potential by the following:

p(t,r,*,y) - - p(r,,y) atr,,y) (48)

where p0(r,.y). representing the ambient (equilibrium) density of the

medium, is, in general, an arbitrary function of the spatial variables r, *, y,

but is not a function of time.

22

Since we have assumed that density is simply a function of depth

coordinate y, and assumed a time-harmonic source, the acoustic pressure

may be written as

p(t,r,+,y) - - j w p0(y) 9(t,rf,y). (49)

Thus, the acoustic pressures in the various media may be expressed as

pI(t,r,+,y) = - j co pI(y) qI(t,r,,y) , (50)

p2&(tr,f,y) - - ( o P2(Y) q"R(t,r,$,y) , (5 1)

PNb(t,r,4,y) - - j o, P2(Y) 92b(t,r,+,y), (52)

p3(t,r,o,y) P - op 3(Y) "(t,r,*,y), (53)

and,

p4(t,r,,y) - co p4(y) qp4(t,r,,y), (54)

where PI(Y), P2(Y), P3(y), and P4(Y) are the ambient densities in media 1, 1I,

Ill, and IV, respectively.

Returning to the specific boundary condition being examined, we set P,

(given by (50)) equal to P2a (given by (51)) at y - ys, and divide out the

corn mon factor of -j w revealing

p I(ys) qp (t,r,,ys) - P2(YS) 92.(tr,+,ys) (55)

Substituting the derived velocity potentials for media I (37) and Ila

(40) into (55) yields

23

pi(Ys) (Bi cos to A, sin no) J,,(k,, r) YI(ys) e122"

= P2(YS) (A 2 cos n+ Ya(YS) + B2 , cos n+ Y2 (Ys)

+ C sin n0Y+a(ys) + D sin n+Y21(ys)) Jn(kr2 r) e 2 . (56)

The time dependence is eliminated by dividing (56) through by the

complex exponential term e 12 ft. Carrying out the indicated multiplications

and factoring yields the following:

IP2(Ys) JO(kr 2 r) Ya(ys) A2. + p2(Ys) Jn(kr 2 r) Y,(Ys) B2.1 cos n4

[P2(Ys) Jn(kr 2 r) Y*,(Ys) Ch1 + P2(Ys) J0 (kr2 r) Y2.(ys) D I sin n+

- pI(Ys) Jn(kr I r) YI(ys) B1 cos n+ + pI(Ys) Jn(kr r) YI(ys) A1 sin n+. (57)

Setting the coefficients of cos n$ on the left-hand side of (57) equal to

the coefficients of cos n+ on the right-hand side of (57), setting the

coefficients of sin n* on the left-hand side of (57) equal to the coefficients of

sin n4 on the right-hand side of (57), and rearranging the equations

generated by this process reveals the pair of equations representing the first

boundary condition (BC #)

P2(YS) Jn(kr2 r) Y;1(ys) A21 + P2(ys) J0(kr 2 r) Y21(ys) B2.

- Pi(Ys) J,,(kr r) YI(ys) B, - 0, (58)

24

and

P2(YS) J'(kr2 r) Y2,(YS) C2. + P2(YS) Jn(kr 2 r) Y2, (ys) D2&

p PI(Ys) J.(kr r) Y-1(Ys) Ai I= 0 . (59)

It should be noted here that (58) and (59) are valid only if the

associated trigonometric function is not identically zero for all values of

azimuthal angle * (for instance, if n - 0, then sin n is identically zero for all

values of + implying that (59) is no longer a valid boundary condition).

The second boundary condition to be applied to this problem is that of

continuity of acoustic pressure across the boundary at y = y.. This implies

P2a(tr,0,y0 ) - P2b(t,r.,yo)

Setting P2 (given by (51)) equal to P2b (given by (52)) at y - yo and

dividing out the common terms reveals

qTa(tr,*,yo) - T b(tr,+,y0 ) (60)

Substituting the derived velocity potentials for media Ha (40) and lib


Jn(kr2 r) [A2a cos n4 Y;5(Y2 ) B2a cos n+ Y2,(Yo) + C2 sin n Ya(Y)

+ D2a sin n+ Y,(yo)I ej ft - Jn(kr2 r)IA2b cos n+ Y;+(Y 0 )

+ B2b cos nf Y2b(yo) + C2b sin n+ Y2b(yo) + D2b sin n+ Y2b(YO)I eixft . (61)

25

Dividing out the common terms eliminates both the horizontal range

and time dependences, revealing, after factoring, the following:

[Y;(Y.) A + Y21(yo) B21I cos n++ [Y+ (Y.) C + Y- (yo) D2 I sin n+

- Y;2b(yo) A2b+ Y2b(yo) B2b] coS n + lY2b(yo) C2b + Y2b(yo) D2b] sin n. (62)

Setting the respective coefficients of cos n+ and sin n+ equal and

rearranging yields the following pair of equations representing the second

boundary condition (BC #2):

Y2+(yo) A2 1 . Y2&(y.) B2a - Y2b(yo) A2b - Y;b(Yo) B2b = 0, (63)

and

Y21(Yo) C2a+ Y21(yo) D2a - Y2b(YO) C2b - Y2b(yo) D2b = 0 (64)

Again, (63) and (64) are valid only if the associated trigonometric

function is not identically zero for all values of azimuthal angle *.

The third boundary condition to be applied to this problem is that of

continuity of acoustic pressure across the boundary at y - YBI" This implies

P2b(tr,+,yB,) - p3(tr,*yE,).

26

Setting P2b (given by (52)) equal to P3 (given by (53)) at y - yB, and

dividing out the common factor of -j w reveals

P2(YB1) "pb(t.r,+,yB,) - P3(YB, ) qp(tr ' yB ) " (65)

Substituting the derived velocity potentials for media 1lb (41) and III


P2(yIB1) Jn(kr 2 r) [A2b Cos n+ Y2b(YB) + B2b cos n0 Y2b(YBI)

+ C2b sin nq Y2b(YB) + D2b sin n+ Y2b(yl)Ie2ft

- p3(yBI ) J,(kr3 r) [A3 cOS n0 Y3(yBI) + B3 cos n0 Y3(YB, )

+ C3 sin no Y;(YB I + D3 sin n4 l Y B )1 eia ft . (66)


complex exponential term ei2dft. Carrying out the indicated multiplications

and factoring reveals the following:

Ip2(yB) Y2 (YB,) A2b + P2(YB) J,(k'2 r) Yib(YB,) B2b] cos n+

Ip2(yB1) J.(k 0)Y y J& 02(y J)(krg DI sin n+

r2 2bYBI)C 2b + P2(YBI) r b 2

27

- [p3(yB,) J (kr 3 r) Y3(y8,) A3 + P3(YB) JL(kr 3 r) Y3(yB) B3 1 cos no

[P3(YBI) J0(kr 3 r Y;(y,1 ) C3 + p3(YB,) J.(kr 3 r)Y(yB 5 D3 ] sin n. (67)

Setting the respective coefficients of cos n# and sin n+ equal and

rearranging yields the following pair of equations representing the third


P2(YB) Jn(k 2 r) 2b(YB) A2b + p2 (YB ) Jn(kr2 Y2t(YB ) B

- P3(yB,) Jn(kr 3 r Y;(y, 1 ) A3 - P3(Y81) Jn(kr 3 r) Y3(yB) B3 = 0, (68)

and

P2(YB1) J(kr2 r) Y2b(Yl) C2b P2(YB1) J(kr2 r) Y~b(yB,) D2b

- P3(YBI) Jn(kr 3 r) Y;(yB1 ) C3 - P3(YBI) Jn(kr 3 r) Y3(y 1 ) D3 - 0 (69)


function is not identically zero for all values of azimuthal angle +.

The fourth boundary condition to be applied to this problem is that of

continuity of acoustic pressure across the boundary at y - YB2. This implies

p3(t.r,*,yB2) - p4(t,r,*.y8 2 )

28

Setting P3 (given by (53)) equal to P4 (given by (54)) at y - yB2, and

dividing out the common factor of -j o reveals

p3(YB2) 3(t.r*Y, 2) - P4(YB2) 94(t,r,,yB2). (70)

Substituting the derived velocity potentials for media III (43) and IV


P3(YB2) Jn(kr3 r) A3 cos nr4 Y + B3 COS 0 Y

+ C3 sin n+ Y;(YB2) + D3 sin n+ Y3(yB2)I ej 2z

p4(YB2) Jn(kr 4 r) IA4 cos n+ Y+(YB 2) + B4 sin n+ Y'(y B2)I e . (71)


complex exponential term ei2'ft. Carrying out the indicated multiplications

and factoring reveals the following:

Ip3(yB2) Jn(kr3 r) Y;(yB2) A3 + (3(YB2) Jn(kr3 r) Y3(yB2) B3 I cos n+

[P3(YB2) J(k 3 r)Y;(yB2) C3 + p3(yB2 ) Jn(kr 3 r) Y(yk H) D3 sin nf

- Ip(y 82) Jn(kr4 r) Y(y 2) A I cos n++ IP4(Y 2) J+(kr4 r)Y(y 2) B4 I sin n*.

(72)

29

Setting the respective coefficients of cos n+ and sin n+ equal and

rearranging yields the following pair of equations representing the fourth


P3(Y82) Jn(kr3 r) Y+(yB2) A3 + P3(YB2) Jn(kr3 r) Y(y 2 ) B3

- P4(YB2) Jn(kr 4 r) YB 2) A4 0, (73)

and

P3(YB2) Jn(kr 3 r) Y;(yB2) C3 + P3(yB2) Jn(kr3 r) Y3(yB2 ) D3

- p4(y 8 2) Jn(kr4 r) Y4(yB2) B4 - 0 (74)


function is not identically zero for all values of azimuthal angle f.

The fifth boundary condition is that of continuity of the normal

component of acoustic particle velocity across the boundary at y - ys. This

implies

Unl(t,r,*,ys) - Un2a(t,r,*,ys)

where:A

U,(t,r*,y) - U1(t,r,y)* - (r*,y), (75)A

Un2(t,r,#,y) - U2,(t,r,#,y)* n,(r*,y), (76)

30

Aand ng(r,#,y) represents the unit vector normal to the boundary at y - Ys.

In order to continue with the evaluation of this boundary condition.

expressions for the velocities and the unit normal vector must be developed.

Recalling that the acoustic particle velocity is simply the gradient of the

velocity potential implies

U1(t,r.+,y) - V q1(t.r,*,y), (77)

where qj1(t.r,,y) is given by (37), and the gradient of 9, expressed in

cylindrical coordinates, is given by

aq(t,r,+,y) A I ai (tr,O,y)A a(t,r,Y)y) AV qc(t,r.f,y) - ar F r * + y. (78)

Performing the indicated partial differentiations results in thefollowing:

dJo(kr, r)

ar tB, cos n+ + A, sin +)r Y (y)e iz -r

dJn(kr I r) d(k I r)(BI cos n++ A, sin n+) Y1(y)e i2 ft -

d(kr, r)

aq1l(t~r,*),y) -dr (,con* 1 kirn)Y1 y r)

d(kr, r)

31

~p(tr'Y)--n B I sin n+ + n A1 cos n#) Jn(kr r) Y-(y) eJ2 f t (80)

and

dY1(y)al,(t'rY) (B 1cos n AI sin n) Jn(kr, r) ei~ft d (81)ay I dy

Substituting (79) through (81) into (77) (and using (78)) yields

djn(ki.I r)

U,(tr,,y) - 1kr, I(B cos n+ + A, sin n+b) Y(y) - rd(kr I r)

+ nr (A, cos n - B, sin n#) Jn(kr, r)Y-(y) A

dY1(y)(B, cos n+ A, sin no JO(kr! 0 dy Y] eiZftt. (82)

Conducting a similar analysis on the appropriate expressions revealsthe following set of equations for the velocities in the remaining media:

U2,(t,r,*.y) - [k, 2 (A 2, cos n+ Y2,(Y) + B2, cos n+ Y2.(y)

djn(kr2 r) AC2. sin n+ Y2,(Y) + D2, sin n* Yz,(y)) r

d(kr 2 r)

+ r1 (C2, cos n+ Y(y) + D2, cOs n Y2 (Y)

32

-A2, Sin r+ y2a(Y) -B2. sinl n+ Y2a(Y)) Jn(kr 2 0)

+ (2.cosnfdY3a(y) + ~ O +dY 2a(y)+(A~~Sfl dy B23csn$ dy

dY+ (Y) dY- (Y)

C2.asin n+ dy + D2 sin n* y Jn(kr r A j~l (83)

U2b(tr,*,y) - lk, 2 (A2b cos nlYb)+ BYb

+C2b sin n+,Y2-b(y) + D2b si n 2b~y d~~r

d(kr2 r)

+ n~ (C2b CO)S I1+ 2Y+,y D2b cos n+* 2Y-

- 2b Sin n+ Y2b(y) - 2b sin n*y2b~y)) J.(k,,2 0)

dY2+b(y) dY2b(y)+ (A 2b COSn0$ y 2 csn dy

dY2+b(y) dYb(y)A+ C2b sinl n+ d + D2b sinl f+ dy )Jn (kr2 r ) y I ejxf (84)

U3(t,,,y) -[kr 3 (A3 cos nfY(y) +B3 cos nfY(y)

33

+ n (C3 cos n Y;(y) + D3 cos n Y3(y)

A3 sin n+ Y;(y) - B3 sin n+Y-(y)) J0(kf r) 4

dY;(y) dY3(y)+(A 3 cosn# + B3 cosn+ dy

dY;(y) dY;(y)+ C3 sinn dy + D3 sinn dy ) Jn(kr3 r)y eIt, (85)

anddJn(kr 4 r)

U4 (t,r,+,y) - (kA4 nos n+ B sid(kr r)

+ (B4 cos n+ - A4 sin n+) Y(y)J.(kr r4r44

dYe(y)

+ (A4 cos n++ B4 sin n+) jn(kr, r) A '] e]2aft (86)

Now, we must turn our attention to deriving an expression for the unit

normal vector. A review of texts covering calculus and analytic geometry

(for instance, Berkey (1988, p. 830), and Leithold (1972, p. 934)) remind us

that if the surface can be expressed as a constant function of all three spatial

variables (i.e., f(r,+,y) - a constant), then the normal vector to this surface at

any point is simply the gradient of the function describing the surface

evaluated at that point. Recalling that both the boundaries ys, YBI' and YB2

were defined to be arbitrary functions of the other spatial variables, let

34

Y - fs(r,+), (87) at the ocean surface (i.e., y - Ys),

y - fD,(r,+), (88) at the ocean bottom (i.e., y - yB), and

y = fB2(r,). (89) at the bottom layer interface (i.e., y - yB2).

Concentrating on the ocean surface for the moment, subtracting fs(r,)

from both sides of (87) yields

y - fs(r,+) - 0. (90)

Letting a new function, xs(r,+,y), equal the left-hand side of (90), we

have an equation of the form

-rs(r,+,y) - y - fs(r,+) - 0. (91)

Taking the gradient of ts(r,+,y) using the cylindrical coordinate system

gradient operator (78) yields

ftS(r,,y) . 1 ars(r,,y) A fts(r,,y) AV*- + yar r Y

V -t(r*,y)= afs(r'€) 1 afs(r ) A

-r " " (92)

Therefore, the normal vector is given by

afs(r*) A 1 afs(r,+) A AN--r - . + +y. (93)ar -r

35

Since we need a unit normal vector, the magnitude of Ns (for which we

use the symbol Ns) must be evaluated. Recalling that the magnitude of a

vector is the square root of the sum of the squares of the individual

components, then Ns can be written directly as

N - r (94)

Therefore, the desired unit normal vector is given by

A I afs(r,)) A I fs(r,) A '

NS a r r ; * }" (95)

Similar analysis reveals the following expressions for the unit normal

(AB (Avectors at the boundaries y - Yen I and y = Ye2 (nB2):

AfB(rf) 1fB )(rA)

^ _r +Y (96)NBI

where

N8,afB(r,+) 2 1 (OfBI(r,+))2

and

Af2 (r,f) afB 2(r,$) ANB

2

36

where

N (7afB2(r +) 2 (fB 2(r)" 2

NB2 - 8r (99)

Returning to the evaluation of the fifth boundary condition, substituting

(82) and (95) into (75), substituting (83) and (95) in'o (76), and performing

the indicated dot products yields

dJn(kr1 r)

= L-r, (BI cos n + A, sin nr) Y-(y) - A

d(kr r)

+ yA, cos n - B, sin n+^ J(krr)Y-(yA

dYI(y)

+ (B, cos n++ A1 sin nf) Jn(k, r) dy I ej2xft

a ( 0fs(r,+) A 1 fs(r,+) A

NS r r r * Y)

- djn(krI r) fr,Unk(t,r,,,y) I - k, B, cos nf + A, sin n ) YI(Y)d(kr 0f(r,)d(kr I r) r

2 (A, cos n+ - B1 sin n+) J,(kr, r) Y-(y)

dY,(Y) , ~

(B,cos n++A, sin n#)Jkr0 dy 3 (100)

37

U,2,(t.r,..y) - [kr2 (A2, cos n+ Y2,(Y) + B2, cos n+ Y;(y)

+ C2 . sin n+ Y,(Y) D2a sin n+ Y1 (y)) dJn(kr2 r)d(kr2 r)

+ C21 cos n* Y;1(Y) + D2 cos n Y21(Y)

- A~a sint n+ Y+1 Y' - B21, Sinn nY2(y)) J(k. 2 0i

dY;.(y) dY2 (y)

+{Aa cos n+ dy B2acos n+ dy

dY_,(y) dy2(y) A+ C2. sin n ) d y + D 2.sin n+) d- JJn(kr2 r) ]e}2k t

Uo,,(t.r. .y) -[- k, ( A2. cos n+ Y+,(y) + B2. cos nf Y-,(Y)

+ C2. sin n+ Y+ (y) + D25 sin n+ Y- (y)) dJn(kr2 r) Of(r)a 2a 2a d(kr2 r) r

ncs n a 2Y2(y) - A2. sin n+ Y2,(Y)-r2 (aCO0Y+) + osn

- B21 sin n*Y2 (y)) Jn(kr 2 r)

38

dY;.(y) dY;.(y) dY2a(y)+ (A2 cosn dy + B2 z cos n+ dy + C2.sin n+ dy

dY2.(y) i!rz+D29 sin n+ y ) Jn(kr2 r)] (101)

Setting Un,(t,r,f,ys) (100) equal to Ua2 (t,r,*,y s) (101), and eliminatingeJ2xftthe common term e yields

-kr (B cos n4 + A, sin n+) YI(ys)r) ard(kr I r) o

n (A, cos n+ - B1 sin n.) J,(kr, r) Y(ys) afs(r,*)r 2 I

dYI (Ys)+ (B, cos n+ + A, sin n+) Jn(kr, r) dy

- kr2 (A 2a cos 41 Y2,(Ys) + B2& cos n+ Y2a(ys)

+ C21 sin n+ Y2,(ys) + D2, sin n+ Y2,(Ys)) dJn(kr 2 r)d(kr 2 r) ar

n (C2. cos n+ Y2,(ys) + D2a cOS n+ Y2a(ys)

A2. sin n+ Y,(ys) - B2. sin n+ Y2 (ys)) Jn(kr2 r) ,,.

39

dY;,(ys) dY2,(YS) dY~a(YS)+ (A 2 cosi n+ + B29COSrl dy + C2.Sinfl d+

dY;a(ys)+ D2. sin n+ dy )Jn(k 2 r)0 (102)

Factoring (102) yields

dJ(kr I r) - f~r+ n___ -a-fs(r,+)

Vkr r)Y(ys) -B I -J 0(kr r) Y-(ys) A,

dY I(ys)+ J(kr r) dy B,] cos nf

+I dJft(kr I r) Y- YS ar.) A,+nJkr0Ds fs(r* BIr d(kr I r) I ar 1 pkr2 r) Iy)a

dY1(y5 )A, in+ J,(kr Ir) dy Asnf

kdjn(kr2 r) Y;() af5(r,+) A rdJn(k 2 r2 0 fs_ r*)k2 d(kr 2 r) OrA~ - kr2 r)0 2

n J(kr0 Y*(y5 ) afs(r,+) Ch- R n krY ( YS af~r D2

40

dYjaY 5 dYjg,(yS)+ J&~k r) d1YS t129 " r),( dy B2aI cos n*

+ kdj,,(kr 2 r) Y(Y) Of~.# k d"'~2 r) 2r2 d YkrY r) Ch r r)~(Y)a2 Hdk2H

t- J.(kf.2 0? Y*a(ys) afs(r,+) Aha + -i Jn(kr 0) Y-(Ys) af B,)12a

dY2.(y5 ) dY2&(YS)+J,(kr 2 r) dy Ch + Jf(kr r) - D . sin n+. (103)

Setting the respective coefficients of cos nf and sin n+ equal and

rearranging yields the following:

dY I(ys) djn(kr I r) - _______

J,(k., r) - B, - kr, Y(y) B Idy d(kr 0r) 1a

nJkr)0 Y- (ys) A, I fs(r,*)

dY;3(yS) dY1a(ys)11(kr r) dy A2a + Jfk(kr2 r) - B2a1r2 dy 2 dy

41

dJn(kr2 ) A dJn(k r2 afs(r*)

Y2&s 22a + 2 Y2(Ys) B25 ]

d(kr2 0 d(kr 2 r)

- ltJ.(kr2 r) Y;(Ys) C2 + J,,(kr r) Y25 (ys) D251 afs(r,, )8*2 2 a (104)

and

dYi(ys) dJn(kr r)

Jn(kr, r) dy AI - y(k I(Ys) A,d(kr! r) Or

n J,(kr, r) Y1(ys) B, afs(r'#)

dY2&(Ys) dY2a(Ys)

2 [Jn(kr 2 dy C2 + Jn(kr 2 r) d D2,l

dJn(kr 2 r) dJn(k, 2 r) _ a rD 1- k Y2a(Ys) C2 kr2 y2 (Y) D2& or

d(kr2 0 d(kr r)

+ Y (ys)A 2n+ Jn(kr 2 r) Y2 (ys) B2 afs( r ' ) (105)


function is not identically zero for all values of azimuthal angle +. It should

be noted here that both (104) and (105) are of the form

42

(LHS term 1)+ (LHS term 2) + (LHS term 3) afs(rq)ar )

-(RHS term)+(RHSterm) or+(R,)erm 3) 0# *

Therefore, we may simplify (104) and (105) by setting LHS term 1

equal to RHS term 1, LHS term 2 equal to RHS term 2, and LHS term 3 equal

to RHS term 3. Performing this analysis and rearranging the resulting

expressions yields the set of six equations representing the fifth boundary

condition (BC #5)

dY ,(ys) dY- (Ys) dY1(Ys)

Jn(kr r) dy Aa + Jn(kr 2 r) dy B2a - Jn(kr, r) BI - 0 (106)r2 dy2 ddy

dJo(kr 2 r) dJn(kr 2 r)kr2 fY2a(Ys) A2a + &r2 Ya()Bad(kr 2 r) d(kr 2 r)

dJn(kr, r)

kr I , YI(ys) B, - 0, (107)d(kr, r)

J,(krr) Y,(ys) C2a + J(kr2 r) Y2(Ys) D2a- J.(k, ,r) Y1(y s) AI - 0 (108)

dY* (ys) dY-2(Ys) dYI(y s )

Jn(kr 2 r) dy C2, + Jn(kr r) dy D2a - Jn(kr, r) dy A, - 0, (109)

43

dJfn(kr 2 r) dJn(kr 2 r)kr2 f1a(Ys) C2a + kr2 - a(Ys) D2a

d(kr2 r) d(kr 2 r)

dJn(kr, r)kr d r Y(ys) A -0, (110)

d(kr I r)

and

Jn(kr 2 r) Y;3(ys ) A2a + Jn(kr2r) Y2 (ys) B2a - Jn(kr, r) Y1(ys) BI - 0 (111)

Here, (107) and ( 110) are valid only if afs(r,+) is not identically zero for8r

all values of range, r, and azimuthal angle, *. Similarly, (108) and (011) are

valid only if afs(r,+) is not identically zero for all values of range, r, and

azimuthal angle, +.

The sixth boundary condition is that of continuity of the normal

component of the acoustic particle velocity at the boundary y - YB1' This

implies

Un2b(t.r,+,ye,) - U,3(t,r,+,ye )

where

U03(t,r,+,y) - U3(t,r,+,y) . , (r,+,y) , (113)

44

and eB,(r,+,y) represents the unit vector normal to the boundary at y - YBI"

Substituting (84) and (96) into ( 12), substituting (85) and (96) into

( 13), and performing the indicated dot products yields

Un2b(t,r,0,y) - 1kr 2 (A2b cos n+ Yab(Y) + B2b COS n Y2b(y)

dJn(kr2 r)+ C2b sin n0 Y2b(Y) + D2b sin n+ Y2b(Y)J r

d(kr 2 r)

+ r Cm cos n* Yb(Y) + D2b COS n+ Y2b(y)

- A2b sin n* Y2(y) - B2b sin n*Y2b(y)) Jn(kr 2 r)

dY2b(y) dY2b(y)+(A 2b cosn+ dy +B2bcosn+ dy

dY+(Y) dY2b(y)

+C2bsinn+ dy + D2bsinn+ d jn(kr2 r) Y e 'Iit

1 OfB 1 (r'+) A af B(r) A.r-{ -+ }NB I a

Un2b(t,r,#,y) = -kr2 ( A2b cos n+ Yah(y) + B2b cos n+ Yb(Y)

+ C2b sin n+ Y',(y) + D2b sin n Y2 (Y)) dJn(kr 2 r) afB,(r,+)

2~ d(kr 2 r) ar

45

r2~ (C~tb cos n4 Y2b(y) + D2bco n+S1 Y2b(Y

-A2bsi n#f nY+bY - B2b Sin n~y2b(y)) Jn(kr r) B,()

dY~b(y) dY~b(y) dYb(y)+ (A 2b COS* + dy + B2b COSn+ dy + C2 b sinn+ dy

dY2b(y) e1~f+D2b sinll+ dy )J(kr2 ) 14

dy 2 NBI

and

U. 3 (t,r,*,y) - I k, ( A3 cos n+ Y+(y) + B 3 cos n+ Y3(

+ C3 sin n+ Y(y) + D3 sin n+ Y(y))dllk 3 HA

d(kr 3 r)

+ 11(C 3 co n iY;(y) + D3 cos n+ Y~

- A3 sin n+ Y(y) - B3 sin n+ Y3(y)) Jn(kr 3 r) 4

dY;(y) dY3(y)

+ A3cosn+d-y- + B3 cos r

dY(y) dY;(y) 4 d

+ C3 Sinn+ d-ay + D3 sinr dy ) 'k3r) Ie'

46

af 81(r,*) A afB(r*) A

NBI ar r

U (t,r,.y) - - (A3 cos n+ Y(y) B cos n+ Y3(y)

+ C3 sin n+ Y'(y) + D3 sin n+ Y-(y)) dJ(kr3 0 afB(r,)d(k r) ar

r 2 (C3 cos n Y;(y) +D3 cos n Y(y)

afB (r,+)

- A3 sin n+ Y3(y) - B3 sin n+ Y3(y)) Ja(kr3 r)

dY;(y) dY3(y)+(A3 o sn* dy +B3 cosn dy

dY;(y) dY3(y) j2xft

+C3 sinn+ dy +D3 sinn dy Jn(kr 3r)le (115)NB1

Setting Un2b(t,r,+,yB,) ( 14) equal to U,3(t,r,+,y, ) (115) and eliminating

eiZ~ftthe common term e yields

N8 ,

47

- k f2 ( A2b cos fl*Y2b(YB1) + B2b COS 0*Y2b(YB 1 )

C2b sin n+Yb(YB,) +lD b(YBI)) d(k 2 r)

r2 2bb sin n+~ r)

- - (Ca,, COS n+ Y~b(yo1 ) + Db CO 2*Yb(YBI)

-A2b sin n+ Y2b(YB,) - B2b sin n~Yb(YB,)) Jn(k 2 r)

dY2*b(YB,) dY-bY, dY+bY 1

+ (A2b COSfn+ dy + B2b COSn+ dbyBI + C2bsin* d42bYB

D2b sinl f+- dy )Jkr)

- -kr 3 (A3 cosn+Y+ B3 cos nfY;( 5

+C3 sin n+ Y;.(YB,) + D3 sin n*l Y3 (B )) din(kr 3 r) 8lI d(kr3 r) or

-n ( C3 cos n+ Y(y. I) + D3 COS n IY(D

- A3 sin n+ Y;(y,1) - B3 sin n+Y3 8 )J(k )fB(r)-(YBI)) J.(k,3 ) -6

48

dY;(YBI) dY3(YBI)

+ (A 3 cos n+ dy + B3 cus n dy

dY;(yB,) dY3(YB1 )

+ C3 sifn* +)+D i+Jn(kr 3 r). (116)dy ~Dsn* dy

Factoring (116) yields

k djnk f2 Y + fB ,(r,+) t2- rdJn(kr 2 r) YaB 12L k 2 d- k 2 bYI r) ____ar -2 2b(YB1I ar Bb~r) 0f d(kr2 0)

n ~ k 1(r,4 - a fB (r ,+ )r 2 k2 0) Y+b(YB,) 1-C2b - r2 Jn r) Y~(B) L

db(YB1) d2b(YB I

+ JA kr 2 0) dy A2b + Jn(kr 2 r) -dy B2bl co~s n+

dJn(kr 2 r)+ af , (r*) djn(k ' 2 0) Y- afB (r,*)+ f2-2b~yq,' ar C2b - r2 -' 2b(YBI) - r D2b1k2 d(kr r)H d(kr 2 r)

2 Jn(kr)2t2 J2bJnBr 2b B,(r,*)br r)b~ J~(kYBr ~2b ,Bb

49

dY2*b(YBI) dY2b(YBI)

+ Jn(kr2 r) dy C2b + Jn(kr 2 r) -dy D2blsin n+

djn(kr, r) afB,(r* ) dJn(kr3 r) - af B(r)k r f' r) YI or A3- k t3 r) Y3(YBI) oar LB3

d~r3 r)d(kr 3

r2Jn(k r3 )Yy) H Y3 r2- Jf,(k HrY3(YB1 ) L1

dY;(yB,) dY3(YBI)

+ Jn(kr, )A kr3 r Iy B3 l cos n+

k dJn(kr3 r) fB,(,)djn(k '3 r ) af B______I-k ~ r3- Yoyd-r 13- "3 -Y-3(YB) 8 r L

d(kr3 r) d(kr, r) r D

3L +3r nkB

iJR(kr 3 r 3(B) Ar3 r) Y fB r,)B

dY;(YB 1) dY(YBI)

+ Jn(k 3 r) -d C3 +JA kr 3 0) dy D1sin n+. (117)

50

Setting the respective coefficients of cos nO and sin no equal and


I~~r )dYzb(YD,) dYib(YB1 )lj(r2 dy A2b + JA kr 2 r) -dy B2bl

djnkr r Y2b(BI)A +kr jn~ '2b ) r) OfB ,(r,+)

tkr 2d(kr2 r) A2b 2 k r 2 r) -~(YI B~b O

n) 2bYB,) d+J(k~ r) B) 2bfB(r0

2 d(k, 2r)

dY;(+ 1 dY(y)

- Jn(kr3 r) dy A3 + Jn,(kr r) dy B31

djn(kr 3r) djn(kr3 r) a 8 r*__YB__ 4 ) A3 + k3 Y) B3 1

d(kr 3 r) d(kr 3 r)

n 1jn(kr r)y (B C3 + j,(kr3 r) -Yy9 ) D31 af (118)

and

51

dYb(YB) dY2b(YBI)

[J,(kr 2 r) dy C2b + J(kr2 r) dy D2bl

dJ(kr2 r) dJn(kr 2 r) 2bfyB) D2b I 8(r*

d(kr2 r) d(kr2 r) ar

af, r(r*))

dY;(YB,) dY3(YBI )

- [J (k r3 0) dy C3 + Jn(kr3 r) dy DJ

dJn(kr3 r) dJn(kr r) afB,(r, )

-k 3 - +(YI C3 + kr3 Y3YI D31 ard(k 3 d(kr3 r)dr 3

af8i(r,4)+ ' Jnk3 rY3('YB1) A3 +Jnkr3 r)3YB|)3 , 19

Again, (118) and (1 19) are valid only if the associated trigonometric

function is not identically zero for all values of azimuthal angle +. Conducting

an analysis of (118) and (119) similar to that of the previous boundary

condition yields the set of six equations representing the sixth boundary

condition (BC # 6)

52

dY2.b(Y 81 ) dY2b(yB1)

.k 2 r) dy A2b +Jn(k r2 0r) y B~

dY3(yB,) dY3(YB1)

J,( ) dy A3 -J,(kr, r) d- B3 -0, (120)

krdJo(kr 2 0r) (B) ~ dj.(k r2 0r)2 2b~B A 2 - )2b(YBI) B2b

k2 d(kr 2 r) dn(kr3 2)

djn(kr 3 r) jkr3)kr3 -Y3(YB 1 ) A3 - &3 -3YB 30 (121)

d(kr 3r) d(kr 3r)

Jn(kr 2 r) Y2+b(YB 1) C2b + Jnkr2 0 Y;b(YB1 ) D2b

Jn(kr 3 r) Y(y 1 ) C3 -Jn(kr, 0)Y(yB, )D3 -0, (122)

53

dYb(YBI) dY2b(yBI)

Jnk2 0 dy C2b + J 2r) dy D2b

dY;(yB,) dY3(YBI)

-jk 3 0 dy C3 - Jn(kr3 r 0 dy D3 -0, (123)

krdJn(kr 2 r) djn(k r2 r))Dk2 2 Y2*b(YBI) C2b. +k&2 d~~)Y2b(YjB1 D2b

d(kr2 r) d~r20

dJn(k r3 r) dJft(kr- r)- kr, d(kr r) V3(Bj C3 - kr ~r -Y3(YS) D3 - 0 , (124)

and

jn(k r2 r) Y2b(Ye, A2b + Jn(kr2 r) Yb(YB, B2b

- Jn(kr 3 r) Y3y, A3 - JO(kr3 r) Y3(y, 1) B3 - 0 (125)

Here, (121) and (124) are valid only i r is not identically zero

for all values of r ange. r, and azi muthal angle, . Si milarly, ( 12 2) and ( 12 5)

54

af,, (r.)are valid only if is not identically zero for all values of range, r, and

azimuthal angle, *.

The seventh boundary condition is that of continuity of the normal

component of the acoustic particle velocity at the boundary y = YB2. This

implies

Un 3(t~r,*,YB2) - Ufl4 (t,r ,,YB2)

whereA

U,3(t.r,*.y) - U3(t,r,.y) • nl2(r,*,y) . (126)

U,4(t,r,f,y) - U4(t,r,*,y) • &s (r,+,y) , (127)

and n 2(r. .y) represents the unit vector normal to the boundary at y - Y 2.

Substituting (85) and (98) into (126), substituting (86) and (98) into

(127), and performing the indicated dot products yields

55

U,3(t,r.+,y) lk !k 3 ( A3 COSn rY3(Y) + B3 cos n+ Y3(y)

+ i)dJn(k '3 r)A+ C3 sin n+ Y+(y) + D3 sin n1 Y-(y)) - r

d(kr 3 r)

113 cos nY(y) +D 3 COS nY(y

- A3 sin n+ Y;(y) - B3 sin n+Y3(y)) J.(kr3 r) 4

dY;(y) dY3(y)+(A3 cosn+ dy +B3 cosn# dy

dY+(y) dY3(y)+ C3 sin n+ dy + D3 sin n+ d ) J,(kr3 r) ] ei2lft

afB2(r'#) A 1 fB2(r,+)

Nar r +YNB2

U,3(t,r,+,y) - - kr3 (A3 cos n+ Y3(y) + B3 cos n+ Y3 (y)

+ C3 sin n+ Y(y) + D3 sin n* Y(y)) dJn(kr 3 r) afB2(r,

d(kr 3 r) r

r2 (C3 cos n+ Y+(y) + D3 cos nsY(y)

afB2(r,+)

- A3 sin n+ Y+(y) - B3 sin n+Y (y)) Jn(kr 3 r)

56

dY;(y) dY3(y)

(A 3 cosn dy +B3 cosn+ dy

dY;(y) dY;(y)C3 inn)dy +D3 sin n* )- n (kr3 r)]"It+r) d NB2 t (128)y 3 NB2

and

djn(kr 4 r)A

Ua,(t,r,),y) - [k, (A 4 cos n+ + B4 sin n+) Y'(y) - rd(kr4 r)

+ !!(B 4 cos n+ A4 sin n+) Y*(y) Ja(kr4 r)

(A4 cos n+ + B4 sin n+) Jn(kr, r) dY;(Y) eiz ft

afB2 (r'+) A af B2(r,+) A

N B2 r 4

dJla(kr 4 r) 8f B2(4,)U,4(t,r,+,y) -- kr,4 (A4 cos n++ B4 sin n+) Y(y) dkr

d(kr 4 r)

n +fe(r,)

r2 (B 4 cos n+ - A4 Sin n+) Y4(y) J.(kr 4 r)

dY;(y) e2xrt

(A4 cos n++ B 4 sin n# J.(kr r) -y = . (129)4 dy NB2

57

Setting U,3(t.r,*yB2) (128) equal to U, 4(t.r,*yB2) (129) and eliminating

e,.2xft althe common term reveals

NB2

- kr3 (A 3 cos n Y3(YB2) + B3 cos n4 Y3(yB2)

dJa(kr3 r) afB2(r+)

+ C3 sin n+ Y3(yB2) + D3 sin n Y3(YB2)j 0d(kr3 r) a

n 3 C+ D cos n# 3(yB2)

afB2 (r,f)

- A3 sin ri Y3(YB2)- Bsin y * (YB2)) Jnkr3 r)

dY;(Y 82) dY3(yB2)

+(A 3 cosn+ dy + B3 cosn dy

dY;(YB 2) dY3(YB2)

+ C3 sinn dy + D3 sin n+ dy ) Jn(kr 3 r)

dJn(kr 4 r)a fB2(r,+)= - kr 4 ( A4 cos n + B4 sin ) Y (yYB2) ~r r

4 Y"(YB2)d(kr 4 r) a

af B(r,+)

r2 (B4 cos n+ - A, sin n+) Y4(yB2) Jn(kr4 r) -

58

dY'y 2

+ fA4 COSn+ + B4 Sin n+) Jn(k, r) 4(B)(130)4 dy

Factoring ( 13 0) yield s

k dJ(k 3 r ) . af B2(r,+) A3 - dJ(k 3 r)efBk r r)3 Y;(YB2) ar n3-kr3 r) Y3(YB2 ) 8r B3d~r3 r)d(k r)

n_____(r+__ afB2 (r,+)

- J,(k 3 r) Y;(B 2 - C3 - 2 Jn(kr, r) Y3(y. 2) -D

dY;(B 2 dY3(yB2)

+J(kr3 r) dy A3 +Jn,(krr) d y B3 osfn+

djn(kr 3 r) af B2(r,+) dJn(kr3 r) af B2(r,+)4 -kr 3 - Y+;YB2) -8 C-3 - &r3 Y3Y2 ar 3

d(kr r) d(kr H)3 3

+Jn,(kr r) A3 + ( r))YA3 +BJ D)r2 '3n(B) +r (kr3 B

dY;( 82 dY3(YB2) D1~

+ J(k, ) dy C3 + JR(k r3 r) dy D1snn

-1k 4 dJn(kr4 r) 8fB2 (r,+) ftnkr 82(r+4d(kr4 r) Y4 (YB2) ar r2 -(B2 r)

0 d- 4(Y 2) A r dJn(kr 4 r) + af B2 (r.*

Jn(kr4 r dy A4J cos n+ I- kr 4 r)B 4B21

iafB 2 (r,+) dY4;(YB 2)

r2 Jft 4 r) Y4(YB2) A4 + Jnk4r) dy in0 (11

Setting the respective coefficients of cos n4~ and sin n4 equal and


[J~k3 )dY;(y 82) dY3(yB)3

lj.k, 0 dy A3 +Jn(k, 3 0i dy B3

d I~ r3 r) yB A3 + nkr3 r) YB2 B31I-aikd(kr 3 0 2 + d(kr r) 0

60

' IJn(kr3 r) Y;(YB2) C3 + r) Y3(r) ) D31 f2

dY 2) dJo(kr4 r) af B2(r,+)-jn(kr 4 r) dy A4 - kr4 -4YB2 ) A4 ar

d(kr4 r)

2 Jn(kr4 r) Y' B4 ~f 2 r)(132)

and

I 0(~ 3 )dY3(YB 2 ) dY3(yB2)

lj(k 3H dy C3 +J& r3 r) dy D31

dj,,(kr r) dJ,(kr r) .ft

dr3 r)dk 3 r)

-r inkrr 3(y. 2 A3 + Jn(kr H) Y3(YB2) B31 2

61

dY+(YB2) dJn(kr 4 r) of B2 (r,)

- Jn(kr4 r) dy B4 - kr4 Y(YB2) B4 ad(k% r) o

+ Rf Jr(k, r) A4(yB) (r +(133)


function is not identically zero for all values of azimuthal angle f. Conducting

an analysis of (132) and (133) similar to that of the previous two boundary

conditions yields the set of six equations representing the seventh boundary

condition (BC # 7)

dYB(Y2 ) dY3(YB2) dYB(y2 )

Jn(kr3r 0 dy A3 + J,(kr r) dy B3 - Jn(kr 4r) dy A4 - 0 134)

dJn(kr 3 r) dJn(kr 3 r)kr3 r)Y(YB2) A3 + kr 3 Y3(yB2 ) B3

d(kr3 r) d(kr3 r)

dJn(kr 4 r)

4-- Y4(YB2) A4 - , (135)d(kr r)

62

Jn(kr3 r) Y(y 2 )C3 + Jn(kr 3 r) Y3(y82) D3 - Jn(kr4 r) (Y.2 ) B4 0 - (136)d(YB2 B4B2 = Y(+6

dY(YB2 ) dYB 2 ) dY(yB 2 )

J,(kr3 r) dy C3 + Jn(kr 3 r) - D3 - J.(k r) - B4 - 0 (137)3 dr3 dy D3 J~r 4 r dy B-,17

dJn(kr 3 r) dJn(kr 3 r)

kr3 d+ r) 2 C3 kr3 Y3(yB2) D3d(kr3 r) d(k' 3 r)

dJn(kr 4 r)

kir4 r) Y4(YB2) B4 -0, (138)4d(kr 4 r)

and

Jn(kr3 r) Y;(y8 2) A3 + Jn(kr3 r) Y a B3 - Jn(kr, r) Y+(y 82 ) A4 -0 (139)

of B2(r,f)

Here, (135) and (138) are valid only if ar is not identically zero

for all values of range, r, and azimuthal angle, +. Similarly, (136) and (139)

Of B2(r,+))

are valid only if is not identically zero for all values of range, r, and

azimuthal angle,*.

63

The final boundary condition is that of discontinuity of the normal

component of the acoustic particle velocity across the boundary at y - yo.

This implies

Un2a(t,r,+,yo ) = Un2b(t,r,+,y o) + [GI cos n+ + G2 sin n+) Jn(kr2 r) eIft, (140)

where:

U, a(t,r,4,y) 0 Aa~~hy •n(r,)y) , ( 141 )

I A

Un2 b(t,r,+,y) - U2b(t,r,+,y) * no(r,*,y), (142)

o(r,+,y) - A (143)

(143) represents the unit normal vector to the boundary at y - y., and G,

and G2 represent amounts of discontinuity. The "prime" superscripts (in Un2a

and U n2b) are used to indicate that these velocities are to be evaluated at y -

yo (This was necessary since the notation Un2a was used in the evaluation of

the velocity boundary condition at the surface, and the notation Un2b was

used in the evaluation of the velocity boundary condition at the boundary y

=yB!).

Substituting (83) and (143) into (141), substituting (84) and (143) into

(142), and performing the indicated dot products yields

64

U (tX.oy) - [kr2 (A2, COS l+ Y1a(y) +B2. cos n# Y1()+ 2 sin nf Y + (Y)

dJ,(kr r)- ~ 2 Afl

* D 2 si ii, Y 2 1 y~j d (k r 2 r) + r i Y 1 Y

+ D2a COS fl* '2a(Y) - A21 Sin fl+ Y2a(y) - B2. sin 0,Y2a(y)I) Jn(kr 2 r)A

dY;1(Y) dY;1(y)+(A2acosno- dy *+B2acosnO dy

dY;1(y) dY2a(Y)AA+ Ch1 si dy + D2. si n dy jf~k r) ' I e i2xft .

U (t,r,o~y) -(A 2, cos dY;1(y) dY21(nady +B 1 csf4 dy

dY2a(y) dY2;1(Y)+ C2. sinl f dy + D2. sin n+ dy kr r) ei~xft . (144)

and

U 2b~tr*y [k l 2 (A 2b COS n+Y2b(y) +B2b COS OYby

dJnl(kr 2 r) A

+ C2b sinl no~ Y2b~y) + D2b sin no* Y2b~y)J rd(k r2 r)

* 1 (C2b COS f Y*bY + 2b COS n+* Y;b~y)

65

+ - A2Ab sin n* Y2b(Y) - B2b sin n*Y2 b~y)J Jn(kr 2 r)*

dY2b(Y) dY2b(y)+(Aabcosn. dy +B2bcosn$ dy

dYab(y) dY2b(Y)A+ C2b sin n+ dy + D2b sin n+ dy J(kr 2r) ] e!2'x y

dYab(y) dY2b(y)Un2b(tr,*,Y)- A2bCOSnl dy +B COS dy

dY+b(Y) dY2b(y)+ C2b sin n+ )y D ) Jg(kr r) ei . (145)dy + D2b sin nf dy

Substituting (144) and (145) into (140), dividing out the common

terms, and evaluating the resulting expressions at y - yo yields

dY2a(Yo) dY2a(Yo) dY2b(Yo) dY2b(yo)dy A,. dy B2a dy A2b- dy Bebcosn,

dYa(yo) \ dY2a(Yo) dY2b(YO) dY2b(YO)dy C2a dy D2a dy Cab- dy Dab) sinn*

-G cos n$+ G2 sin n . (146)

Setting the respective coefficients of cos n+ and sin n equal and

rearranging yields the following pair of equations representing the eighth

and final boundary condition (BC #8):

66

dY2;(Y 0 ) dY28(y0 ) dY2b(Yo) dY2b(YO)dy A2+ dy B2 - dy A2b - dy B2b) - GI, (147)

and

dY28(Y0) dY2a(y0 ) dY*b(yo) dY2b(yo)

{ dy Ch dy D2a- dy C2b- dy D2b)-G 2 . (148)


function is not identically zero for all values of azimuthal angle *.

F. SUMMARY OF BOUNDARY CONDITION EQUATIONS AND THEIRVALIDITY

To summarize, the boundary condition equations which must be

satisfied for our general waveguide model are as follows:

P2(YS) J(kr2 r) Y+8(Ys) A28 + P2(YS) Jn(kr2 r) Y2a(Ys) B2a

-pI(ys) Ja(kr Ir) YI(ys) B - 0, (149)

which is valid only if cos n+ is not identically zero for all values of $.

P2(Ys) J.(kr 2 r) Y2;(ys) C2h + P2(Ys) Jn(kr 2 r) Y2,(ys) D2a

- PI(ys) Jn(krI r) Yi(ys) A, - 0, (150)

which is valid only if sin n+ is not identically zero for all values of *.

67

Y2,(Yo) A2a + Y2;(yO) B2a - Y2b(yo) A2b - Y2b(Yo) B2b - 0, (151)

which is valid only if cos n+ is not identically zero for all values oft.

Y;3 (y.) C2t + Y2(aYo) Da - Yzb(YO) C2b - Yzb(Yo) Dab - , (152)

which is valid only if sin n+ is not identically zero for all values of $.

P2(YBI) Jn(kr 2 r) Y2b(yBI) A2b + p2(yBI) Jf(kr 2 r)Y 2b(ye1 ) B2b

- P3(YBI) Jn(kr 3 r) Y(y,) A3 - P3(YBI) J(kr 3 r) Y3(ye,) B3 = 0, (153)

which is valid only if cos n+ is not identically zero for all values of *.

p2(YBI) Jn(kr2 r) Y2b(YB) C2b + P2(YBI) Jn(kr2 r) Yzb(yl,) D2b

P3(YB) J.(kr 3 r) Y(yB,) C3 - p3(yBI) Jn(k' 3 r) Y3(YB1 ) D3 - 0, (154)


p3(yB2) Jn(kr 3 r) Y*(yB2 ) A3 + p3(ye2) Jn(kr3 r) Y3(yB2 ) B3

- P4(Ya2 ) Jn(kr 4 r) Y4(yB2 ) A4 - 0, (155)


68

P3(YB2) Jn(kr3 r) Y3(YB2) C3 + P3(YB2) Jn(kr 3 r) Y3(yB2) D3

- P4(YB2) Jft(kr 4 r) Y4(ye 2) B4 - 0, (156)


dYa (Ys) dY2a(Ys) dY1 (ys)Jn(kr 2 r) dy A2a + Jn(kr r) dy B2a - Jn(kr, r) dy B, -0, (157)


dY23 (Ys) dY-2 (Ys) dYi(Ys)J&~r r) dy C2 + Jn(kr2 r) d D2y- Jn(kr, r) dy A - 0 0158)dy Ddynkrr 2 1 O 18

which is valid only if sin n is not identically zero for all values off.

dJn(kr 2 r) dJn(kr 2 r)r2 Y2a(Ys) A2a +kr 2 2a(Ys) B2a

d(kr 2 r) d(kr 2 r)

dJn(kr r)

h- k I YI(Ys) B, - 0, (159)d(kr, r)

which is valid only if cos n and ar are not identically zero for all values

of r and .

69

Jn(kr2 r) Y;2(ys) C2a + Jn(kr 2 r) Y2.(ys) D2a - J(kr I r) Yi(ys) A, - 0, (160)

which is valid only if cos n and afs(r,+) are not identically zero for all values

of r andS.

dJn(kr 2 r) dJn(kr 2 r)kr2 YzkYs r)C 2 + kr2 Y2a(yS) D2&

d(kr2 r) Y3 y)d(kr 2 r)

dJn(krI r)

- kr I YI(ys) A, - 0, (161)d(kr I r)

which is valid only if sin n* and afs~r) are not identically zero for all valuesa~r

of r and + .

Jn(kr 2 r) Y2,(ys) A2a + J0(kr 2 r) Y2,(ys) B2a- JD(kr I r) YI(ys) B, - 0, (162)

which is valid only if sin n+ and are not identically zero for all values

of r and.

70

dY2b(YBI) dY2b(YBI)

Jf(krr) dy A2b + Ja(kr r) dy B2b

dY3(yB,) dY3(YBI)

Jn(kr 3 r) dy A3 -Jn(kf 3 r) dy B3 0 - (163)


dY2b(YBl) dY2b(YBl)

Jn(kr2 r) dy C2b + JO(kr 2 r) dy D2b

dY3(yB,) dY3(YBI)

Jn(kr 3 r) dy C3 -J(kr3 r) dy D3 -= . (164)


dJn(kr 2 r) dJn(kr2 r) -

kr)Y;b(y5 1 ) A2b + kr2 Y2b(YBI) B2bd(kr2 r) d(kr2 0

dJn(kr 3 r) djn(kr3 r)r3 d(kr Y(y) A3 - kr3 d(kr 3) (y) B3 - 0, (163)

3 )3

71

OfB(r*)which is valid only if cos n+ and are not identically zero for alla~r

values of r and *.

Jn(kr 2 r) Y2b(YlB) C2b + J (kr2 r) Y2b(YBJ) D2b

- Jn(kr3 r) Y;(YB,) C3 - Jn(kr3 r) Y3(ye) D3 = 0, (166)

afB, (r,+)which is valid only if cos n+ and are not identically zero for all

values of r and*.

dJn(kr2 r) dJn(kr 2 r)kr 2 0 2b(YBI ) C2b + kr 2 d(k2 r) Y2bYBI) D2b

d(kr2 r) dk 2 r

dJn(kr 3 r) dJn(kr3 r)- kr 3 d(kr Y(YBI) C3 - kr 3 ATk3 r) Y3(YBI) D3 - 0, (167)

afB,(r,*)which is valid only if sin n+ and are not identically zero for allar

values of r and*.

72

JA(r2 r) Y2 BI) A2b * Jn(kr 2 r) Y2b(YB1 ) B2b

- Jn(kr 3 r) Y(yB,) A3 - Jn(kr 3 r) Y3(YB) B3 - 0 , (168)

8fB,(r, )which is valid only if sin n# and are not identically zero for all

values of r and *.

dY3(YB2) dY3(YB2) dY4(YB2)

Jn(kr 3 r) dy A3 + Jn(kr, r) dy B3 - Jn(kr 4 r) dy A4 - 0 (169)

which is valid only if cos n is not identically zero for all values of *.

dY;(YB 2) dY3(YB2) dY(YB 2)Jn(kr 3 r) dy C3 + Jn(kr4 r) ddy B4 -0, (170)


73

dJn(kr 3 r) dJn(kr 3 r)Y(YB2 ) A3 , kr 3 k3(YB 2 ) B3

d(kr, r) d(kr 3 r)

dJn(kr4 r)4

kr4 Y4(Y ) A4 - 0, (171)d(kr4 r)

8fB2(r, )which is valid only if cos n$ and are not identically zero for allor

values of r and*.

Jf(kr 3 r) Y;(YB2) C3 + J.(k, 3 r) Y3(yB2) D3 - Jn(kr4 r) Y4(YB2) B4 = 0, (172)

afB2(r,+)which is valid only if cos n* and are not identically zero for all

values of r and*.

dJn(kr 3 r) dJo(kr 3 r)d(kr3 r) d(kr 3 r)

dJ0(kr4 r)4 ,4

kr,4 Y4 (YB2) B4 = 0, (173)d(kr4 r)

74

Ofe 2 (r.0)which is valid only if sin n$ and are not identically zero for allor

values of r and *.

Jn(kr3 r) Y;(yB2) A3 + Jn(kr3 r) Y3(yB2) B3 - J(k' 4 r) Y (yB2) A4 - 0, (174)

afB2 (r.+)

which is valid only if sin n$ and are not identically zero for all

values of r and *.

dY2a(Yo) dY21(yo) dY2b(yo) dY2b(yo)y A2a + d B2a A2b- y 2b) -GI, (175)

dy dy dy dy(1)


dY2a(Yo) dY2a(Yo) dY2b(Yo) dY2b(YO)(~ ~ D2b) - G2 , (176)dy C2a + dy D2a dy Cm- dy

which is valid only if sin n# is not identically zero for all values of *.

G. DIFFERENCES NOTED DUE TO ARBITRARY BOUNDARY SHAPE

Before going on to verify that the set of derived general boundary

condition equations reduces to a well-known and well-documented set of

boundary condition equations for a very specific set of waveguide conditions,

the interesting and somewhat unexpected appearance of J,(kri), i - 1, 2, 3. 4

75

terms in some of the general boundary condition equations needs to be

discussed. These terms cannot be eliminated in the general case of

arbitrarily shaped boundaries because the radial component of the wave

number is not constant, and in fact, it depends upon the orientation of the

local normal vector to the surface.

To show that this is true, we will begin with the specific case of a planar

boundary. As shown in Figure 3. the vector wave number k may be resolved

into its component vectors, k. and k, with respect to the coordinate axes r

and y, that is,

k = k'r + ky. (177)

Planar boundaryr

K7

k

y

Figure 3. Planar Boundary Wave Number Vector Decomposition

Evaluating the geometry reveals the following pair of equations to describe

these component vectors:

A Ak=-kcos~r- kr, (178)

andA A

ky - k sin 0 y - kyy, (179)

76

where k is the magnitude of the vector k, 0 is the angle oetween the vectorA A.

k and the r-axis, r is the unit vector in the radial direction, y is the unit

vector in the y direction, kr is the magnitude of the radial component, and ky

is the magnitude of the depth component.

Now we will explore the more general case shown in Figure 4. Again the

k vector may be decomposed into its component vectors with respect to two

arbitrary boundary

rr

yy

Figure 4. Generalized Boundary Wave Number Vector Decomposition

very different sets of coordinate axes, the standard r and y axes, and the r'

and the y' axes, which are oriented based on the local normal vector. When

decomposed with respect to the r and y axes, the components may be

expressed as indicated in (178) and (179). However, when decomposed with

77

respect to the r' and the y' axes, the resulting components may be expressed

as follows:

(0+ A A

kr k cos (+)' - kr' , (180)

andA A

ky - k sin (0 + 0) y' - ky, y", (181)

where k is the magnitude of the vector k, 0 is the angle between the vectorA

k and the r-axis, 0 is the angle between the r and r' axes, r' is the unitA

vector in the r' direction, y' is the unit vector in the y' direction, kr' is the

magnitude of the component in the r' direction, and ky. is the magnitude of

the component in the y' direction.

Using the appropriate trigonometric identities, (180) and (181) may be

rewritten as

A

kr. = k (cos 0 cos P - sin 0 sin )r' , (182)

andA

ky. - k (sin 0 cos 0 + cos 0 sin P) y' (183)

Equations (182) and (183) may be simplified further by carrying out

the indicated multiplications and using (178) and (179), revealing

78

A

k r . - (kr cos- ky sinp)r , (184)

and

ky = (k sin + ky COS) (185)

Equation (184) shows that the component of the wave number along

the tangent plane (i.e., in the r' direction) at any point depends on the angle

0, and therefore, on the specific point along the arbitrarily shaped boundary

at which the vector is to be evaluated. Therefore, this component is not

constant, and must be maintained in the boundary condition equations.

H. VERIFICATION OF INITIAL RESULTS

We will now show that the boundary condition equations derived in

this section reduce to the well-known and well-documented set of boundary

condition equations for the following classical waveguide problem: Assume

that there are only three fluid media, not four, and that

All sound speeds are constant, i.e.,

" cl(y) - cf." c2(Y) - c2, and- c3(y) - c3.

All ambient densities are constant, i.e.,

* PI(y) - P1.• P2(Y) - P2, and

" P3(Y) - P3.

79

All boundaries are planar and parallel, i.e.,

" ys(r*) = 0," y0(r,+) - yo, and

" YB , (r,+) - D.

These conditions represent a waveguide made up of three layers. The

flat boundary at y - 0 separates a semi-infinite medium (medium I:

- 0 S y z 0) and a finite medium (medium II: 0 % y % D) of (perhaps)

different specific acoustic impedances. The flat boundary at y = D separates a

finite medium (medium II: 0 z y z D) and a semi-infinite medium (medium

Ill: D s yx + co) of (perhaps) different specific acoustic impedances. These

conditions imply that the following arbitrary constants may be set equal to

zero for the reasons indicated:

* B3 (no wave reflected in negative y direction)* D3 (no wave reflected in negative y direction)* A4 (medium not modeled)" B4 (medium not modeled)

Also, n - 0 since plane, parallel boundaries remove angular dependence.

These conditions also imply that the wave number k and the

propagation vector component in the y direction k. are constant in a given

fluid medium. In this case, the solution to (28) is known, and can be written

as follows:

Y(y) -Ay e vik + By e'kYy (186)

80

Thus, we may set the arbitrary functions Yt (y) and Y-(y) in our

previous work as follows:

YI(y) Ay ekYy (187)

and

Y- (Y) - By e Y . (188)

Also, since all of the boundaries are plane, parallel surfaces, af , af

ar' ar

afBI

and - are identically zero for all values of r and *. Thus, the following

boundary condition equations have been invalidated for the reasons

indicated:

* (150) is invalid because sin nO (for n - 0) is identically zero for all valuesof *,

* (152) is invalid because sin n+ (for n - 0) is identically zero for all valuesof +,

S(1 54) is invalid because sin n$ (for n - 0) is identically zero for all valuesof +.

* (1 55) is invalid because medium IV is not being modeled,* (156) is invalid because medium IV is not being modeled,* ( 58) is invalid because sin n+ (for n - 0) is identically zero for all values

of *,

S159) is invalid because afs(r,*) is identically zero for all values of r and *,

81

*(160) is invalid because is identically zero for all values of r and*,a+

*(161 ) is invalid because both sin n+ and afs(r'+) are identically zero fora~r

all values of r and +,

(162) is invalid because both sin n+ and a are identically zero for

all values of r and *,* (164) is invalid because sin no (for n - 0) is identically zero for all values

of 0,

afB (rf)

* (165) is invalid because is identically zero for all values of r andar

afB1 (r,$)

0(166) is invalid because is identically zero for all values of r and

afB1 (r,#)

* (167) is invalid because both sin n and are identically zero for8)r

all values of r and *,afB ,(r,+)

• 168) is invalid because both sin n+ and are identically zero for

all values of r and *,* (169) is invalid because medium IV is not being modeled,* (170) is invalid because medium IV is not being modeled,* (171) is invalid because medium IV is not being modeled,* (172) is invalid because medium IV is not being modeled,* (173) is invalid because medium IV is not being modeled,* (174) is invalid because medium IV is not being modeled, and* (176) is invalid because sin n+ (for n - 0) is identically zero for all values

of *.

82

Thus, the original set of 28 equations in 17 unknowns has been reduced

to a set of six equations in six unknowns. The next step will be to evaluate

each of the remaining equations in turn so that they may be compared with

the equations developed by Ziomek (1991) for this particular waveguide

problem.

The first of the remaining equations is

P2(ys) Jn(kr 2 r) Y2(Ys) A2& + P2(Ys) Jn(kr2 r) Y2,(ys) B2a

- PI(ys) Jn(kr i r) YI(ys) B, = 0. (149)

The Bessel function dependence of (149) may be eliminated by virtue

of the fact that, in this simple waveguide problem, the radial component of

the propagation vector is the same in all three media (implying that the

Bessel functions may just be divided out). Recalling that the densities are

constants and that the value of y at ys is identically zero, (149) becomes

P2 Y2a() A2a + P2 Y2a(0 ) B2a - P Y (0 ) Bi -0. (189)

Substituting the appropriate functions of y into the simplified

expression ( 89), and noting that el ° is equal to unity yields the following:

P2 A2a + P2 B2 a - p, BI - 0. (190)

83

This equation is the same as that derived by Ziomek (199 1, equation

(3.9-25)) for this boundary condition.

Conducting a similar analysis on ( 51) yields

e-1ky2 YO A2 e ky2 Y A B2a - e A2b - e+jky2 YO B2b - 0. (191)

This equation is the same as that derived by Ziomek (1991, equation


Equation (153) reduces to the following (after additionally noting that

B3 has been set equal to zero)

P2 e -kY2 D A2b + P2 e +j Y2 B2b - P3 e Y3 A3 - 0. (192)



Equation (157) becomes

ky2 A2a - ky2 B2a + ky BI = 0. (193)



84

Equation (163) reduces to the following (after additionally noting that

B3 has been set equal to zero):

-"k D *k D -"k

ky2 e -iY2 A2b - ky 2 e itY2 B2b - kY3 e iky3 A3 - 0. (194)



In order to evaluate (175), let G - k r (as suggested by Ziomek ( 1991,2x

equation (3.9-35) and following)). Substituting yields

j ky 2 e-kY2 y 0 A2a ik 2 e+ity2Yo B2a

+ j ky2 e iky2YO A2b - j k 2 e+jk y2 B2b - (195)

This reduces to

A2a + e 0 y2 B2a + e A2b-e B - + 0J .096)

85



Thus, we have shown that the theoretically derived set of equations for

a general waveguide problem reduces to the set of equations expected for

the classical waveguide problem. This provides us with the confidence to go

on with the solution for the unknown arbitrary constants.

86

IV. SOLUTION FOR THE UNKNOWN ARBITRARY CONSTANTS USINGSYMBOLIC ALGEBRA CAPABILITIES OF Mathematica

In Section III of this thesis, we developed a set of 28 boundary

condition equations in the 17 unknown constants. The purpose of this section

is to generate a solution to this system of equations for the general

waveguide case.

A review of (149) through (176) reveals that the coefficients of these

unknown constants are, in general, complicated expressions involving depth-

dependent densities, range-dependent nth order Bessel functions, and as yet

unspecified depth-dependent velocity potential functions (i.e., the Y* and Y_

terms). In order to maintain the generality of the generated solution, we will

require either many long hours of tedious algebra involving manipulations of

these complicated expressions (with the high probability of algebraic errors)

or a computer program capable of conducting such manipulations directly on

these symbolic expressions. Fortunately, Mathematica for the Macintosh

computer (version 1.2.1 f33 (enhanced)) is the one such program available to

us at the Naval Postgraduate School, and therefore will be used to generate

the general solution desired.

The first step in this process will be to program Mathematica to solve

for the unknown constants for a very specific set of waveguide conditions.

By doing this, we will gain experience in using the program and confidence

that the program output is reliable. For this work, we will use a three media

87

waveguide with plane, parallel boundaries. Using vector-matrix notation, a

compact system equation may be written for this (or for any other) case as

follows:

Az-b, (198)

where A is the matrix of coefficients, z is the column vector of unknown

constants, and b is the column vector of known constants.

For the three media waveguide with plane, parallel boundaries, these

vector-matrix quantities are defined as follows:

A- aIj aj,2 aj.3 0 0 00 a2,2 a2,3 a2.4 a2,5 00 0 0 a3,4 a3,5 a3,6a4,1 a4,2 a4,3 0 0 0

0 0 0 a5,4 a5,5 a5,610 a6.2 a6.3 a6,4 a6. 5 0

where

a, 1 - - p1 (ys) YI(ys) (199)

al,2 = P2(Ys) Y;1(Ys) (200)

a1,3 = P2(YS) Y2&(Ys) (201)

a2.2 - Y;,(Yo) (202)

a2,3 - Y2a(Y.) (203)

a2,4 - - Y;1)(yo) (204)

a2,5 - - Y2b(yo) (205)

88

a3,4 - P2(YB,) Y*2b(YBI) (206)

a13 5 - P2(YB,) Y~b(YB1) (207)

a3,6 - -P3(Y]B 1) Y;(YBIj) (208)

dY I (Ys)a4 ,1 - dy (209)

dY +a(ys)a4,2 - dy (210)

dY2a(YS)a4.3 - dy (211)

d2b(YBI)a5,4 - dy (212)

dY2b(YBI)

a5,5 - dy (213)

dY;(yB31)

a5 ,6 -- dy (214)

dY*a(Y.)a6,2 - dy (215)

dY2a(Y0 )a6,3 -w dy (216)

dY~b(yO)a6 ,4 - dy (217)

89

dY2b(YO)a6.5 dy (218)

x - [BI A2a B2a A2b B2b A3IT (219)

where the superscript T indicates the transpose matrix operator (indicating

that x is a column vector),

b - [0 0 0 0 0 GI] T (220)

where the superscript T indicates that b is a column vector, and

G(221)

It should be noted here that we have defined the matrix A in a very

specific manner. Each row of A represents one of the valid boundary

condition equations for the specific waveguide being studied. These appear

in the order presented in Section Ill. For the three media waveguide with

plane, parallel boundaries, row I of A contains the coefficients found in

(149). Row 2 contains the coefficients found in (151), and so on. Row 6

contains the coefficients found in (175). For simplicity's sake, we have used

generic elements, such as al,2, to replace the more complicated expressions.

In addition, zeros have been used to indicate that the appropriate unknown

constants do not appear in a specific boundary condition equation. We will

use this convention in the work which follows.

90

In this three media waveguide case, A is a six by six square matrix.

Therefore, the solution to ( 198) may be written directly as

x - A- ' b. (222)

wrnere the superscript -I indicates tne inverse matrix operator.

Using Wolfram (1988) as a programming reference guide, a

Mathematica "notebook" was created to solve this three media waveguide

case using the solution technique expressed in (222). The Mathematica code

required to perform this task is as follows:

a - ((alcl, alc2, alc3, 0, 0, 0),(0, a2c2 a2c3, a2c4, a2c5, 0),(0, 0, 0, a3c4, a3c5, a3c6},(a4cl, a4c2, a4c3, 0, 0, 0),(0, 0, 0, a5c4, a5c5, a5c6),(0, a6c2, a6c3, a6c4. a6c5, 0)):

b = (0, 0, 0, 0, 0, GI);

x - (Inversefa).b

In developing this code, we continued to utilize the generic matrix

elements described earlier. Two subtle differences in the notation used in

the code from the notation discussed earlier need to be pointed out. First, we

have represented the elements of' matrix A (for example, al, 2) as individual

variables (the corresponding variable name would be alc2). This slight

deviation in notation was used because subscripting as defined by

Mathematica would not have been useful for our purposes. In this revised

notation, the small case letter "c" represents the comma in the element name.

91

This deviation was required because the program uses the comma to

separate individual array elements. The second notational comment refers to

the fact that a lower case letter "a" was used to represent the matrix A. This

was required to conform with Mathematica's notational convention, which

reserves names beginning with capital letters for built-in functions.

Unfortunately, running this code resulted in halted execution due to a

singularity error. We surmise that the problem occurred when the program

was attempting to take the inverse of the matrix A. Luckily, Mathemalica

has a built-in function, LinearSolve, Which evaluates (198) directly if the

matrix A is a square matrix. Thus, the following revised code was written:

a - ((alcl, alc2, alc3, 0, 0, 0).(0, a2c2, a2c3, a2c4, a2c5, 0),(0, 0. 0, a3c4, a3c5, a3c6),(a4cl, a4c2, a4c3, 0, 0, 0),(0, 0, 0, a5c4, a5c5, a5c6),(0, a6c2, a6c3, a6c4, a6c5, 0));

b - (0, 0, 0, 0, 0, GI);

LinearSolve[a,b]

The revised code ran successfully. The output of this code is the desired

vector x. When Mathematica functions such as Factor, Cancel, and Simplify

were applied to the output, the same result was returned, indicating that the

program was satisfied that the output was as simple as it could make it.

Closer inspection of the output revealed that each of the six elements was of

the form

92

num! (223)II ' denom

where x, represents the first element of the vector z (in the three media

waveguide with plane, parallel boundaries, this element is the unknown

constant BI), num1 represents the numerator expression for the first

element, and denom represents the denominator. Fortunately, all of the

elements of the output vector have a common denominator. This inspection

also revealed that some algebraic manipulations could be manually

performed to simplify the expressions somewhat. Thus, the robustness of the

symbolic algebra functions of Mathem/tica is at best questionable.

We will now present the results of the program for the three media

waveguide with plane, parallel boundaries. The first step will be to simplify

the results manually in order to generate generic expressions for the

unknown constants in terms of the generic elements. Second, we will

substitute (199) through (218) and (221) into the generic expressions to

reveal general expressions for these unknown constants. Finally, we will

assume constant speed of sound and constant density and show that the

general expressions formed from the Mathematica output are the same as

those derived by Ziomek (1991) for the classical waveguide case. This

verification will be conducted in the following order: A2a, B2a, A2b, B2b, B1,

and A3. We will demonstrate this entire process for the unknown constant

A2. only, and simply present the results for the other five unknown

constants.

93

The first output expression to be explored will be the common

denominator, denom

denom - - a1,3 a2,5 a3,6 a4,1 a5,4 a6.2 + al., a2,5 a3,6 a4,3 a5,4 a6,2" a 1.3 a2,4 a3,6 a4,1 a5.5 a6,2 - a Ij a2,4 a3.6 a4 .3 a5 .5 a6,2" a 13 a2,5 a3.4 a4 .1 a5,6 a6,2 - a1,3 a2,4 a3,5 a4,1 a3,6 a6,2- al', a2,5 a3,4 24,3 a3.6 26,2 + al., 2, 23.5 a4.3 25.6 a6,2+ 21,2 a2.5 23,6 a4,1 25.4 a6.3 - 21,, 22,5 23,6 24,2 a5,4 a6.3- a1,2 a2.4 23,6 a4,1 a5, 26,3 +al, 2.4 a3,6 24.2 a5,5 26,3- a 1,2 22,5 23.4 24,1 a5,6 a6,3 21a,2 a2,4 a3.5 a4,1 a5,6 26,3+ al., a2,5 a3,4 24,2 a5,6 26,3 - a11 2, a3,5 a4 .2 a5.6 a6.3- a 1,3 a2,2 a3,6 N4, 25,5 26,4 +a1,2 a2,3 a3,6 a4,1 a5.5 a6.4- al,, a2,3 a3,6 24,2 25.5 26,4 +al, 2,2 a3,6 24,3 a5 26,4" a 1,3 a2,2 a3 .5 a4.1 a5,6 a6,4 - 2,2 2,3 23,5 a4.1 a5,6 26,4" al., 2,3 a3.5 a4,2 25,6 26,4 - a,, I22,2 23.5 24,3 a5,6 26,4

" a 1.3 a2,2 a3,6 24,1 25,4 26,5 - 21,2 2,3 23,6 a4.1 a5,4 26,5" a1 ,, 2,3 a3,6 24,2 25,4 a6,5 - 2jlj a2,2 a3,6 a4,3 25,4 a6,5

- 2,3 22,2 a3,4 a4,1 25,6 26,5+ 21,2 2,3 23,4 a41 25,6 26,5al., 2,3 23,4 24,2 25,6 a6,5 2a, 2,2 23,4 a4 .3 a5,6 a6,5 (224)

Factoring (224) reveals

denom - al., a2,5 a4,3 26,2 (a3,6 a5,4 - a3.4 25,6)- a1.3 2,5 24,, 26,2 (23,6 25,4 - a3,4 25,6)+ a.3 2,4 24,, a6,2 (23,6 91. - a3,5 25,6)- al,, a2,4 24,3 a6,2 (23.6 a3,5 - 23,5 a5,6)+2a,2 a2,5 a4,1 26,3 (a3.6 25,4 - 23,4 25,6)

- al., 2,5 24,2 26,3 (23,6 25,4 - 23,4 25,6)+ ,1 2,4 24,2 26,3 (a3,6 a5,5 - a3.5 a5,6)-212 2,4 a4.1 a6,3 (213,6 a5,5 - 23,5 a5,6)

a+ .21,23 3a4 .1 a6,4 (a3,6 a5,5 - a3,5 25,6)- 2,3 a2,2 24,, a6,4 (a3,6 25,5 - a3.5 a5,6)

+ al, 2,2 24,3 26,4 (a3.6 a5. - 23,5 25,6)

- 111 2,3 14,2 a6,4 (a3.6 a5.5 - a3,5 a5,6)+ al,3 22,2 a4,l 26,5 (23,6 a5,4 - 23,4 25,6)

1121,2 a2,3 a4.1 26,5 (213,6 a5,4 - 3,4 925,6)

94

+ al, a2,3 a4.2 a6,5 (a3,6 a5,4 - a3,4 a5,6)- aI, I a2,2 a4,3 a6,5 (a3 .6 a54 - a3.4 a5.6) -(225)

Collecting com mon ter ms yields the f ollowing generic expression f or the

common dlenominator:

denom - (a3 .6 a5,4 - a3,4 a5,6) [al, a2,5 a4,3 a6,2- a1,3 a2,5 a4,1 a6 .2 + al,2 a2,5 a4,1 a6.3 - al., a2.5 a4.2 a6.3+ a1,3 a2,2 a4.1 a6.5 - a1,2 a2 .3 a4,1 a6,5 + al., a2,3 a4.2 a6,5

- a1,1 a2.2 a4,3 a6.5] + (a3,6 a5,5 - a3,, a5,6) l, 3 a2,4 a4,1 a6.2- al., a2.4 a4,3 a6,2 + al, a2,4 a4,2 a6,3 - al, 2 a2,4 a4,1 a6.3+ a1,2 223 a4,1 a6,4 - a1,3 a2,2 a4,1 a6,4 + a1,1 a2,2 a4.3 a6.4

- al', a2,3 a4,2 a6,41 (226)

Substituting (199) through (2 18) into (226), and using the f acts that

ys- 0 and Y - D reveals

dY (D) dyb(D)

denm G(2 0D Y+b(D) dy -P3(D) dy Y;(D))

dY+2a(yo) dY2a(0) dY 1(0) dY;1(y0)X [p1(0) Y-1(0) ddy 2b(yo) -P2(

0 ) ddy Y2a(0) Y;b(y0'l

dY,(0) dY2a(Yo) dY2,(0) dY2a(Y,)~P2( 0 ) dy Y+a(0) dy Y~b(yo) -PI1(0)1 (0) dd y 2b~yo)

95

dYI(O) dY~b(yo) dY1(O) dY-2b(yO)+ P2( 0 ) d-y- Y2a(y.) Y~a(o) dy -P2(

0 ) dy Ya (0) Y1( 0)d

+P1()Y( dY(O) dY2b(yo) I a dY2a(O) dYb(YO)Ip()~O d-y- Y2a(Y@) dy -P 1(O) Y (O) Y*(Y-) dy dy

dY;(D) dY-2b(D)+ GO2() Y~b(D) dy- P3(D) dy Y;(D))

1P2(O) dy d 3()~(oP()iO y d ~(O

dY,( 0) dY~a(y) dY,(0) dY2 ()

p(0Y()dy dy Y2.O b(yo)O dy 1 (0) Y b(yO)

dY,() dYYb(YO dY 1(0) dY~b(YO)(0) Y-(0Y1 ~o (0) P2 -ay- Yj(y 00))-y Y dyo

dY 1((0) dYb(YO) dY 1( 0) dYb(yo)+ P2()(0) Y~a(O) dy2,( -dy -P2(0)Y (O dy 2(O Y2 (0) dy

(227)

Simplifying (227) reveals

96

p 1(0 Y () dY;a(y 0) dY2(O) dY2',(o) dY; 1,(yo)denm-p1 O)j()(dy dy dy dy

dY 1(0) dY;a(y0) dYa(Yo)

dy (dy Y~()~()dy

dY;(D) dYb(D

X2b~yO) (P2(D Y2+b(D) d y- - P30D dy Y30)

dY;(D) dY~b(D)- 2b(YO) (P2(D) Y2b(D) dy P3(D) dy Y+(D)1

1PI~) Y(O)(Y+ dYa(O) dY~a(O)-~~ 2a()Y()(YV dy -dy Y2a1(yo))

-P2(0) dyO Y+ (aY. Y- (0) - Y4a(0) Y-a(Yo))J

dy 2b(~d) y- - P3(D) dy Y3

dY*2b(yo)) Y- dY;(D) dY2b(D)

dy ( 2(D Y() dy - PO) dy Y;(D))1. (228)

97

Equation (228) is the final general form of the Mathematica output

common denominator. If we now assume constant speed of sound and

constant density in a specific medium, the depth-dependent functions in

(228) become complex exponentials and the denominator becomes (using the

subscript c to ikncate the constant speed of sound assumption)

denom -- j 2 e-kY3 D

X(pk -P2 k (p2 k - p 3 k ) e-kY{Pky 2 -2ky ky3 Y

+ (p, ky 2 + P2 k ) (P2 ky3 + P3 ky2 ) e + ky 2 D . (229)

Now that the denominator has been simplified, we'll concentrate on

obtaining expressions for each of the unknown constants in the order stated

above. The first constant is

A2a - num2 (230)Aa=dehorn

where

num 2 = GI I- al, 3 a2,5 a3.6 a4,1 a5,4 + a1,, a2,5 a3,6 a4,3 a5,4+ a1.3 a2.4 a3,6 a4 1 a5.5 - a, I a2.4 a3.6 a4.3 a5.5+ a1.3 a2,5 a3,4 a4, 1 a5.6 - al, 3 a2,4 a3,5 a4, 1 a5.6- a1,1 a25 a3.4 a4.3 a5,6 + a1,1 a2,4 a3,5 a4,3 a5,61 . (231)

98

Factoring (231) and collecting common terms yields the following

generic expression for the numerator of Aha:

nuM2 - G, [(a,., a2.5 a4.3 - a1,3 a2.5 a4.1) (a3,6 a5,4 - a3.4 a5.6)

+(a,, a2.4 a4.3 - al, 3 a2,4 a4.1) (a3,5 a5,6 - a3,6 a5,5)1 (232)

Substituting appropriate expressions into (232) yields the following

general expression f or the numerator of A 2a:

num2 -k ( 1 0)Y_ dY2a(O) dY 1(0)nu 2 x'p();O y -2 0 dy Y2a(O))

dY;(D) dYb(D

X l2b'YO' GP20' Y2b(D) dy P3(D) dy Y+'")

dY;(D) dY~b(D) Y() 23-Y~b(yo) (P2(D) Yb(D) dy -P3(D) dy Y;D)3 23

Again, making the constant speed of sound and constant density

assumptions and substituting the appropriate depth-dependent expressions

allows us to write the numerator of A2. as (using the subfscript c to indicate

the COWSIMI speed of sound assumption)

99

num -kL (k P2 k ) e-i 3D2c 2x' Y2 yI

X I(P 3 k -P2 k Y )ikY Y o -It Y D+ (P2k 3+P3k Y2) eikY2 YO e ik D

(234)

Thus, the generaresultl for A2. is

- dY2a(O) dY 1(0)

A2 a (pi()YI(0) d y P2( 0 ) dy Y;(o))

dY;(D) dYb(D

~ I~b(Y) ( 2 Yb(D) dy- P3(D) dy Y;(D))

+dY;(D) dY2b(D)-Y~b(yO) GO2() Y2b(D) dy-P 3(D) dy Y+(D))I/

dY;3(y0 ) dY23(0) dY+,(0) dY23 y0

fp()Y() dy dy d y dy

dY (0) dY;3(y0) dY3 (y0

dy (dy y 2a(0)-Y 3 (0) dy

100

dY;(D) dY*bDX IY~bYO) t 2(D b(D) d y - P3(D) -dy Y;(D)

dY;(D) dY2b(D)

- 2b(YO) (P2(D) Y~b(D) dy - P3(D) dy Y;(D)) 1

- [p0i~0 (~ayodY2,(O) dY2,(o)I a dy dy Y2a(Y)

-P2(0 ) dY1(O) (Y+ (Yo) Y- (0) - Y* (0) Y- (Y.))I

dYib(YO) dY(D) dYb(D

dY~b(YO) dY;(D) dY~b(D)

dy (P2(D) YUbD) P3 dy Y+(D))I (35

For constant speed of sound and ambient density. A 2areduces to

101

A 2 (p . k P2 k y _k ) e D

x[(~k~-~2~ )kyY 0 it~ D ~ik - e+jky, D]/

-ke+lkYY e-DkY2D + (P2 k D+ P3 ) e kY2YO

-j2k Y3D (pk2 P2k ) (P2 k - P3 k2 ) e-it Y2D- 2 Vk2 - 1 Vy k 3

+(p, ky2 + 2 ky1 ) (P2 ky3 + P3 ky2 ) e+jkY2 D . (236)

We must now verify that (236) simplifies to the well-known and well-

documented expression for this unknown constant. Eliminating the common

te -itY Dterm e 3 Dand dividing numerator and denominator by (p ky2 + P2 k )

yields

102

(PI k 2- P2 k Y)

A 2&c k %=3kY -P2 kv Y3e 2Y e-1 2 D

4x k Y2(pk Y2+P 2 k )

+ (2 Y +P3k 2 -k y YOe 1k Y2D

+PI )e Y22 e P22 kJ/

(p, k p2 P2k 1)(2k3-P3k2)e' 2D

+ (2 Y (+p3k -jk )D* 2 27

If we %jefine a reflection coefficient at the boundary between medium

two and medium one as suggested by Ziomek (1991, equation (3.9-54)) as

follows:

(p, k 2- P2 k )

R1- (238)

(p, k Y P2 k )1

103

then, (237) becomes

A 2 %-j kr 2 (P3 k - P2k ) e+IkY2 YO e -I1ky2 D +

4% kY2

(p2 k + p3 k Y) e-1Y2 e 1k 2 D2

[R21 (P2k3 -P 3k2 )e-ky 2 D +(pk +pk )etk Y2 D (239)

Rearranging the denominator and dividing both numerator and

denominator of (239) by (P2 ky3 + P3 ky2 ) reveals

A2& jkr R2 1

4g kY2

(p3 k -p 2 k )ky2 - ky

x I e+lk Y2 y etky -2k D + e-1ky2Y° e 2k 2 D 1

(P2 k3 + P3 k2 )3 2

104

(p3 k - p 2 k )

le+jk y2 R21 e-jk2 . (240)

(P2k3 + p3 k )

If we define a reflection coefficient at the boundary between medium

two and medium three as suggested by Ziomek (199 1, equation (3.9-55)) as

follows:

(P3 k2 - P2 k )2 Y

R23 - (241)

(P3 ky 2 + P2 ky3 )

then (240) becomes

A2ac - j kr R21 IR2 3 e+ lky2YO D -1k y k

4x k2

ie+iky2 D - R21 R23 e ]. (242)

105

Dividing the numerator and denominator of (242) by e +t2 reveals

A2 c . i kr R21 IR23 eIk Y2YO eY 2 Y2 D + e 2 Y2 Y /47g k y2

e-12kyD

I- R2 , R23 e Y2 D. (243)

Factoring e-ky2 YO out of the numerator and rearranging yields

A2c .j kZR21 I + R2 3 e "'2 YO e12kY2 D I ekY2YO /

4x kY2

Ii -R21 R23 e Y2 D (244)

Rearranging (244) provides us with the following desired expression:

106

I I+ R23 e-k Y2(D - Y0)1e-Ik Y2 YO

A~C- R2 U kr. (245)

I I - R2, R23 e -'2 Y2D kY

This is exactly the equation derived by Ziomek (1991, equation (3.9-50)) for

this unknown constant for the classical waveguide case.

For the unknown constant B2., Mathematici provided the following

flumerator:

num3 - G, (a192 a2,5 a3,6 a4 .1 a5,4 - a,,, a2 ,5 a3,6 a4 .2 a3,4- a 12 a2,4 a3,6 a4,1 a5 .5 + a I,1 a2,4 a3,6 a4.2 a5.5- a 1,2 a2,5 a3 ,4 a4 ,1 a5 .6 + a 1.2 a2 .4 a3,5 a4,1 a5,6+ a,., a2.5 a3.4 a4.2 a5.6 - a,., a2 .4 a3.5 a4,2 a5.61 (246)


,generic expresston for the numerator of B2,:

num 3 - G, [(a,,, a2,5 a4,2 - a1.2 a2,5 a4,I) (a3.4 a5,6 - a3,6 a5,4)+ (a,1. 1 a2.4 a4,2 - a 1.2 a2.4 a4, I) (a3.6 a5,5 - a3,5 a3,6)] . (247)

Substituting appropriate expressions into (247) and using (228) yields

the following general expression for B2,:

107

dY;5(O) dY1(0)

dY'2b(D) dY;(D)

x [2b(yo) GO dy Y;(D) - P2(D) Y'bD d

+ ~bYO ~ 2() ~bD)dY;(D) dY~b(D)j0) ~ ~ d -p()Y-bD p3(D) dY Y(D))1

dY;a(Yo) dY~a(0) dY;8(0) dY; 3.(Yo)

dy dy dy dy

dY1(O) (dY!,(Yo) dY2,a(Yo)1

P2y0) dy y 2a(o)-Y2&(0) dy

Iy2b(YO) (P2(D Y~b(D) dy(D - P3(D) dy2bD Y"(D))

dY;(D) dY~b(D)Y2b(YO) tP2(D) Y2b(D) d3p()dyYD)

dY2,(0) YO

-, l(0) Y-(0) (Y;1 (y.) dy - dYa(

108

-P2(0) dY,( (Y.) Y-a(0) - Y" (0) Y- (Y.))1

dYbY) dY(D) dY"bD

dYb(y) dY(D dYb(D)x(P 2(D) Y2*b(D) d-y- - P3(D) dy Y*(D))1(2)dy

For constant speed of sound and ambient density, (248) becomes

k -1k DB2 = (PIk~ +P2 k )e Y"3

X I(P3 k -P 2 k Y ) e4 k 2 YO eilk2 D +(2k3+p3k 2e-1k Y2 YO +k Y2DI

-j2 k Y2e- k3 D (p, k p2P2k l) (P2 k p3P3k 2) e- Y'2D

+ (p, k +P2 k Y )(P2 k +P3 k )2e+ k Y2D )*(249)

109

Using the definitions of R21 and R23 presented earlier, (249) may be

reduced to

R23 e-j2ky2 (D-yo e-1kY2YO

B -- kr. (250)B2ac =4x

1I - R21 R23 e D kY2

Multiplying (250) by RA B2 may be written as

1B2s, - i2' A2a" (251)

This is exactly the equation derived by Ziomek (199 1, equation (3.9-51)) for


For the unknown constant A2b, Mathemat/ca provided the following

numerator:

num4 - G, I- aj, 3 a2,2 a3,6 a4,1 a5,5 + al, 2 a2,3 a3,6 a4,1 a5,5- a1.1 a2.3 a3,6 a4,2 a5,5 + a1,1 a2.2 a3,6 a4.3 a5.5+ al, 3 a2,2 a3,5 a4, 1 a5,6 - al, 2 a2,3 a3,5 a4,1 a5.6

+ a1,. a2,3 a3,5 a4, 2 a5,6 - al,. a2,2 a3,5 a4.3 a5,6 ] . (252)


generic expression for the numerator of A2b:

110

nuM4 - G, [a1,3 a2.2 a4,1 - a1,2 a2,3 a4 .1 + a,., a2,3 a4,2 -

al., a2,2 a4,31 (a3.3 a3.6 - a3,6 a5.3) (253)


* the following generzl expresion for A2b:

[YY) dY 1(0) ()O1()0) dYja(O)A2b - 2 ~2(o P2(0 ) dy Y+ao-ioY() d

dY 1(0) dY~a(O)

-Yayo(2 ()dy Y2a(Op()iO dy

dY~b(D) dY;(D)GO(() d- Y;(D) -P2(D) Y~b(D) d

I pO)YO)(dY+2a(yo) dY2a(O) dY2*a(o) dY~a(yo))

dy dy dy dy

dY 1(0) dY a(yo) dY2a(YO)

dy dy Y21(a(2& dy

dY3(D) dY*2b(D)X LT2b1yO) G~2 0D Y2b(D) dy "(') dy Y3(iJI

dY*(D) dY-2b(D)-Y~b('y 0) (P2(D) Yb(D) d-y- - P3(D) dy Y;-(D)I

dY2a(O) dY2,(O)

- I~(O Y() ~dy -dy

-P()dY,(O) (Y+ (Y0) Y-a() - Y~a(O) '45 (y0 ))

dy p 2()Yb(D) d y- - P3(D) dy Y;D)

dY~b(yO) dY;(D) dY~b(D) t

- y GO (P2w b(D d -P3(D) dy Y*(D)jj (254)


A2bC - kL e'jk 3 D I(pI k Y P2 kY )e~2 Y edk Y2D

112

+ (pk 2- P2 k Y ) e-1 Y2 0Oe *i 2 DI(P2 k Y +P3 k )/

j kY2e-k 3 ((pI k2-P2 ky P2 3- P3 kY ) e jk'2

+ (p, k2 +P2 k ) (P2 k +P3 k2 ) e~j2 D)] . (255)


reduced to

I R21 e-1 2 YO+ e+ IkY2 YO

A~b- (256)

11 - R21 R23 e-j2k Y2 D kc2



113

For the unknown constant B2b, Ma4emaitca provided the following

numerator:

num5 - G, [a1 .3 a2.2 a3,6 a4,1 a5,4 - al, 2 a2,3 a3,6 a4,1 a5,4+ aI,I a2,3 a3,6 a4,2 a5,4 - a1,, a2,2 a3,6 a4,3 a5,4- al, 3 a2,2 a3,4 a4,1 a5,6 + a1 ,2 a2,3 a3,4 a4,1 a5,6- a1,1 a2,3 a3,4 a4,2 a5,6

+ a1,1 a2,2 a3,4 a4,3 a5,6] (257)


generic expression for the numerator of B2b:

num5 - G, [al, 3 a2,2 a4,I - al, 2 a2,3 a4.1 + a,., a2,3 a4 ,2 -aI,1 a2,2 a4,3] (a3,6 a5,4 - a3,4 a5,6) (258)


the following general expression for B2b:

dY 1(0) dYza(0)

B 2n 2 P2(0) dy Y,(O)-P1(0)Y dy )

dY1(O) dY2a(0)

-Y2&(Yo)(p2(O) dy Y21(0)-p 1(0)Y1(0) dy

dY;(D) dY.b(D)

x tP(D) Y2+b(D) dy P3( D) dy Y3 (D)}

114

dY~a(yo) dY~a( 0) dY;,( 0) dY;3(y0 )

Ip()Y0(dy dy dy dy

dY1(O) dY~a(Yo) dY2 (Y.)

P240 - dy Y~a(0)-Y2a() d

dy p() -dy YD

dY(D) dY~b(D)

- 2bYO P2(D Y+bD dy-P 3(D) dy Y(D))

dy - dy 3~(Y)

dYb~O)dY(D) dYb(D)YXbdy (P2D Yb(D) dy-p 3(D) dyY(D))

dY~b(Yy) d() dY ()

dfbyo()~ bD dy p3(D) dyY(D))I 29

x G20 11+


B2bc- k e-1ky 2 D e- iky3 D (p k + P2kY ) e+jky 2 Y

2x y ]2

+ (p ky2 - P2 ky ) ky2 2 k y3

r 2k2 -jk 3D ((, k 2-P 1)(2kY 3kY -1k Y2D[-j2k e-ik3D {(plk a-p 2 kv )(p2 ky -p3 kv2 )e tky2

Y2 Y)(P 2 +p3 ) e+jky2 D (260)+ (pjky +p~ )P2 k +p¥ k ky 2


reduced to

[R2, e- kY2 Y + e'lkY2 YO

B2b kr R23 e , (261)

I -R 2 1 R2 3 e' , 2 ] k /2

116

or, rewriting,

B2bC - A2bC R23 e12 Y'2 D (262)



For the unknown constant B1, Mfath emaica provided the following

numerator:

num1 - G, [a1 ,3 a2,5 a3.6 a4,2 a5,4 - a1,2 a2 .5 a3.6 a4.3 a5,4- at1.3 22A a3,6 a4,2 a5,5 +a1.2 a2,4 a3.6 a4.3 a5.5- al, 3 a2,5 a3.4 a4,2 a5,6 + a1,3 a2.4 a3,5 a4,2 a5.6+ a1,2 a2,5 a3,4 a4,3 a5,6 - a1,2 a2,4 a3,5 a4,3 a5,61 .(263)


generic expression7 f or the numerator of B 1:

num I - G I I(a 1,3 a2.5 a4.2 - a 1.2 a2.5 a4,3) (a3,6 a5,4 - a3.4 a5,6)(a,, 3 a2.4 a4.2 - a1,2 a2.4 a4,3) (a3.5 a-,6 - a3,5 a5,5)] . (264)


the f ollowing general expression7 f or B 1:

jk dY2,(O) dY2a(O)

2n ~P2(0) (aO dy dy Y29()

117

dY;(D) dY2.b(D)x [Yt~(Y) ( 2 Yb(D) dy -P3(D) dy Y+(D)

dY2b(D) dY;(D)

+ Yb(YO 3 dy Y()-P 2(D) Y~b(D) )I/

dY2a(y.) dY2a(O) dY;3(O) dY2,(Y,)

1p()YO(dy dy dy dy

dY1 (0) dY;1(y.) dY2a(Yo)

dy ( dy Y -(0)-Y+11(0) dy

dY -(D) dYb(D)

IY21(Y0 ) Gp2 0D Y2b(D) dy - NO) dy Y+;(D))

- ~bYO ( 2() ~bD)dY;(D) dY~b(D)Y2+b(YO)d -P(D Y~ )P 3(D) dy Y;(D)}I

-p 1p(0) Y-(0) (Y+ayo dY2 () dY2,()I 2ayo)dy dy Y2a(Yo))

-P2(0) dY (0) (Y+ (yO) Y- (0) - Y;1(O) Y-a(yo)) I

118

dY2b(yo) dY3(D) dYb(D)[ dy 0Y(D) dy - p3( dy Y;(D)

dY~b(YO) dY;(D) dY b(D)dy p2(D) Y2b(D) dy - p3(D) dy Y;(D))]. (265)


-" D +jkyy e-1kyD

B I.k(2p 2 k ) y3 D(P3 -P2k )eB = 2x Y2 y2 Y3

(P3 k y2 + P2 ky 3 ) e yYe Y2 )

ik-j D *kk D

[-j2k e- ky 3D (() - P2 k )(P2 k Y 3 k ) e-kY 2Dky2 kY2 Yl Y3 Y

+(pk 2+P 2 k )(P 2 k3 +p 3 k )e+kY 2D )] (266)

Using the definitions of R21 and R23 presented earlier, and defining the

transmission coefficient at the boundary between medium two and medium

one T21 (see Ziomek (199 1, equation (3.9-58))) as follows:

119

2 P2 ky2

T21 - , (267)p! k :p 2 kpI kY2 ! 2 kY 1

(266) may be reduced to

II + R23 e*2 YO e-J2 ky2 D J -Jky 2YO

"c 4 kr T2 (268)BIC= 4x 2

[1 - R2 1 R23 -12k D2

Multiplying (268) by R - allows us to write in the formR21 IC

B1 = Bhe T21 . (269)



Finally, for the unknown constant A3, Mathematics provided the

following numerator:

num 6 - G, [- al, 3 a 2.2 a 3,5 a4 ,1 a5,4 + al 2 a2 ,3 a 3.5 a4,1 a5,4- a1, a2,3 a3,5 a4,2 a5,4 + a1,, a2,2 a3,5 a4,3 a5,4

120

+ a1,3 a2,2 a3,4 a4,1 a5,5 - al, 2 a2.3 a3,4 a4,1 a5,5+ a,1, a2 .3 a3 ,4 a4 .2 a 5 - a,., a2,2 a3,4 a4,3 a5,51. (270)


generic expressio for the numerator of A3:

num 6 - G, [(a,, 3 a2,2 a4,1 - al,2 a2.3 a4,1 + a1, a2,3 a4,2

- aI, ! a2,2 a4,3)1 (a3.4 a5,5 - a3,5 a5,4). (271)


the following general expression for A3:

dY2b(D) dY2b(D)

A3~L 2 D Yb dy dy Y2b(D))

dY1 (0) dY2a(0)2aY;(yo) {P2(o) dy Ya(o) -P (o) Y1(0) dy }

dY1(0) dY2a(0)

-Y a(Yo)(P 2(0) dy Y2.(O)-(o)Y(0 ) y (0)

dY~a(yo) dy23(0) dYja(0) dY2a(yo)

dy dy dy dy

dY,(0) dY¥2(yo) dY 2.(yO)

dy ( dy Y2a()Y a(o) dy

121

IY~(Y) p2 D)Y~D)dY;(D) dfb(D)2b 2b(~d) y- - P3(D) dy Y;(D))

dY3(D) dY2b(D)

-Yb(yO ( 2 bY(D) dy-P 3(D) -dy Y;(D))

-[p,(O) Y_(O) (Y dY; 5(O) dY2&(O)dy -dy

-P2(0 ) dY(O (Y+ (YO) Y-a(0) - Y" (0) Y-a(yo))I

dY2b(YO) dY;(D) dYb(D

dYb(yo) dY;(D) dY2b(D)

dy G2 2b(D d-y - P3(D) 3 +() (272)


A3 -tr(2P2 k Y ) j(pk Y+P 2 k ) e + Y2 0

122

-JkyY+(Pj k Y2 - P2kI )e Y2 /

j2 ky2 3D ky 2 ky 2 k 3 k 2 D

+(pjky2 + p2 ky1)(P2 ky3 +p3 ky2 ) e i2 )] . (273)

Using the definitions of R21 and R23 presented earlier, and defining the

transmission coefficient at the boundary between medium two and medium

three T23 (see Ziomek (199 1, equation (3.9-59))) as follows:

2 P2 ky

T23 - 2 (274)

P3 ky2 + P2 ky3


123

JR21 eiky2 Yo + eIky2 YO e-jky2 D

A _ kL T23 . (275)A3c =4xr

e-J2ky 2 D I -"[1I- R21 R23 e[~'Y Y2 e-ItY3D

(275) may be rewritten as

e-Jky2 D

A 3 M A2bC T23 (276)

e-jkY3 D



To summarize the results for the three media waveguide with plane,

parallel boundaries, generic expressions for the numerators of the unknown

constants are given by (232), (247), (253). (258), (264), and (271). The

generic expression for the common denominator is given by (226). The fact

that these constants have a common denominator may be useful when

constructing a modular program for numerical calculations involving these

constants. For the general case, in which speed of sound and density are

124

arbitrary functions of depth, the unknown constants are given by (235),

(248), (254), (259), (265), and (272). Again, it should be noted that the

denominators in these expressions are exactly the same, offering the same

advantages in modular programming. For constant speed of sound and

density, the unknown constants are given by (245), (251), (256), (262),

(269), and (276).

The fact that Mathematica's output can be shown to be equivalent to

results derived in a completely independent manner gives us some

confidence in the answers arrived at by Mathematica's LinearSolve function

and symbolic processing. We will now program Mathematica to solve the

four media waveguide problem assuming all boundaries are planar and

parallel. By obtaining constant sound speed, constant density expressions for

the resulting eight unknown constants and making judicious selections of

transmission and reflection coefficients, we can mathematically "remove" the

fourth medium and verify that the four media case correctly reduces to the

three media case. This will offer us more confidence in the program's output,

and provide some meaningful results.

For the four media waveguide problem, the vector-matrix quantities of

(198) are defined as follows:

125

A - al.1 a1,2 al,3 0 0 0 0 00 a2.2 a2.3 a2.4 a2.5 0 0 00 0 0 a3,4 a3,5 a3,6 a3.7 00 0 0 0 0 a4,6 a4,7 a4,8a5.I a5.2 a5,3 0 0 0 0 00 0 0 a6.4 a6,5 a6,6 a6.7 00 0 0 0 0 a7,6 a7,7 a.80 as,2 a8,3 a8,4 a8,5 0 0 0

where

a1,1 = - PI(Ys) Y1(Ys) (277)

al.2 - P2(Ys) Y2a(Ys) (278)

a1,3 - P2(YS) Y;a(Ys) (279)

a2 2 - Y2j(Yo) (280)

a2,3 - Y2,(Y0) (281)4

a2.4 - - Y2b(Yo) (282)

a25 - - Y2b(Yo) (283)

a3,4 - P2(YeI) Y2b(YB) (284)

a3,5 - P2(YBI) Y2b(YBe) (285)

a3.6 - P3(YeI) Y;(Ye ) (286)

a3,7 - P3(YeI) Y3(yeI) (287)

a4.6 - P3(YB2) Y3(Y 2) (288)

126

a4.7 -P3(YB 2) Y3(Y 2) (289)

a4 -~~4 2 (290)

dY I(ys)a 5 1 - dy (291)

a 2-dY* (yS)(2)

a 3 -d2ays(23

a5.2 - dby) (294)

dY2 (y)

a 5 - dy (295)

dY;b(YBI)

a6 ,4 - dy (296)

dY 3b(YBI)

a6,5-- d (297)dy'

a6,6 - -dy (298)

d3(yB2)

a7,7 - dy(299)

dyB2

127

dY(YB2)

a7 8 - dy (300)

dY* (Y.)

a8 ,2 dy (301)

dY2a(Y0 )a8,3 - dy (302)

dY2b(Yo)a8 ,4 - dy (303)

dY2b(yo).8,5 = - dy (304)

II1 1 131 A2a B2a A2b B2b A3 B3 A4IT (305)

where the superscript T indicates the transpose matrix operator (indicating

that z Is a column vector),

b - 10 0 0 0 0 0 0 G,IT (306)

where the superscript T indicates that b is a column vector, and

G, - - k r (307)

2x"

Since the matrix A is an eight by eight square matrix for this four

media waveguide problem, the solution to the system can be written as

described in (222). The Mahemabca code required to perform this task is as

follows:

128

a - ((aidl, alc2, alc3, 0, 0, 0, 0, 0),(0, a2c2, a2c3, a2c4, a2c5, 0, 0, 0),(0, 0, 0, a3c4, a3c5, a3c6, a3c7, 0),(0, 0, 0, 0, 0, a4c6, a4c7, a4c8),(a5clI, a5c2, a5c3, 0, 0, 0, 0. 0),

a (0, 0, 0, a6c4, a6c5, a6c6, a6c7, 0),(0, 0, 0, 0, 0, a7c6, a7c7, a7c8),(0, a8c2, a8c3, a8c4, a8c5, 0, 0, 0))

b - (0, 0, 0, 0, 0, 0, 0, GO);

z - (InverselaJ).b

This code resulted in halted execution due to a singularity error in the

evaluation of the matrix inverse. Luckily, the LinearSolve function can be

used to solve (198) directly for this problem because A is an eight by eight

square matrix. The following modified Mathematica code was developed:

a - ((aidl, alc2, alc3, 0, 0, 0, 0, 0),(0, a2c2, a2c3, a2c4, a2c5, 0, 0, 0),(0, 0, 0, a3c4, a3c5, a3c6, a3c7, 0),(0, 0, 0. 0, 0, a4c6, a4c7, a4c8),(a5cl, a5c2, a5c3, 0, 0, 0, 0, 0).(0, 0, 0, a6c4, a6c5, a6c6, a6c7, 0),(0, 0, 0, 0, 0, a7c6, a7c7, a7c8),(0, a8c2, a8c3, a8c4, a8c5, 0, 0, 0))

b-(0, 0, 0, 0, 0, 0, 0, GO);

Line arSolve[a,b I

This revised code ran successfully, yielding results in the same format

as the three media waveguide problem discussed earlier [see (223)1. Again,

when Mathemotics functions such as Factor, Cancel, and Simplify were

129

applied to the output, the same result was returned, indicating that the

program was satisfied that the output was as simple as it could make it. This

fact confirms our suspicions that the symbolic algebra routines contained in

Mathematica lack sufficient sophistication for this application.

We will now present the results of the program for the four media

waveguide problem. The analysis of these results will be conducted in a

manner similar to the analysis conducted for the three media waveguide

problem. That is, the results will be simplified manually in order to generate

generic expressions for the unknown constants in terms of the generic

elements. Then, (277) through (304) and (307) will be substituted into the

generic expressions to reveal general expressions for these unknown

constants. Next, we will assume constant speed of sound and constant

ambient density, and develop a set of expressions for the unknown

constants. Finally, we will make some judicious assumptions regarding the

reflection coefficient (at the boundary between medium three and medium

four), the transmission coefficient (at the boundary between medium three

and medium four), and the location of the boundary YB2 (thereby

mathematically removing the fourth medium) to show that the constant

speed of sound, constant density expressions for the four media waveguide

problem reduce to the results of the three media waveguide problem

already verified. This verification will be conducted in the following order:

A2a. B21, A2b, B2b, A3, B3- BI, and A4. We will demonstrate this entire process

for the unknown constant A25 only, and simply present the results for the

other seven unknown constants.

130

The first output expression to be explored is the common denominator,

denom

denom = - a1,3 a2,5 a3,7 a4,8 a5,1 a6,4 a7,6 a8.2

"1,1 42,5 4 .4,8 a6. 7.6 as.2 + al.3 a2,4 a3.,7 a4,8 a5,1 a6,5 a7,6 a, 2- a,,, a2.4 a3.7 a4,8 a5,3 a6,5 a7,6 a8,2 + a1,3 a2.5 a3,4 a4,8 a5,1 a6.7 a7,6 a8.2- a1.3 a2.4 a3,5 a4.8 a5,1 a6,7 a7.6 a8.2 - a,', a2, 5 a3.4 a4,8 a5,3 a6,7 a7,6 a8.2+ a,., a2.4 a3,5 a4,8 a5.3 a6.7 a7.6 a8,2 + al. 3 a2.5 a3.6 a4,8 a5,1 a6,4 a-7. 7 a8,2- a,., a2.5 a3,6 a4,8 a5.3 a6,4 a7,7 a8.2 - a 1.3 a2,4 a3.6 a4,8 a5,1 a6,5 a7.7 a8.2+ a,', a2,4 a3,6 a4,8 a5.3 a6.5 a7,7 a8.2 - a1.3 a2.5 a3,4 a4,8 a5,1 a6.6 a7.7 a8.2+ a1. 3 a2,4 a3,5 a4,8 a5,1 a6,6 a7.7 a8 .2 + a,., a2.5 a3 4 a4,8 a5.3 a6.6 a7, a8,2- a,., a2 4 a3.5 a4.8 a5,3 a6.6 a7.7 a8.2 + a1,3 a2,5 a3,7 a4,6 a5.1 a6,4 a7,8 a8.2- aj, 3 a2.5 a3,6 a4,7 a51 a6,4 a7.8 a8,2 - a,,, a2,5 a3.7 a4,6 a5,3 a6,4 a7,8 a8.2+ a,., a2,5 a3,6 a4,7 a5.3 a6,4 a7,8 a8,2 - a1,3 a2,4 a3.7 a4,6 a5 .1 a6,5 a7,8 a8,2+ a1.3 a2.4 a3.6 a4,7 a5,1 a6.5 a7,8 a8,2 + a1,. a2,4 a3,7 a4,6 a5,3 a6.5 a7,8 a8,2- a,,, a2.4 a3.6 a4.7 a5.3 a6,5 a7,8 a8.2 + a1.3 a2.5 a3,4 a4,7 .a5,1 a6,6 a7.8 a8.2- a1.3 a2, a3.5 a4,. a5,1 a6.6 a7,8 as,2 - a,., a2.5 a3 4 a4,7 a5.3 a6,6 a7. 8 as.2+ a,., a2,4 a3,5 a4,7 a5,3 a6,6 a7.8 a8,2 - a1,3 a2,5 a3,4 a4.6 a5,1 a6,7 a7 .8 a8.2+ al. 3 a2,4 a3,5 a4,6 a5,1 a6,7 a7,8 a8,2 + a,., a2.5 a3,4 a4. 6 a5.3 a6,7 a7,8 a8.2- a,,, a2,4 a3,5 a4,6 a5,3 a6,7 a7.8 a8,2 + a1.2 a2,5 a3.7 a4,8 a5.1 a6,4 a7,6 a8.3- a,,, a2,5 a3,7 a4,8 a5,2 a6.4 a7,6 a8,3 - a1,2 a2,4 a3.7 a4,8 a5,1 a6,5 a7,6 a8,3+ a,,, a2.4 a37 a4.8 a5.2 a6,5 a7,6 a8,3 - a1.2 a2,5 a3,4 a4.8 a5 .1 a6,7 a7.6 a8.3+ a1,2 a2,4 a3,5 a4,8 a5.1 a6,7 a7,6 a8.3 + a,,, a2.5 a3,4 a4,8 a5,2 a6,7 a7,6 a8,3- a,,, a2 4 a3.5 a4. 8 a5,2 a6,7 a7,6 a8,3 - ?1. 2 a2.5 a3,6 a4,8 a5,1 a6,4 a7.7 a8.3+ a,., a2,5 a3,6 a4,8 a5,2 a6,4 a7,7 a8.3 + al. 2 a2,4 a3.6 a4,8 a5.1 a6,5 a7,7 a8,3- a,,1 a2.4 a3,6 a4,8 a5,2 a6.5 a7.7 a8,3 + a1.2 a2.5 a3. 4 a4.8 a5,1 a6,6 a7,7 a8,3- a1,2 a2,4 a3,5 a4,8 a5.1 a6,6 a7,7 a8.3 - a,,, a2,5 a3.4 a4.8 a5.2 a6,6 a7,7 a8.3+ a,,, a2,4 a3,5 a4,8 a5,2 a6,6 a7,7 a8,3 - a1.2 a2,5 a3,7 a4.6 a5 .1 a6,4 a7,8 a8.3+ a1.2 a2.5 a3,6 a4.7 a5,1 a6,4 a7.8 a8,3 t a1,1 a2.5 a3,7 a4.6 a5.2 a6.4 a7,8 a8.3- a,,, a2,5 a3,6 a4,7 a5.2 a6,4 a7,8 a8,3 + al. 2 a2,4 a3.7 a4,6 a5,I a6,5 a7,8 a8,3- a1.2 a2,4 a3.6 a4,7 a5.1 a6,5 a7,8 a8,3 - a,,, a2,4 a3,7 a4,6 a5,2 a6,5 a7,8 a8.3+ a,1, a2.4 a3,6 a4.7 a5,2 a6.5 a7.8 a8,3 - a1.2 a2.5 a3.4 a4.7 a5.1 a6.6 a7.8 a8.3+ a1,2 a2, 4 a3,5 a4,7 a5,1 a6,6 a7,8 a8,3 + a1,. a2,5 a3,4 a4,7 a5,2 a6,6 a7.8 a8.3- a1,, a2,4 a3,5 a4,7 a5,2 a6,6 a7,8 a8,3 + al,2 a2,5 a3.4 a4,6 a5,1 a6.7 a7,8 a8 ,3- a1,2 a2,4 a3,5 a4,6 a5,1 a6,7 a7,8 a8,3 - a1,1 a2,5 a3,4 a4,6 a5,2 a6,7 a7,8 a8,3+ a, a2.4 a3.5 a4,6 a5,2 a6,7 a7,8 as, 3 - al, 3 a2,2 a3,7 a4,8 a5,1 a6,5 a7,6 a8,4+ a1,2 a2,3 a3,7 a4,8 a5,1 a6.5 a7.6 a8.4 - a,., a2.3 a3 7 a4.8 a5.2 a6.5 a7,6 a8

131

+ a,., a222 a3,7 a4,8 a5 .3 a6,5 a7 .6 a8,4 + a1,3 a2,2 a3,5 aO, a5 ,1 a6 .7 a7,6 Pa3,4- a1,2 a2,3 a33 a4,8 a5.1 a6 .7 a7,6 a8 .4 + a,., a2.3 a3,5 a4 .8 a5,2 a6,7 a7,6 a8,4- a,., a2,2 a3,5 a4,8 a5,3 a6.7 a7,6 a8,4 + a 1 .3 a2 .2 a3,6 a4 .8 a5 .1 a6,5 a7.7 a8 .4- a1.2 2.3 a3 .6 a4,8 a5 a6 .5 a7,7 a&.4 +' a,. a2 .3 a3,6 a4,8 a5,2 a6,5 a1,7 a8,4- a,,, a2,2 a3.6 a4,8 a5 .3 a6,5 a7 .7 a&.4 - a1,3 a2 .2 a3,5 a4 ,8 a5,1 a6,6 a7,7 a8,4" a1,2 a2,3 a3,5 a4,8 a5.1 a6.6 a7 .7 a8 .4 -a,., a2.3 a3.5 a4,8 a5,2 a6 .6 a7 .7 a8 .4" a,', a2,2 a3 ,5 24, a5,3 a6,6 a7,7 28,4 + a 1,3 a2 .2 a3,7 a4.6 a5.1 a6.5 a7 ,8 28,4- a 1,2 a223 a3,7 a4,6 25.1 a6.5 27,8 28,4 - 21,3 2.2 a3 ,6 a4,7 a5, 1 a6.5 a7 .8 a8,4+ 21,2 a2 .3 a3,6 a4,7 a5.1 a6.5 a7 .8 a8,4 + a,., a2,3 a3 .7 a4 ,6 a5,2 a6.5 27.8 28,4- a,., 22,3 23,6 24,7 25,2 26,5 a7,8 a8.4 - a,., 2,2 23,7 a4 ,6 a5 .3 26,5 27,8 28,4+ a,., a2 .2 23.6 24,7 25,3 a6.5 a7 ,8 28,4 + 21.3 2,2 a3,5 a4,7 a5.1 26,6 27,8 a8.4- 21,2 23 a3,5 24,7 a5,1 a6,6 a7,8 28,4 + a,., 22,3 a3,5 24,7 a5,2 26,6 27,8 28.4- a2.1 a2,2 a3,5 24,7 25,3 26,6 27,8 28,4 - a1.3 22 23,5 a4 ,6 a5,1 26,7 27,8 a8,4+ a21.2 a2,3 23,5 24,6 a, 2 6,7 a7,8 28,4 - a,., a2,3 a3 ,5 24,6 25,2 a6,7 27,8 28.4+ a,., 2,2 a3,5 24,6 25,3 26,7 a7,8 28.4 + a 1,3 2,2 a3 .7 24,8 a5,1 a6,4 a7,6 28,5- a1.2 2,3 23,7 24,8 a5,1 a6., 27,6 28,5% + a,., 22,3 23,7 a4,8 a5,2 26,4 27,6 28,5- a,,, 2,2 23,7 24, 25,3 26,4 27,6 28,5 - 2,3 a2,2 23,4 24,8 a5.1 26,7 27,6 28,5" 21,2 a2,3 23,4 24,8 a5.1 a6,7 27,6 28,5 - 21,1 22,3 23,4 24,8 a5,2 26,7 a7,6 28,5" a,,, 2,2 23,4 a4,8 25,3 26,7 a7,6 28,5 - aj,3 2,2 23,6 24, 25,1 26,4 27.7 28,5" 21.2 2,3 23,6 a4,8 a5.1 26,4 27,7 28,5 - a,,, 2,3 23,6 24,8 25,2 26,4 a7,7 28,5" a,,, 22,2 23,6 24,8 25,3 26,4 27,7 28,5 + a213 22,2 23,4 a4 ,8 25,1 26,6 27,7 28.5- a21,2 22,3 23,4 24,8 a5,1 26,6 27,7 28,5 +a,., 2,3 23,4 24,8 25,2 a6,6 27,7 28,5- 2,., a2,2 23,4 24,8 25,3 26,6 47.7 28,5 - a,,3 2,2 23,7 a4,6 a5,1 26,4 27,8 28,5

+ 21,2 a2,3 23,7 24,6 1ij 26,4 27,8 28,5 + j,3 2,2 23,6 24,7 a5,1 26,4 27,8 28,5

- 2 1,2 a2,3 a3,6 24,7 a5.1 26,4 27,8 28,5 - a,,, 2,3 23,7 a4,6 a5,2 26,4 a7,8 28,5+ a,., 2,3 23,6 24,7 25,2 26,4 27,8 a8,5 + 2,., 2,2 23,7 24,6 25,3 26,4 27,8 28,5- a,., a2,2 23,6 24,7 25,3 26,4 2*7,8 28,5 - 21,3 a2,2 23,4 24,*7 a5.1 a6,6 a7,8 28,5" 21,2 22,3 a3,4 24,7 a5,1 26.6 27,8 a8,5 - a 1 2,3 23,4 a4,7 a5,2 a6,6 a7,8 28.5" a,., 22,2 a3,4 24,7 25,3 a6,6 27,8 28,5 + a,3 2,2 23,4 24,6 a5,1 26,7 27,8 28,5- a21,2 2,3 23,4 24,6 a5,1 26,7 27,8 a8,5 + 2,., 22,3 23,4 a4,6 25,2 26,7 27,8 28,5- a,,, 2,2 a3,4 24,6 25,3 26,.7 27,8 28,5 . (308)

It is readily apparent that (308) is a rather complicated expression

involving 128 terms. In order to simplify the algebra somewhat, we have

divided the expression into four distinct subexpressions, such that

132

denom - denomA + denomB +denomc + denomD , (309)

where denomA is the sum of terms involving the element a8,2, denomB is the

sum of terms involving the element a8,3, denomc is the sum of terms

involving the element a8,4 , and denoMD is the sum of terms involving the

element a8,5. This simplification allows us to work with 32 terms at a time,

and yields the following generic expressions for the various parts:

denoMA - 8,2 (al,3 a5,1 - a,., a5,3) [a2,5 j(a4,8 a7,6 - a4,6 a7,8)x ( a3,4 a6,7 - a3,7 a6,4) + (a4.7 a7 .8 -a 4,8 a7,7) ( a3,4 a6,6 - a3,6 a6.4) I

- 2,4 [(a4,8 a7,6 - a4,6 a7,8) (a3,5 a6,7 - a3.7 a6,.5)+ (a4.7 a7.8 - a4,8 a7,7 ) (a3 ,5 a6.6 - a3,6 a6.5) 11 , (310)

denomB - a8.3 (a,,, a5.2 - al,2 as,,) [a2,5 t(a4, a7.6 - a4.6 a7,8)x (a3,4 a6,7 - a3.7 a6.4) + (a4,7 a7,8 - 248 a7,7 ) (a3,4 a6,6 - a3,6 a6,4)]

-a2.4 1(a4.8 a7,6 - a4,6 a7,8) (a3,5 a6,7 - 23.7 a6,,)+ (a4,7 a7,8 - a4.8 a7,7) ( a3.5 a6.6 - a3,6 a6.5)1 (311)

denomc - a8,4 [ 6a4,7,6 - a4.6 a7,8) (2a3,5 a6,7 - a3 7 a6,5)+ (a4,7 a7,8 - a4,8 a7,7) (a3,, a6,6 - a3,6 a6.5)]

x(a2,2 (a,, 3 a,1 - a,., a5,3) + a2.3(a,., a352 - a1,2 a5.1)) , (312)

and

denomD - a8.5 1(a4,5 a7,6 - a4,6 a7,8) (a3,4 a6,7 - a3,7 a6,4)+ (a4,7 a7,8 - a4,& a7,7) (2a3.4 a6,6 - a3,6 a6 4jJ

x(a2,2 (a,,, a5 .3 - aj,3 a5.1) + a2.3 (21,2 a5. - a,., a52)) (313)

133

Substituting (310) through (313) into (309) and combining like terms

yields the following generic expression for the common denominator:

denom - UO(a, 27.6 - a4,6 a17,-8 ) (a3.4 a6,7 - a3,7 a6.4)

(a4.7 a7.8 - a4,8 27,7) (a3.4 a6.6 - a3.6 a6 j4)x [(a2,5 a8 .2 - 2,2 a8,5) (al,3 a5,1 - a,', a5.3)

+ (2,3 a8.5 - 2.,5 a8,3) (a1,2 a5.1 - 21,1 a5.2)11

- [I( a4,8 a7,6 - a4,'6 a7,3 ) (a3,5 a6,7 - a3,7 a6 5)

+ (a4.7 a7.8 - a4.8 a7.7) ( a3.5 a6.6 - a. 65

x1(a2,4 28,2 - a2,2 a8,4) (21,3 a5,1 - a,,, 25,3)

+ (a2,3 a8,4 - 2,4 28.3) (21,2 a5,1 - a,., a5.2)11 (314)

Substituting (277) through (304) into (314), using the facts that Ys - 0,

Y - DI, and B2- D2, and performing some algebraic manipulations results

in the following generpI expression for the common denominator:

rdY+(D 2) dY'(D2)

denom - I (p3 D2 ) Y(D2 ) dy - P4(D) dyY(D2)

x~3DI dy Y;(DO)- P2(DI)Y20(D) dy

dY3(D2) dY4(D2)(P4(D2 dy Y+(D 2) - p3(D2) Y3(D2) dy

134

I, d2b(DI) dY;(D)JX tp3(DI) dy Y;(DI) - P2(Dj) Y~b(D,) d

~ I (;~Y 0 dY b(YO) dY*a(yo)

21 dyo - dy Y2b(YO)

x~p(OiY~O)dy 2 0) dyYI(O))

+ (p 1 O) b ) - P2 (Y) dTY 2 (o)

dY~a( (O) dY(yo)x~p(Oy( YbYO -2Y) dy Yao)

dYa(D) dY()- [(p 1 0)Y-1D dy) -P2(0) dy Y(D))

x ~dy Y(j-p(d,) Y 2b(,

dY*(D2) dY(D 2)

4 33

dY2b(DI) dY;(D,)X dY(D) Y;(D1) - P2(D1) Y2b(D)dy ]xy dy(l) d

dY2b(yo) dY2,(Yo)

X[(,;* +°2a dy -dy Y2b(YO))

dY2a(O) dY1(O)x~p1 (O)YP(O) dy -P2( 0 ) dy Y2(O))

dY2 b(YO) dYb(yo)( dy Y2b(Y°)-Y2aJY°) -d-y

d OIpi(O)Y O) y2 Yj+(O))]. (315)p 1(O ) Y -P2( 0 )dy dy

If we now assume constant speed of sound and constant density in a

specific medium, the depth-dependent functions in (315) become complex

exponentials and the denominator becomes (using the subscript c to indicate

the constant speed of sound assumptoan)

136

denomc -2k eky4 D2 [(pk + P 2 ) (p 3 k -P2 k )

x (p,4 ky 3 - P3 ky4 ) e+iky 2 DI e-jky 3 D2 e+{ky3DI

+ (pik +P2k (3 2 P2 k3 p )

(P4 ky3 + P3 ky4 ) e' jky2DI e+ kY3 e-k Y3

-(p, ky2 -P2 ky, ) G3 ky2+ P2 k )

X (p4 k - p 3 k ) e-k' 2D e )' Y3 D2 e +kY3 DI

- (pky2 -P 2 ky I ) 3 ky2 -P2 ky3 )

X (P4 k + P3 k 4 ) iky2D I kY3 D2 eiky3 DI (316)

137

Now that the denominator has been simplified, we'll concentrate on

obtaining expressions for each of the unknown constants in the order stated

above. The first constant is

A2a -' um 2 (317)A =dehn '

where

num 2 - G, 1- a1,3 a2,5 a3,7 a4,8 a3,1 a 6.4 a7 6

+ al.1 a2,5 a3 .7 a4,.8 a5 .3 a6 ,4 a7 .6 + a1,3 a2 ,4 a3,7 a4,8 a5 .I a6 ,5 a7,6

- a,., a2 .4 a3.7 a4,.8 a5. 3 a6.5 a7.6 + al, 3 a2 5 a3.4 a4,8 a5., a 6.7 a7 .6

- at, 3 a2.4 a3.5 a4,8 a,. 1 a6,7 a7.6 - a,., a2,5 a3.4 a4,8 a5,3 a6,7 a7,6+ a,., a2,4 a3,5 a4,8 a5.3 a6,7 a7.6 + al,3 a2.5 a3,6 a4.8 a5,1 a6,4 a7.7- al. 1 a2,5 a3,6 a4 .8 a5,3 a6.4 a7.7 - a1.3 a2,4 a3 .6 a4,8 a5,1 a 6,5 a7 .7* a1 .1 a 2.4 a3.6 a4.8 a5.3 a6 .5 a7 .7 - al. 3 a2 ,5 a 3.4 a4 .8 a5 .1 a6.6 a7.7

* a1, 3 a2 .4 a3.5 a4 ,8 a5. 1 a6,6 a7 .7 + a, 1, a2 .5 a 3.4 a4 .8 a5 .3 a6,6 a7.7

- a,., a2.4 a3,5 a4,8 a5,3 a6 .6 a7 .7 + al. 3 a2.5 a3.7 a4.6 a,,I a6,4 a7 .8- a1,3 a2 ,5 a3 .6 a4,7 a5,1 a6,4 a7 .8 - a,., a2 .5 a3 .7 a4 .6 a5. 3 a6,4 a7 .8

+ a,., a2 .5 a 3,6 a4 .7 a5,3 a6,4 a7,8 - al, 3 a2 .4 a3,7 a4, 6 a5,1 a 6,5 a7,8+ a 1.3 a 2,4 a 3,6 a4 .7 a5 .1 a6,5 a7 .8 + a,, a 2.4 a 3,7 a4 ,6 a5. 3 a6.5 a7 .8

- a,, a 2,4 a 3,6 a4,7 a5,3 a6,5 a7 .8 + al. 3 a 2,5 a 3,4 a4,7 as,, a6,6 a7,8- a1.3 a2,4 a3.5 a4 .7 a5,1 a6 .6 a7 .8 - a,., a2,5 a3.4 a4.7 a5,3 a6,6 a7 .8+ a,', a2,4 a3,5 a4 .7 a5,3 a6,6 a7,8 - a1 , 3 a2,5 a3.4 a4. 6 a5,1 a6.7 a7,8

" a1.3 a2.4 a3 ,5 a4,6 a5,1 a6,7 a7 .8 + a,., a2.5 a3.4 a4 .6 a5,3 a 6,7 a7 .8

- a,,, a2 .4 a 3.5 a4 .6 a5.3 a6,7 a7,8 1 - (318)


generic expressioan for the numerator of A 2s:

num 2 - G, (al. 3 a5 ,1 - al.1 a5 .3 ) [a 2 ,5 (a 4,.8 a7. 6 - a4,.6 a7.8 )

x(a3,4 a6 .7 - a3.,7 a6,4) + (a4,7 a7.8 - a4,8 a7,7) (a3,4 a6.6 - a3,6 a16.4)I

138

21 a2 4 1 (a4 '8 a7.6 - a4,6 a7 8) (a3,5 a6.7 - a3.7 a6 5+ (a4,7 a7.8 - a4.8 a7,7) a33 a6.6 - a3.6 a6.5) (319)


the following generalexpression for the numerator of A2a:

dY~a(0) dY,(O)nu M2 1(p,(o)0y() P2( 0 ) y Y(o))

2x dy 2ay)

dY'(D2) dY;(D 2)X L-Y2,(Yo) I(P3(D2 YD 2 ) dy -P4 (D2) dy Y(D2))

dY*2b(DI) dY3(D1 )(p3(D.) dy Y3(D) - P2(D D) dy

dY3(D2) dY4(D 2)+ dy Y+(D 2) -p(D 2) Y3(D2) dy

dY2b(D1) dY'(D1 )xp3 DI) dy Y;(D) - p2(DI) Yb(D3) dy ]

dY+(D 2) dY;(D 2)

+Y2b(Yo) I(P3(D2) Y+(D2) d -p4(D2) dy Y (D2 ))

139

dY2b(DI) dY3(D0)xI{p 3(D ) dy Y3(DI)- p2(D )Yb(DI) dy

dY3(D2) dY*(D 2)+ (P4(D2) dy Y4(D 2) - P3(D2) Y(D 2) dy

dY b(DI) dY3(DI)

GOp(D) dy Y(DI) -PNMI) YbD1) dy (320)

Once again, making the constant speed of sound and constant density

assumptions and substituting the appropriate depth-dependent expressions

allows us to write the numerator of A2. as (using the subscript c to indicate

the constant speed af sound assumption)

-Jkr (I -P 'i y t D

num2¢ - 2 P ky2 2 kyI e

2x 2)

X 3 ky2 + P2 ky 3 ) G3 ky4 - P4 ky 3 )eY2 D e +1ky2YO e~ y3 D2e+ y k DI

-(G3 ky2- P2 ky3) (p3 ky4 + P4 k yD) e- Y2DI e#ItY2 e +ItY3D2 e-I y3DI

140

+ (pk 2- P2 k Y ) (P3 - p4 k Y3) e+ikY2 D I e-ikY2 YO eIk Y3 D 2 e+jk Y3 D I

-(p3 k2 +P k Y3 ) ( 3 k 4 +p kp Y ) 2 DI -k2O 0 4kY D 2 -k Y 3D I

(321)

Thus, the generalresult for A2a is formed by dividing (320) by (315)

and is given by

dY2((0) dY,(0)A 2 .-k(pt(°)Y (0) dy -2(°)-dy 2a(0))

dY+(D 2) dY+(D 2 )x L- 2bY.o) ((D) Y;(D2 ) dy -P (D2 ) dy Y;(D2 ))

dY2b(D-) dY3(DI)(P3(DI) dy Y3(DI)- P2(DI)Yb(DI) dy )

dY3(D2) dY4(D 2)(P(D) dy Y+(D 2 ) - p(D)Y(D) y )

141

dY* (DI) dY*(D,)

x(P3 O dy Y;(DO)- P2(DI) Y2b(D,) dy

dY*(D 2) dY;(D 2)

+ 0(Y) I(p 3(D2 ) Y;(* -p4( 2 dy Y+(D 2

dY21b(DI) dY3(D,)

X (PIp() dy Y3(D) - P2(Dj) Y2b(DI) dy

dY3(D2) dY4(D2)

(P4(D2) dy Y+(D 2) - P3(D2) Y ( 2)-a;)

dY2b(D,) dY;(D,)

xfp(DOI)Y()- D)Yb) dy

dY(D 2) dY;(D 2)

(P(D) 3 ) dy -P4(D) dy Y 2 ))

dYb(* dY3(D,)

x (p 3(Dj) dy Y3(D)-P 2(DI) Yb(DI) dy)

dY3(D2) 4Y(D2)+ (P(2 dy Y;(D2) - p3(D2) Y3(D2) dy)

142

dY2b(D,) dY+(D1 )X (PO(I) dy Y;(D1)P(D bD) d3

x~~ ~ (yP2diby)d~() Y2b(I))dy - d

dY2a(O") dYy,()

d2a(dyo dY 2b(yo)

x ( o)Y dy Y2 (YO) - Y j dy0 )

( a~) P2 1(0 ) yx~pjO)Y() dydy Y2a(O))I

Y(D 2 dY( 2 )

- Rp3(d2 Y 2 dyo - p(D2) dy YD)

dY() dY3 D(0)

xf~3(,) dy Y3 D)P(IbDdy

dY3 (D2) dY;(D2)(P(2 dy) dy(D)P4( ( D2)D 2 dY+ (2)

1434

dY2b(DI) dY;(D1)Xilp(D 1) -d IId-P2D) 21

dY+bYO dta(yo)

x d~~2by,-d) ~ bY)

(Y+ (() d YO2ao~jO dy P()dy Ya()

dY~a(Yo) dY2"b(yO)+ (dy Y2*b(YO) - y2a(Yo,) dy )

dY* (0) dY1l(0)(ip (O) Y-10 aY(O)A dy322)jj

dy P2( 0) 2a (322)

For constant speed of sound and ambient density, A 2areduces to

2ac 2 Y22a

x [(p3 k Y2+ P2k Y3) (P3 Y - P4 k )3 eik Y2 DI e+ik Y2YO e-Ik 3D 2 e+k Y D I

G(3 k2 P2 k3 ) (G3 k4 + p4 k 3)eikY2 DI e +kY 2 Yo e+iky 3 D2 e-"k Y3 D

+(G3 k 2- P2 k Y3) G3 k Y4-P4 k Y3) eokY e%°'k 2Y e;' k Y3 D e+ °

+, ( ID, jYo D2 jkD

G3 k Y2+ P2 kY3 ) (p3 ky4 + p4 ky3 ) e+lky2 D e -ikY2YO e+lky3 D2 e-ikY3 D /

[2 e-ikY4D[(plk +p2 k )(pk -P 2 k)

y2 ky 2 Yl kP2 Y3

/ ~*i D1 -"k .2 *k D1

x (p4 ky 3 - P3 ky4 ) e+iky2 e ky3 D2 e+ Iky3

+ (p, k p2 k ) (pG3 k Y 2 k 3)

(P4 ky 3 + P3 k y4 ) e +ky2DI e+kY3 e-'ky3 D1

145

(p k ky2 - P k Y1G k Y2 + P2ky3

2 e- k3 e+ k3X(P4 k Y3 - p3 k ,4 ) e-Jk Y2 DI e-kY I 3D

- (,k y2 - P2kY 3k Y2 -p2ky3 )

-i D 2(P4 k Y3+ 3 k Y4) e'1kY2 D I e+ ikY3 e-1k Y3 D 1 (323)

Eliminating the common term e 4 dividing numerator and

denominator by GI k 2 + P2 kYI ), and using the definition of R21 given by

(238) yields

Ah - - R2 1

4xkY2

x [(p3 k y2+ P2 ky3 ) (G3 k y - P4 ky3 ) e-iky 2 DI e+iky 2 YO e-Iky 3 D2 e+ ky3 D

146

(G3 k y2 - P2 k y3 ) (G3 k y4 + p4 k y3 ) e'k Y2 D i e +IkY2 YOe + I y3 D e- 'k y 3D

+ (p3 k Y -p 2 k Y )(G 3 k p4 P4k 3) e l2 DIe-' 2 YOe1k 3D 2 e-Ik Y DI

- jk D2 +k D

+(P3 k - 2 k Y (P k Y-P k 4) e+ I2D I e-' 3 e 3 - + k(2 + P2k 3) (4 k + p k 4) e+ItD, e-y2 Yo e I 3D2 kJ 1, , D

[(P3J -o2k )(Io, -pk, );* ; e" e 3

+( i 2ky ) (py k,, 3 k ky 2 D +k 3 2eIyD

-R 2, GP3 kY2 + P2 k y3) (4 ky3 -P3 k y4) e-Ik '2 D e- kY3 D2 e +kY3 D)

R21 (p3 ky2 -P2 ky3) G4 ky3 +P3 k,4 )e Ik'Y2 DIt e-ityD|

(324)

147

Dividing numerator and denominator of (324) by (p3 k + P2 k ), andky2 Y

using the definition of R23 given by (241 ) yields

-- jkr i(P3 k 4 -p 4 k 3) e Y2D I e+jky2 YO e-JkY3 D 2 e +kY34Y 3

4nkY2

- R23 GP3 k y4 + P4 k y3 ) e - ik y 2 D I e +itY2 YO e+*Jky3 D2 e - jky 3 D I

-k e2 -ik D.

+ R2 3 P3 ky4 - P4 ky3 ) e+it y2DI e2jky2YO e-it Y3 e Y3 D

- (G3 ky 4 4 P4 k3) e+iky 2 D I ° iY 2 YOe+ikY3 D2 e-jkY3 D

[R 23 (G4 k 3 P3 k , ) e+it y2DI e-iky3 D2 e+iky3DI

+ (P4 ky3 + P 3 ky4 ) e +Ity2 DI eY3 e'Y 3

148

- R21 (p4 ky3 - P3 ky4 ) e -Jky2DI e '1ky3D2 e 3'YDI

~ -jk1 .jD 2 -i~D 11

R21 R23 (p4 k , + P3 k ) e-jk2 D Ie e 3 e -j 3 j] (325)"k 3 Y4

Dividing numerator and denominator of (325) by (P4 ky3 + P3 ky4 ) .

defining a reflection coefficient at the boundary between medium three and

medium four, R34, as

(4 k 3 - P3 k ,)

R34 , (326)(G4 k + P3 k y)

and multiplying through by the -1 appearing in the numerator yields the

following desired expression:

- +jkr R21 [R 34 e 'jky2 (D1 Yo ) e-ly3 (D2 - D1 )

4xkY2

149

+R23 e-i (D- YO ) e-ik Y3 (D 2- D )

+ k (DC YO) e-jk 3(D2 - D1) + +jky2(D C YO ) .k(D2- DI+R2 3 R3 4 e e+k2 e - e j2y3 /

R2 3 R34 e-ky3 (D2" DI ) e+ ky2 Di + ekY3 (D2" DI ) e+jky 2 DI

- R2 1 R34 e Y3 2 I e-ky2D 1 (D2 - DI) -j (327)R21 R3e 2 R2 1 R23 e ek3 C'k2 I] 37

Now that we have derived the equivalent classical expression for the

four media waveguide problem for the unknown constant Aa&, we must

verify that it reduces to the proper expression if the fourth medium is

removed mathematically. In order to conduct this evaluation, we must

assume the following:

D2 -D, - D (that is,y 82 = Y = YBy)' (328)

and

R34 - 0 . (329)

Substituting conditions (328) and (329) into (327) reveals

150

4Tnky2

e j 2D _ R21 R23 e -i 2 D (330)

Equation (330) is equal to equation (242). Since (242) has already been

verified, we can conclude that our derivation of the solutions for A 2a' and

hence, A2&c for the four media waveguide problem is correct.

Continuing with the analysis of the four media waveguide problem code

results, for the unknown constant B2., Mfatheouatica provided the following

numerator:

num3 - G, 1aj, 2 a2,3 a3,7 a4,8 a5,1 a6 .4 a7,6-a1,1 a2,5 a3.7 a4,8 a5,2 a6,4 a7,6 - al, 2 a2,4 a3.7 a4,8 a5,1 a6,5 a7,6

" al,1 a2,4 a3,7 a4,8 a5.2 a6,5 a7,6 - a 1.2 a2,5 a3,4 a4, a5,1 a6 .7 a7 .6" a 1,2 a2,4 a3,5 a4,8 a5 .1 a6,7 a7 .6 + a,,1 a2,5 a3,4 a4, a5,2 a6,7 a7.6-- a,,, a2,4 a3,5 a4,8 a5 .2 a6,7 a7,6 - a j.2 a2,5 a3,6 a4, a5,1 a6,4 a7,7+ a,1, a2,5 a3,6 a4,& a5,2 a6,4 a7,7 + a1,2 a2,4 a3,6 a4, a5,1 a6,5 a7.7- a,,, a2,4 a3.6 a4,8 a5,2 a6,5 a7,7 + a 1,2 a2,5 a3,4 a4,8 a5,1 a6,6 a7,7- a1,2 a2,4 a3,5 a4, a5,1 a6.6 a7,7 - a,,, a2,5 a3,4 a4,8 a5,2 a6,6 a7 .7" a,,, a2,4 a3,5 a4, a5.2 a6,6 a7,7 - a 1.2 a2,5 a3.7 a4.6 a5,1 a6,4 a7 .8" a 1.2 a2,5 a3,6 a4,7 a5,1 a64 a7.8 + a,1, a2,5 a3,7 a4,6 a5.2 a6 .4 a7,8- a,,, a2.5 a3.6 a4,7 a5,2 a6,4 a7.8 + a 1,2 a2.4 a3.7 a4,6 a5,1 a6,5 a7 .8- a 1.2 a24 a3.6 a4,7 a3,1 a6 .5 a7 .8 - a,,1 a2.4 a3.7 a4,6 a5,2 a6,5 a7.8+ a,., a2,4 a3.6 a4,7 a5,2 a6,5 a7,8 - a1,2 a2.5 a3,4 a4,7 a5 .1 a6 .6 a7,8+ a 1,2 a2.4 a3,5 a4,7 a5,1 a6.6 a7,8 +a,,1 a2,5 a3.4 a4,7 a5.2 a6.6 a7,8

- aI,1 a2 ,4 a 3,5 a4,7 a5,2 a6,6 a7,8 + a1 ,2 a 2,5 a 3 ,4 a4 ,6 a5, 1 a6 ,7 a7 ,8- a ,2 a2.4 a3,5 a4,6 a5,t a6,7 a7.8 - aI., a2,5 a3.4 a4,6 a5.2 a6.7 a7 .8+ a, 1 a2,4 a3,5 a4.6 a5,2 a6,7 a7 ,81 . (331)


generic expression for the numerator of B2a:

num 3 "G, (a,, a5,2 - al, 2 a5,,) a2.5 (a4.s a7,6 - a4.6 a7,8)

X (a3.4 a6.7 - a3.,7 a6,4) (aU4,7 a7.& - a4.8 a7,'7) (a3.4 a6,6 - a3,6 a6.4) I-a2.4 ((a4,& a7,6 - a4,6 a7,S) (a3,5 a6,7 - a3,7 a6 5)+ (a4,7 a7,8 - a4.8 a7,7 ) (a3.5 a6,6 - a3.6 a6,5)Il. (332)


the following general expression for B2a:

dY(O0) dY;,(O)B 2x dy )

dY(D 2) dY;(D 2)x I,-Y(Y) I(P3 (D2 ) Yy(D2) -dy D2) (D2))

dYbX(( 1) dY3(DI){p3(D3) dy Y2(DI)-P2 (D,)Y (D,) -dy

dY3(D2 ) dY;(D2 )+ (P4(D2) dy Y+(D 2) - p3(D2) Y(D2) d )

152

dY* (D,) dY(D,)

4 ~ dy 3;D1 P 2()b(DI) dy

dY*(D 2) dY; (D2)Yb(yo) I (P3 D2Y;D3 dy -p( 2) dy Y4( 2))

dY2b(D,) dY3(D1)x(OI) ) )Y

dy Y3(D,) -P2(DI)Y2b(D,) dy

dY 3(D2) dY4(D 2)(p(2 dy Y+(D 2) - p( 2) Y3(D2) d

4Y P(DD dy()

X (POI)dYb() Y(DI) -P2(D) Yb)- dy 111/

dY*(D 2) dY;(0 2)II(P3(D2)Y.w2) dy P4(D2) dy Y;(D2))

dY.2b(DI) dY3(D1 )X (POW) -)Ydy Y3(D)-P2(D1)Yb(DI) d

dY (D2) dY(D 2)

+ P4 dy Y4(D2) -P3(D2) YO(2)d

153

dY2'b(DI) dY3(D1 )x (P3(D) dy Y;(D,) - p2 D)Yb(DI) d

dY~b(YO) dY.a(y.)2a dy -oi Y2b(YO)

dY2a(O) dY1l(0)x (p 1(0) y 1() dy P2(0) dyY~a(O))

dY2a(ylo) -dY 2b(yo)

(dy - 2b(yo)2o) - dy

dY;,(O) dY1l(0)xl~p (01() dy P2( 0) d Y+)

dY(D 2) dY;(D2)

3IP3 D)Y( 2) y -P4(D2) dy Y(D2))

dY2b(D,) dY3(D,)X (POI dy Y3(DI)-P 2(DI) Y2b(D,) dy

dY3(D2) dY4(D2)+ (P4 D2) d- Y;(D2) -P0( 2) Y3( 2 d

1 54

dY~b(Dj) Y()

x~3(j) dy 3;D)-P(D) 2b(DI) dy

dYb(YO) dYa(yo)

2& 3( 0 dy - y Yb(YO)

x p]0)Y_ dY~a(O) dY 1(0)

x~~ 1(O) 0) dy - P2(0 ) dyY 2a(O))

dY2a(YO) dY~b(ylO)

dy Y2b(YO) - Y22(y0) -dy

dY~a(O) dY1 (0)

x (P (0)Y1 0)d P2( 0 ) dy Y (0))1J (333)


I'~krpk pk )It D2

X L(P3 k Y P2 k 3) G3 k Y4- P4 k ) e-l 2 DIe+I 2Y e-l k3 D 2 ejk Y3D

- (p, k Y - P2 k Y3) (P3 k Y + P4 k Y ) efk Y2 DI e ik Y O *jk 3 D 2 e-ik D1

+ (P3 ky2 - P2 ky) G 3 kY4 -P4 k Y3 2 D I -lk D2 eik3D

-(p, ky2 +P2 k ) (P3 ky, +p k3) e*iky2DI e-ikY2YO e+kY3 D2 e-it 3 Dt ]

e i2jk 4 l ( p k 2 +P2 )(G k 2-P

X(P4 k P3 k) e+iy2DI e'y3D2 e+jky3DI

+ (p, k Y2 + P2 ky I ) G3 ky 2 2 +Pky3 )

X (P4 k + P3 k ) e kY2 DI e+ ItY3 D2 e-tY3 DI

x y3 Y4

- 6k )(G3 k 2 k 3

156

( p4 k3 P3 k ) e-'kY2 DI e kDY3DDI

(p k y - Pk y )G -P

- (pk& -P 2 k i ) k2 -2 k 3

x (pk + P3 k ) e - ky 2 DI e +1ky 3 D2 e-kY3 D 1 (334)x 3 k 4

Using the definitions of R21, R23, and R34 presented earlier, (334) may

be reduced to

B2a - +jkr [R34 e-'ky 2 (DI- YO ) eJky3 (D2" DI

4xkY2

+ 2 -'k y2 (DI- Yo 0 ) 1 (D 2" D I+ R2 3 e e+2 ey3

+ R2 3 R34 e+kY2 (D1- Y ) - (D2 - D I) e4k 2(D 1- y a) e+jk 3(D 2 " D I

157

LR23 R34 e" k 2 DIe jt3 (D2DI+e4 j 2 DIe * 3(D2DI

R,R34e -k2 D I jkY (D 2- D I -R2 , 1 V2D R23 ej 3 (D 2 - D 1 (333

Using conditions (328) and (329), (335) may be reduced to

B~a - jkrLe +1k 2 (D .Y)+R23 e- Y2 (D-0 1-

4xk

I e lkY D _R,R23e -k2 D (336)

-it yFactoring e Y2 0 out of the numerator of (336), and dividing the

numerator and denominator of (336) by e+ jk 2 Dreveals

158

B2RC - -+jkr 1( + R23 C-j2k Y (D - y 0 e-1k 2Y

I -2 2 12 e2 D"(32


verified, we can conclude that our derivation of the solutions for B 2a' and

hence, for B~cfor the four media waveguide problem is correct.

For the unknown constant A2b, Mfalhema~ica# provided the following

numerator:

num 4 - G, [- a1,3 a2.2 a3.7 a4,& a5,1 a6,5 a7,6 4+ a 1.2 a2,3 a3,-7 a4,8 a5,1 a6,5 a7,6 - a,., a2,3 a3,7 a4,8 a5,,2 a6,5 a7,6+ a,,, a2,2 a3.7 a4,8 a5.3 a6,5 a7,6 + a 1.3 a2.2 a3,5 a4.8 a5,1 a6,7 a7,6- a 12 a2.3 a3,5 a4,8 a5,1 a6,7 a7,6 + a I, a2.3 a3.5 a4.8 a5,2 a6,7 a7,6- a,., a2,2 a3,5 a4 .8 a5,3 a6.7 a7,6 + a 1.3 a2,2 a3,6 a4,8 a5,1 a6,5 a 7- a3,2 a2,3 a3.6 a4,8 a5.1 a6,5 a7,7 + a,,, a2,3 a3.6 a4,8 a5,2 a6,5 a7,7- a,,, a2,2 a3,6 a4,8 a5,3 a6,5 a7,7 - aj, 3 a2,2 a3,5 a4,8 a5,1 a6,6 a7,7" a 1,2 a2,3 a3,5 a4,8 a5,3I a6,6 a7,7 - a,., a2,3 a3,5 a4,8 a5,2 a6,6 a7,7" a,,, a2,2 a3,5 a4,8 a5,3 a6,6 a7.7 + a 1,3 a2,2 a3,7 a4,6 a5,1 a6,5 a7.8- a 1.2 a2.3 a3,7 a4,6 a5,1 a6,5 a7,8 - a1,3 a2,2 a3,6 a4 .7 a5.1 a6.5 a7,8+ a 1.2 a2.3 a3,6 a4 .7 a5,3I a6,5 a7,8 + a,,, a2.3 a3.7 a4,6 a5,2 a6,5 a7,8- a,,, a2.3 a3.6 a4,7 a5.2 a6.5 a7,8 - a,., a2,2 a3,7 a4 .6 a5,3 a6,5 a7.8+ a,,, a2.2 a3,6 a4.7 a5.3 a6.5 a7.8 4 a1,3 a2.2 a3,5 a4,7 a5,3 a6,6 a7,8

a a1,2 a2,3 a3,5 a4,7 a5.1 a6.6 a7,8 + a,,, a2.3 a3,5 a4,7 a5,2 a6,6 a-7,8-a,,, a2,2 a3,5 a4,7 a5,3 a6.6 a7,8 - a1,3 a2.2 a3.5 a4.6 a5,1 a6.7 a7,8

159

" a1,2 a2,3 a3 ,5 a4,6 a5,1 a6,7 a7,8 - al,1 a2,3 a3,5 a4,6 a5 ,2 a6,7 a7.8+ aI., a2,2 a3,5 a4,6 a5,3 a6,7 a7,81 (338)


generic expression for the numerator of A 2b:

num4 - G, [(a 4,8 a7,6 - a4,6 a7.8) (a3,5 a6,7 - a3,7 a6,5)+ (a4.7 a7.& - a4.8 a-, 7) (a3 .5 a6.6 - a3.6 a6.5)]

x [a2,2 (a1 ,3 a5 1 - al,, a5.3) 4 a2,3 (al,1 a5,2 - a1,2 a5,). (339)


the following general expression for A2b:

dY (D2) dY-(D 2)-2- (33(D2) d -(D2) dy Y(D)

A 2n 02)t P4(D2) YD2))

dY2b(DI) dY3(DI)( p3(D,) dy Y3(Dt) - p(DI)Y;b(DI) dy .

dY3(D2) dY4(D2)

G0(4( 2) - Y+(D 2) - P3(D2) Y(D 3 2) dy

dY2b(Dl) dY(D 1 )X (PODI) d Y;(D 1) - P2(DI) Y2b(DI) 3D)

160y

160

dY2a(o) dY 1(0)IY; (Y.) (P 1(0) y 1(0) d - P2( 0) dy Y(0))

dY (0) dYja(O)

+Y a~yO,) (P2(0) dy + a(O) - P,(O) YI(0) d

dY+(D 2) dY;(D 2)

i (p3(D2 Y;(D 2) dy - P4(D2) dy Y+(D 2))

dY+2b(DI) dY3(D,)

x tp(D,) dy Y3(D,) - P2(Dj) Yb(DI) d

dY30 2 ) dY+(D 2)+ P4D2 Y(D 2) - P0(2) Y ( 2))

dY2+b(DI) dY+(DJ)

x~ 3D) dy 3dy

dY2b(yo,) dY~a(yo,)2a dy - dy Y2b(yo)

dY2a(O) dY 1(0)x (p (0) Y-1(0) dy -P2( 0) dy Y a(O))

161

dY2,(y0, yY y) dY~b(yo)+ (dy Y2bYO - 2a 0o dy

dY+2(o) dY-,(0)x G1 O) Y1(0) d2 P2( 0) dy Y2a(O)JI

dY+(D 2) dY+(D 2)

- (pD2 ~ 2 )dy-p( 2) dy Y(2

dY2b(DI) dY3(D,)

x [p(D1) dy Y3 D) -P 2(Dj) Y~b(DI) dy

dy 3(D2) dYQD 2))+ (p,(D2) dy Y;(D2) -POOD 2 Y3(D2) dy

((D)dY~b(DI) Y+D)P(I - dY;+(D)

xtp3 (D3 dy Y(,-D) 2b(D,) dy

x (Y(Y 0)dY~b(yo) dY~a(yo) by)dy -d

dY- (0) dY1(O)

x~pjO)Y(0) d-y- P2( 0) -dyYa()

162

dY 2(YO)dY;b(YO)

dy - 2b(yo) -Y2&(YO) dy )

dY*a(0) dY (0)(0) (0)- P2(--- Y+(0))1l (340)


A2b = jkr e4ik Y2 D I e'kY4 D 2 U(p3 k Y- P2 k )2n 2 3

X (G3 k Y4- P4 k Y3) elk Y3 D2 ejk Y3 D1

x(P3 , k 2 k ) ei 2Y (p , Y2 + P2 ) eiy 3 D k2 Y

12 k Y e-l 4 D21(k 2 + P2 k Yl)(G 3 k 2 -P2 k~ )

163

x G4 ky3 - P3 ky4 ) e -1ky2DI e k y3D2 e * ky 3 D

(p, k Y2 P2 k Y1) GP3k 2+P2 Y

" (P4k Y3+P3' k4)e*i 2 DIe *1 3 D2e1 Y3DI(p, ky 2 - P2 k y I P Y2 +2k y3)

y 3 2

S(P4 ky3 + P3 ky4 ) e j kY2 DI e

2jkY3 eJkY3 DI

4- P3 k ) ky 2 D1 ky3 D2 k i 341Y3

X(P4 k3+ P3 k4) ejk y2 DI e + Y3 D 2ejk y3 D!1 (341)

Using the definitions of R21. R23, and R34 presented earlier, (341) may

be reduced to

164

-j2 , (D2- D )]A2b C - Jkr [I,+R23 R34 e-jk D-D

4xkY2

x 1 C k 2YO R2 1 e °1 ] /

ej 2ky3 (D2 - DI ) -i2k (D2 - D ) -j2k D1

[R23 R34 e+1- R2, R34teIt3 -R21 R23 e21y ]

(342)

Using conditions (328) and (329), (342) may be reduced to

A2b - +Ikr le i Y2Y0 + R21, e ! ,2 /

4xkcY2

11- R2 R23 e 2 1. (343)


verified, we can conclude that our derivation of the solutions for A 2b' and

hence, for A2bC for the four media waveguide problem is correct.

165

For the unknown constant B2b, Ma/hematica provided the following

numerator:

num 5 - G, [al3, a2,2 a3,7 a4 ,8 a5 ,1 a6,4 a7, 6

- a1.2 a2 .3 a3,7 a4,.8 a5 .1 a 6.4 a7 ,6 + a,., a2,3 a3 .7 a4,8 a5, 2 a 6.4 a7.6- a,., a2 .2 a3,7 24.8 a 5.3 a6.4 a7 ,6 - a1, 3 a2.2 a3 .4 a4,8 a5 .1 a6.7 a7,6+ al, 2 a2 .3 a 3.4 a4.8 a5,1 a 6,7 a7 ,6 - a,., a2,3 a3 4 a4,8 a5, 2 a6.7 a7.6+ a,., a2 .2 a 3 4 a48 a5,3 a6,7 a7.6 - a1.3 a2,2 a3,6 a4,8 a,1 a 6,4 a7,7+ al. 2 a2 .3 a 3,6 a4,8 as5, a6.4 a7 .7 - a,., a2 .3 a 3,6 a4,8 a 5,2 a6,4 a7 .7+ a,., a2 .2 a 3,6 a4 .8 a5 .3 a6A a7,7 + al. 3 a2.2 a 3, a4,.8 a5, a6.6 a7 .7- a .2 a2 .3 a 3.,4 a4 .8 a5,3 a6,6 a7 ,7 a , a2.23 a3.4 a4 .8 a5, 2 a6.6 a7,7

- a,., a2.2 a3, a4,8 a5.3 a6.6 a7,7 - aL3 a2,2 a3,7 a4.6 a5.1 a6.4 a7,&+ al 2 a2.3 a3.7 a4,6 a5.1 a6.4 a7.8 + a 1.3 a2.2 a3,6 a4.7 a5,1 a6,4 a7,8

- a.2 a2.3 a3.6 a4,7 a5, a6,4 a7.8 - a,., a2,3 a3,7 a4,6 a5,2 a6,4 a7,8+ al. I a2.3 a3,6 a4.7 a5,2 a6,4 a7.8 a., a2.2 a3.7 a4,6 a5,3 a6.4 a7,8- a,,, a2,2 a3,6 a4,7 a5.3 a6.4 a7,8 - a.3 a2,2 a3,4 a4,7 a5,1 a6.6 a7.8" a).2 a2,3 a3.,4 a4,7 a5, a6,6 a7.8 - al., a2,3 a3.,4 a4.7 a5,2 a6,6 a7,8

" z,I a2,2 a3,4 a4,7 a5,3 a6,6 a.8 + al,3 a2.2 a3,4 a,6 a5 ,1 a6,7 a7.8- 01.2 a2.3 a3,4 a,6 a5,1 a67 a7,8 a12, a.3 a3.,4 a4,.6 a5.2 a6.7 a7,8

- a1 , a2,2 a34 a4, 6 a5,3 a6,7 a7,81 . (344)


generic ej,,ression for the numerator of B2b:

num5 - G I(a,. a7,6 - a4,.6 a7 ,8 ) (a3,4 a6,7 - a 3,7 a6.)

S(a.7 a7,.8 - a4,.8 a7,7) (3,. a6,6 - a3.6 a,4)1

x 1a2,2 (al,, a5,3 - al.3 a5,j) + a 2,3 (al, 2 a,1 - al, a5,2) ] . (345)


the following generalexpression for B2b:

j66

dY'(D2) dY;(D 2)B2b - 2x E I (P3(D2 ) Y;( 2 d+ 4 34 Y;(D2)

dY~b(D,) dY3(D1)(P3D) dy Y- (D)- P2 (D1) Y*b(D 1) dy'

dY3(D2) YD)-P( 2)dY4(D2)(P(2 dy Y(D2 4 p( 2) Y;(D) dy

dYb(DI) dY*(D,)x (p3(D,) d- Y;-(DI)-P 2 (DI) Y'bI dy

dY 1(0) dY a(O)

21Yay)((O dy Y2a()p() 1 O dy)

dY2*a(o) dY1l(0)

y2a(Yo) (p 1(O) Y 1(0) dy dy20)- Y a(O))] /

dY*(D 2) dY;(DM2 1It(P3wYw2 d p(D2) Y 3 2)

dY2-b(DI) dY;(D1)

X tp(O) dy Y3(D1) - P2(D ) Yb(I dy

167

((D)dY3(2 4D)-P OY(D2) dY4(D 2)dy (D2 - 3(D) YD 2)dy

dY*2b(D,) dY;(D1 )X (p3(DI dy Y;(D,) -P2(DI) Y-b(DI) d3

dY~b(yO) dY' (y.)

2a~y~(c dy -dy Y2b(Y)

dY~a(O) dY 1(0)x(P1(0) Y1 (0) dy- P2(0) dy Y a(O))

dy23(Yo) dY~b(yo)

dy Y2b(YO) - Y2a(Yo) -dy

dY'a(o) dY1l(0)

x~py~O dy dy 2~~o)

dY+(D 2) dY'(D2 )

-(P3(D2)Y3w 2) dy -P4(D 2) dy Y+(D 2))

dY2b(D,) dY3(D1)

x~3 D) dy Y3 (D)POODY~b(DI) dy

168

+(4D)dY3(D2) dY4(D2)(p~2 y- Y+(D 2) - POOD2 Y(D 2) dy

dY~b(DI) dY+(D1 )X (P3(Dj) dy Y;(D0)-P 2 (DI) Y-b(DI) dy3

dy 3 d 1yd

dY*a(yO) dY (o)

dY&(YO) dYb(YO)( dy ~ybYO- aYo dy )

dYa() dY 1()x~pjO)Y(0) 2) dy Y Y-(o))1 36

dy dy


B~c !LjAe-itY2D -ik4D 2 *I k +P2 k

169

x (p4 k 3 p3 k Y) etY 3 D e *ky3 DI

+ (p k) P2k ) (p, +k 2) p2kyD e+Ity 2 o

[ 2(p k Y 2 k I 4 D 2 [ y(Yp k P2 k ) ( P k Y P] /-It 2 )e y)o(p~k+pk p)e

(P4 k y3 - P3 k y4 ) e eit D2 e- 3 e t

+ (pk Y2+P 2 k Y )G 3 k~ p2P2k )

(P4 k Y3 + P3 kY4 ) e +ity 2 DI e+iky 3 D2 e'ky3D

170

( p, 2 P2k 1) (P3.k Y2+ P2, k)

(P4 ky3 P3 ky4 ) e- Y2 DI e-y3D2 ekY3 D

X (P4 ky +P3ky4 )eky 2D I e- ky3 D2 e'Iky 3D ] (347)

Using the definitions of R21, R23, and R34 presented earlier, (347) may

be reduced to

B2b - +jkr [R23 + R34 e-2k y3 (D2 - D )

4nk Y-12k ~ .k ye-12kyD

x1+ R2 1 e -j2k y2 y e. I y 2YO /

171

IR23R34e-j2k y3 (DU2- D I)IR2 3 R34 eY+ D1 l

e -i2k y2 D i -2k y (D2- D I ) e -Ji2k~y D 1J34- R21 R34 e 1kY 1kY D2 - R P21 R23 e-j.2 (348)

Using conditions (328) and (329), (348) reduces to

B2bc +jkr R2 3 e +k 2 Y +R21-i2 D e/21y2 D

4xk

[1 - R2 1 R23 e 2ky ]. (349)

Equation (349) is equivalent to equation (261). Since (261) has already

been verified, we can conclude that our derivation of the solutions for B2b,

and hence, for B2b ¢ for the four media waveguide problem is correct.

For the unknown constant A3, Mathematica provided the following

nu merator:

num 6 - G, 1a1, 3 a2, 2 a3,5 a4,8 a5 .1 a6 .4 a7, 7

- a1 , 2 a 2.3 a3,5 a4,8 a5 .1 a6 4 a7 ,7 + al.1 a2 3 a3, 5 a4 ,8 a5,2 a6 4 a7 .7- a, 1, a 2.2 a3.5 a4, 8 a5 .3 a6 4 a7,7 - a1 .3 a2.2 a3.4 a4,8 a5 .1 a6 .5 a!,-

+ aL 2 a 2,3 a3,4 a4 8 a5 .1 a6 .5 a7 .7 - a1 I, a2, 3 a3, 4 a4 .8 a5,2 a6,5 a7,7Sa I., a2, 2 a3, 4 a4 ,8 a5,3 a6,5 a7,7 - a 1 .3 a2,2 a3,5 a4,. a5,1 a6 4 a7,8

172

+ a1.2 a2.3 a3,5 a4,7 a5,1 a6,4 a7 .8 - al.1 a2.3 a3,5 a4,7 a5,2 a6 4 a7,8+ a I,, a2,2 a3,5 a4,7 a5.3 a6,4 a7,8 + a1.3 a2,2 a3.4 a4,. a5,1 a6.5 a7.8- al, 2 a2.3 a3,4 a4,7 a5, 1 a6,5 a7,8 + al,, a2,3 a3,4 a4,7 a5,2 a6,5 a7,8- a1 .1 a2,2 a3,4 a4.7 a5.3 a6,5 a7,8

1 • (350)


generic expression for the numerator of A3:

num 6 - G, (a4,9 a 7.7 - a4 ,7 a7 8) (a 3,5 a6,4 - a3,4 a6.5)

x ta 2 ,2 (a,. 3 a5,1 - a1.1 a5,3) + a2,3 (a1,1 a5,2 - a1,2 a5.1)] (351)



A3 -kc (p 3(D2) Y(D) 4(D2) dY3(D2)Y(D2)

2nd - p4(D2) dy 4

dY (b(DI) dY2b(DI)xp2(D1) dy Y2b(D) - P2DY2,D) dy

dY2a(0) dY 1(0)

2a(Y () dy -P2( 0 ) dy Yao) 2aYo'

dY1(0) dY;a(O)

dy Y2a(0) - P1(o) Y1(0) dy ) /

173

dY*(D 2 ) dY;(D 2)[(p3(D2 )Y4D dy - P4(D) dy Y 2))

dY.2b(DI) dY3(D1)

xGp(D) dy Y(D) - P2(DI) Yb(DI) d

+ (,D)dy2 Y4(D 2 ) - p3(D2) Y3(D2) dy4(2

xflp(D1 dy Y;(I)-P(DIY~bDI)dy

dY~b(yo) dYYa(Yo)

x dyI -d dy (DO yO)

x~i()~O d P2(D0 ) dY*2 () d

dY~a(Y o) dY2 b(yo)

dy dbyO - y jaYo dyo

dY~a(O) dY1 (O)

,(p() Y,(O) dy P2(0 ) d y

174

dY'(D2) dY;(D2)(pr2Y(D)dy -p( 2) dy Y(D2))

dY 2b(DI) dY3(D1 )x p(OO dy Y3(DI)- P2(Dd)Y2b(DI) dy

+(4D)dY3D)42 POY3(D2) dY4(D2)

+ dy -Y'(D) p(D)Y() dy

dY~b(DI) dY'(D)X (POID) dy Y;(D1 ) - P2(D1) Y2b(DI) d

dY;2b(Y0 ) dY~a(yo)

2a~y~Y dy dy 2bY)

dY2a(O) dY 1(0)

dy d

dY2a(Yo,) dY.2b(YO)+ (dy - 2b(YO) - Y2a(Yo) -dy

dY+a(0) dY1,(0)x~pjO1Y(O) dy - P2(0 ) dyY~a(O)IIl (352)

175


A 3 C - 2 x e Y3 2 e-= ' 4 2 [(2 P2 k 2 ) (P4 k + k

x l(p, k 2- P2k Y ) e-% ° 2 O (, k Y2+ P2k, )Y11 e 2 i

ky2 e kY2 2Y ky p2 Y3

x ( 4 ky3 - P3 ky4 ) e+y2 D e - t 3 D 2 e+Ik Y3DI D

+2k ek2(p, k Y2+ P2 k Y G 3 k P2 k )

(P'I ky3 P3 k y4 ) eiky 2DI e +ky3 D2 e y3

- (p, k 2-k )PGkk 2 + k 3

176

~~~- "t-k D2 *kD

X(P4 k -p3 ) e-11Y2 DI e kY3 e +Y3 DI

x ky3 - ky )

X (P4 ky3 + P3 k y4 ) e-iky2DI e+k y3D 2 e-iky 3D ] (353)

Using the definitions of R21, R23, R34, and T23 presented earlier, (353)

may be reduced to

A3C - +jkr T23 le4 k' '2 + R2, e 2YO I e- ky D e4ikY3 D2 /

4xkY2

R23 R34 e Y3 2 + e (D2 - D R21 R34 ek e(2 - ) 2k 2D

-R21 R23 eAy3 (D2 - DI ) ei2ky2 D (354)

Using conditions (328), (329), (354) may be reduced to

177

A3C +jkr T23 le +| y 2 Y R21 e i y2 YO I e-iky 2D e +kY3 D

4xkY2

-|2kyD

II- R21 R23 e 2 (355)


verified, we can conclude that our derivation of the solutions for A3, and

hence, for A3C for the four media waveguide problem is correct.

For the unknown constant B3, Mathemabia provided the following

numerator:

num7 = G, I- al,3 a2,2 a3,5 a4, a5,1 a6,4 a7,6+ al,2 a2,3 a3,5 a4,8 a5,1 a6 4 a7.6 - a,1 a2,3 a3,5 a4,8 a5.2 a6,A a7,6+ a, a2,2 a3 ,5 a4,& a5,3 a 6.4 a7.6 + a,3 a2,2 a3,4 a4,8 a5,1 a6,5 a7 ,6- a1,2 a2,3 a3,4 a4,8 a5,1 a6,5 a7,6 + a,1, a2,3 a3,4 a4,8 a5,2 a6,5 a7,6- a, a2,2 a3,4 a4,8 a5,3 a6,5 a7,6 + al,3 a2,2 a3,5 a4,6 a5,1 a6.4 a7 ,8- al. 2 a2,3 a3,5 a4,6 a5,1 a6.4 a7.8 + a,,I a2.3 a3,5 a4,6 a5,2 a6, a7,8- a1,1 a2,2 a3,5 a4,6 a5, 3 a,4 a78 - al,3 a2,2 a3,4 a4,6 a5,1 a6.5 a7,8+ a1,2 a23 a3,4 a4,6 a5,1 a6,5 a7 ,8 - a1,1 a2,3 a3,4 a4,6 a5,2 a6,5 a7,8

+ al,, a2,2 a3,4 a4,6 a5,3 a6,5 a7 ,81 . (356)


generic expression for the numerator of B3:

178

rnUM 7 - G, (a4 6 a7.8 - a4,8 a7,6) (a3.5 a6,4 - a3,4 a6,5)a2,2 (a1,3 a5,1 . a5 1.1 a5.3) 4 a2,3 (a 1,1 a5,2 - a, s1) (357)


the following gemerilexpression for B3:

(4w)dY;-(D 2) dY*(D 2)B3 - y Y(D2) - 3D2) YO (D2) d

dYb(DI) dY2b(DI)X (P2(Dj) dy Yb(DI) -P2(D1) Y+bD) d

dY2a(O) dY 1(0)[Y4(Y,,) G(0) Y (0) dy - P2( 0) dy Y~a(o)) Y2a(Yo)

dY (0) dfa(O)

X~20 dy 2~()p(),O dy )

dY -(D2) dY(D 2)

It(P3(D2)Y4 D2 d -p( 2) dy Y.(D2))

dY2+b(DI) dY3(D,)dy Y3(D)P 2(D)Y-b(DI) d

179

+(4D)dY3(D2) Y(2 PO Y3D)dY4(D2)

(p4Wi d Y;D 2 )- p(D2 Y3 D2)dy

dYtb(DI) dY*(D1 )x~p3D1 ) dy Y;(D,) -P 2(Dj) Y'bD)d

dY~b(YO) dY2a(yo)

2a dy - dy Y2b(YO)

dY2a(O) dY 1(0)x p 1(O) Y-1(0) dy P2(0 ) dy Y a(O))

dY2a(Yo) dY~b(yo)

+ dy Y2b(yO) - 2.aYo) dy

dY'a(0) dY1,(0)x(p 1 (0) Y-1(0) dy - P2( 0) dy Y~a(O)H

dY(D 2) dY(D 2 )- t(~3 2 )~ d -P4(D2) dy Y 2))

dY~b(DI) dY3(D1 )

xfl3 D) dy Y3(D,)-P2(D,)Y2b(DI) dy

dY3(D2) dY4(D 2)+ 4 d y) _ Y4(D 2) - P3(D2) Y3( 2 d

dY~b(DI) dY*(D,)

X (PO(D) dy Y;(DI) -P 2(D,) Y-bI dy

dy dy - y b())-

dY'a(O.) dY()

x p(OY-) dy -dy Y(o))

dY~a(Yo d (O))

dy Y2+(yo) -Y2(YO) d by-

dY+ (0) dYl(0)

xtp (0)~() dy - P2( 0) dy Y a(O))1l (358)


B3 Ae- kY3 D 2 e-jkY4 D2 (2P 2 k ) (P4 k -P3 k)3c-2n Y 3- Y

i(p, k - P2 k , ) e-Y2 O + (p, k,,2 + P2 k, l ek # l /

[2 kY2 ek Y2 + PkY kPy2 ky

X (P4 k y3 -P3 k Y4) e 2 e- 3 D2 e 3

+ k pY 2 + P2 k ,) ( 3 k Y 2 k )"' 2 "D

G4 k Y3 + P3 kY4) e *Y2 I e +WY3 e D3

-(p, k,,2- P,2 k,,, G,, k, Y2 , P k )

x (p, kY3 - P3 ky4 ) ei ty 2D eik Y3 D2 e#iky 3 DI

( p-p,, k - P2 k, ) G 3 k(2 -p 2 k )

182

X (pG4 + P3 k 4 ) e-jkY2 D1 e+ItY3 D2 e-kY3 D 1 (359)


may be reduced to

.jk f41 y-1k y -"k D _.k

B3C - +Jkr R34 T23 Ie+Ity2 YO + R21 e 2YIe ky2DI e k 32 /

4gkY2

R23 R34 e-1kY3 (D2- D I e + e (D2- D R 1 -jkY3 (D2 - D!) e-12ky2 D I

[R2 3 34 e 3 2 jey3 (2 - 1 - 2 ky2Die 3.

-R 2 1 R23 eI3 (D2 - DI) e1 2kY2 D 1 (360)

Using conditions (328), (329), (360) may be reduced to

B3c - 0. (361)

Equation (361) is exactly what one would expect for a semi-infinite

medium (i.e., no wave propagation in the negative y direction). Thus, we can

183

conclude that our derivation of the solutions for B3 , and hence, for B3C for

the four media waveguide problem is correct.

For the unknown constant B1, Mathematica provided the following

numerator:

num I - G, [a1 ,3 a2,5 a3,7 a4,8 a5,2 a6,4 a7,6- a1 ,2 a2,3 a3,7 a4,8 a5 ,3 a6,4 a7 ,6 - al, 3 a2,4 a3,7 a4 ,8 a5 ,2 a6,5 a7 ,6+ a1 ,2 a2,4 a3,7 a4,8 a5,3 a6 ,5 a7 ,6 - a1 ,3 a2 ,5 a3,4 a4 ,8 a5,2 a6,7 a7 ,6* a ,3 a2,4 a3,5 a4 ,8 a5 ,2 a6 ,7 a7 ,6 + a1 ,2 a2,5 a3,4 a4,8 a5 ,3 a6,7 a7 ,6- a ,2 a2,4 a3,5 a4 ,8 a5 ,3 a6,7 a7 ,6 - a ,3 a2,5 a3,6 a4 ,8 a5 ,2 a6,4 a7,7+ a1,2 a2,5 a3 ,6 a4,8 a5,3 a6,4 a7,7 + al, 3 a2 .4 a3 ,.; a4 ,8 a5,2 a6,5 a7, 7

- al, 2 a2,4 a3,6 a4,8 a5,3 a6,5 a7,7 ,, a1 ,3 a2,5 a3,4 a4,8 a5,2 a6,6 a7,7- a ,3 a2,4 a3, a4,8 a, 2 a6,6 a7,7 - al, 2 a2 ,5 a3,4 a4,8 a, 3 a6,6 a7,7+ a1 ,2 a2,4 a3,5 a4 ,8 a5 ,3 a6,6 a7,7 - a1 ,3 a2 ,5 a3,7 a4 ,6 a5,2 a6,4 a7 ,8+ a1 .3 a2.5 a3.6 a4,.7 a5.2 a 6, a7,8 " al. 2 a2 .5 a3 .7 a4,6 a, 3 a6.4 a7,8- a1,2 a2,5 a3.6 a4,7 a5 3 a6.4 a7 .8 * a1 ,3 a2 .4 a3,7 a4 .6 a5.2 a6,5 a7,8- al. 3 a2 4 a3 .6 a4 '7 a5,2 a6,5 a7 .8 - al,2 a2,4 a3.7 a4,6 a5.3 a6,5 a7,8+ a1,2 a2,4 a3,6 a4,7 a5, 3 a6.5 a7,8 - a1.3 a2 .5 a3.4 a4 .7 a5,2 a6.6 a7,8* a 1 .3 a2,4 a3.5 a4,7 a5 .2 a6.6 a7.8 + a 1,2 a2 .5 a3.4 a4,7 a5.3 a6,6 a7 .8- al.2 a2,4 a3 .5 a4,7 a5,3 a6,6 a7,8 + a1,3 a2 ,5 a3,4 a4,6 a5,2 a6 .7 a7,8- a1 .3 a2.4 a3 .5 a4,6 a5, 2 a6,7 a7,8 - a1,2 a2 .5 a3.4 a4 .6 a5.3 a6,7 a7,8+ a1,2 a2,4 a3,5 a4,6 a5,3 a6,7 a7,81 (362)


generic expression for the numerator of B 1:

num1 = G, (a1 .3 a5.2 - al. 2 a5,3 ) [a 2 .5 [(a 4.6 a7 ,8 - a4 ,8 a7 .6 )

x (a 3,4 a6,7 - a3,7 a6,) (a4',8 a7,7 - a4,7 a7,8) (a 3,., a6,6 - a3,6 a6,4) I

- a2,4 (a4 6 a17 .8 - a14,8 a7,6) (a3,5 a6,7 - a3 .7 a6,5)+ (a4, a7.7 - a4,7 a7,8) ( a3,, a6,6 - a3,6 a6,5) 1J (363)

184


the f ollowing greenel expression f or B 1:

dY* (0) dY~a(O)

Bd(2( -Y(2 ) -dP(0DY20XI 2xOIP(2 dy (D)pD)YD)dy

dYb * dY(D

0Y( 2) 4Y;2)2+ Y-(p 3)(P2) -Y -(D)- P(D2)2dy- Y 2 ))

dY2*b(DI) dY(D 1 )X (P3(Dj) dy Y;(D 1 ) - P2(DI) Y2 01D) d

dY(D 2) dY3(D 2)- Y 3b(Y ) Y34 D2 d ;( - P(D2) Y(D 2)dy)

dY201D) dY3( 1

dY (D2)D; dY(

dy 4 2D ) - P3(D) Y(D) d

185

+ ( 3D Y( 2)dYQD 2) dY3(D2)0)dy -P4(D2) dy Y 2 ))

dY~b(DI) dY+(DI)X 60 dy Y;(D,) -P2(D,) Y2b(D,) dy

dY4+(D 2) dY+(D2 )

II(3(D 3Y(D2) d -P4(D2) dy YZ(D2))

dY2+b(Dl) dY3(D,)

"p 3GO,) dyY 3(D1 ) - P2(Dj) Y-2b(DI) dy)

dY3(D2) 4Y ( 2)

(P4(D2) dy Y;(D2) - POOD2 Y3(D2) d

dY42b(DI) dY(D 1

" 1.p3Dj) dy Y (D) - P2(D) Y2b(DI) 3dy )

dY~b(YO) dY~a(yo)-

2 (;y dy - dy YbY)

dY;1 (0) dY, (0)x (pj(O) Y (O) dy P2(0) dy Y~a(o))

186

dY2a(Yo) dY~b(yO)

+(dy- Y~b(YO) - ,;(Yol) -dy

dY~~o) dY1j(0)x ( 1(0 Y1 0)dy~a P2(0 ) dy o)

dY4(D2) dY3(D2)(p(D 2)Y;0 2) ~ p4(D2) dy Y4(D 2))

dYib(DI) dY3(D1)

x (30Jo dy Y0(1) - P2(DI)Yib(DI) dy

dY3(D2) dY4(D2)(P4 dy) Y;(D2) - P3(D2) Y3(D2) dy

dY2b(DI) dY;(D,)

x~3 D) dy Y;DOP2(DI) Y2b(DI) dy

dY~b(yOl) dY~a(yo)I(Y+ (Y,) d -dy,

(PI~) Y-dY 2&(O) dY 1(0)Y-()

x ~ 1(O)() y - P2(0)d d Y2O)

187

dY2a(Yo) dY2b(YO)

dy Y2b(YO) -Y2a(Yo) dy

dY;,(0) dY,(0)(p,(O)Y(o) dy Y"(o))11. (364)

For constant speed of sound and ambieny density, (364) becomes

BI =j'-(2P 2 k ) e-l Y4D

X [3 k y2 + P2 k ) (P4 3 k4e-lkY e-a2Y e-lk 3 2 ek Y3DI

+(G3 k 2 -P2 k )3 (G3 k Y4 + P4 ks, Y3ej Y e+k Y2 YO e 1k Y3 D 2 eIk Y3 D I

+ (p3 kY2 -P 2 ky), ky - P3 k ) e-iky2 DI eiky2 YO e-iky3 D2 eikY3 DI

188

+(p, k , P k3) (p k + P4 k3) ekY2 DI ek Y2 YO eilkY3 D/2 3 D

12k, ik 4 D2 e- y D2 (p , k2 + P2 k ) (G k 2 -p 2 kv )

+k(p k Y2 k Y G Y2 k Y3k +jk D I jk D2 e+jky3 DIS(p k -P 3 k ) eky2 3 ek y3

X(P)4 k +3 P3 k Y4) e'lk y2 DIe'jk y3D 2 e 1k Y3DI

x y3 Y4

(py k y P2k ) G3 ky 2 + P2 ky 3 D

(p, kv -Pk )(~ ' e )e189

(O k 2 p3 k y, ) (p2D eky 3D"2 eky 3D

189

(p, k , p3 ) e-'k2D e1 Y3 D2 e-'kY3 D 1 (365)


may be reduced to

+jkrB¢ -C T21

4xky2

x[e.pky2yo [R34 e-j2ky2D e-2ky3 D +R23e-2ky2 D +

e-kY2YO [I1 R3e 34 (2 -y3 (D2 /

-j2ky3 (D - DI ) e-2ky3 (D 2 - DI ) e-j2ky2 D[R23 R34 e + 1 - R2 1 R3 1e4 '

- R21 R23 e -1

2k 2 • (366)

Using conditions (328) and (329), (366) reduces to

190

B IC- +kr [ 1 D jkT21 IR23 e-k Y2 De +j 2 0O + e- Y2 0 O /4xk

Y2

I -R21 R3e-Jk2 1. (367)


verified, we can conclude that our derivation of the solutions for B V and

hence, for Bic for the four media waveguide problem is correct.

For the unknown constant A4, Mothemailk provided the following

flumerator:

num8 = G, [a1,3 a2,2 a335 a4,7 a5.1 a6,4 a7,6- a 1,2 a2,3 a3.5 a4,7 a3,1I a6,4 a7,6 ,a 1, 1 a2,3 a3,5 a4,7 a5,2 a6,4 a7.6- a, 1, a2,2 a3.3 a4,7 a5,3 a6,4 a7,6 - a1,3 a2.2 a3,4 a4 .7 a5,1 a6,3 a7,6" a 1.2 a2.3 a3,4 a4.7 a5,1 a6,5 a7,6 - aI,1I a2,3 a3,4 a4 .7 a5.2 a6,5 a7 .6" a,1, a 2 .2 a 3.4 a4,7 a5.3 a6,5 a7.6 - a1,3 a2,2 a3.5 a4.6 a5,1 a6.4 a7,7" a1,2 a2,3 a3,5 a4,6 a5.1 a6.4 a7,7 - a,1, a2.3 a3,5 a4.6 a5.2 a6,4 a7.7" a,,1 a2.2 a3.5 a4,6 a5,3 a6,4 a7,7 4 a1,3 a2.2 a3.4 a4,6 a5.1 a6.5 a7.7

a a1,2 a2,3 a3,4 a4,6 a5,1 a6,5 a7,7 + a I, a2,3 a3,4 a4.6 a5.2 a6,5 a7,7-a,., a2,2 a3,4 a4,6 a5,3 a6,5 a7,7] (368)


generic ezpressjon7 for the ?umerwtor of A4:

191

nums - G, (a4. a7,6 - a4,6 a7,7 ) (a 3.4 a6.5 - a3,5 a6,4)

x [a2,2 (a,., a5,3 - aI,3 a5,1) - a2,3 (a1, 1 a5,2 - a1,2 a5.1)] (369)



dY;(D 2) dY3;(D2),4 -kr p3(D2) d Y(D 2)- p(D 2 ) Y(D2) dy )2x dy Y33 d

dY2b(DI) dY*b(D,){p 2(DI) Y2b(DI) dy - P2(DI) dy Y2(D))

dY1(0) dY a(O)[Y;(yo) (p2(o) dy Y22(o) - P(O) Y(0 dy

dY1(O) dy;a(0)-Y2(Yo) (p2(0) d- Y;(O) - p1(0) Y,(0) y ) /

dY(D 2) dY(D 2)(p3(D 2 Y (D2 ) dy -p 4(D2 ) dy Y4(D2 ))

dY2b(DI) dY3(D)x{p3(Di) dy Y3(D)- P 2(D,)Y+2b(D,) dy )

192

dY3(D2) dY4(D 2)+(P4 D2 ) dy Y(D 2) - POOD2 Y3(D2) d

dy 4P d

dY2b(DI) dY*(D1 )

x33D d Y;D)P2(DI) Y2b(D,) dy

dY2b(yo,) dY(yo,)

2a dy -dy Y2b(yo)

dY~a(O) dY 1(0)x(pIOY-(0oP() dydy Y2a(O))

dY2&(y.) dYib(yol)

dy Y2b(yo) -1 Y2a(y0) dy

dY*a(o) dY1,(0)

x~1()~()dy dy 2

dY4(D2) dY;(D2)S(P 3(D2 ) Y;+(DO) dy PO p()dy +( 2 )

dY~b(DI) dY3(Dj)

X (OI)dy Y3(DI) P2(Dd)Y2b(DI) d

193

dY3(D2) dY'(D2)+(4D)-dy Y4(D 2) - P0(2) )3D2

+ Y(D2 )dy

dY b(DI) dY+(D)

x(OO d Y;(D,) -P2(D,) Y-bD) d

dY~tb(yo) dY*a(yo)

SI(Y (y 0 dy - dy Yb(YO))

dY21(O) dYj(0)x (p,(O) Y1 (O) d - P2( 0) -dy Ya.(O))

dY2a(Yo) dYb(yo)+(dy Y-2b(yO) - 2a(Yo) dy )

dY~a(O) dY1l(0)dyP1 0 Y10 P2( 0) dy Y1(0))1 (370)


194

A 'k(2, k )(2 , kAcw 2x Y2 y3

x[(p k2 - P2,k ) e"kY2 YO + (,, k , + P2k ) e ikY2 Y /

2k 2e-kY4D21 (p, k P ) (P3 k -P2k

X (p4 kY3 -P3 k y4) e ikY2 e- lY3 e lkY3OD

( k Y P2 k ) ( k Y P2 k )

X (P4 ky3 + P3 ky Y) ekY2D e3 2 e-lkY3 D

(p, k Y -k ) G k 2+ k )3

(P4 -P3k ) e , k2 Y3D2 e ikY3D

195

- (pl k y2 - P2 k y1 ) GP3 ky2 - P2ky3

x (G4k~ Y3+P 3 k ) e1Y 2 D e j 3 D 2 ejk Y3 Dli. (371)

Using the definitions of R21, R23, R34, and T23 presented earlier, and

defining the transmission coefficient at the boundary between medium three

and medium four as follows:

2 p3 ky

T34 k (372)P4 k Y3 + P3 k y4


A4 = - T23 T34

4nky2

[j!kY2 YO R -jky2y o -k2D -ik (D2 - DI e D2

196

e-12k (DI- D ) -j2k D -j2k (D2- D )

IR23 R34 e 3 + 1 - R21 R34 e Y2 e 3 2 I

-R 2 1 R23 e 2 D1 . (373)

Using conditions (328), (329), and letting

T34 1, (374)

and, as a result,

k -k (375)y4 Y3

(373) reduces to

A4C - jk T23 lekY2YO + R21 ;kt~ 0 2 e kY2 e+kY3 D /

4xkY2

[1- R2 1 R23 e-2k Y2 D (376)

Equation (376) is equal to both equations (275) and (355). Since (275)has already been verified, we can conclude that our derivation of the

197

solutions for A4 , and hence, for A4C for the four media waveguide problem is

correct.

To summarize the results for the four media waveguide with plane,

parallel boundaries, for the general case, in which speed of sound and

density are arbitrary functions of depth, the unknown constants are given

by (322), (333), (340), (346), (352), (358), (364), and (370). As in the three

media waveguide case, the denominators in these expressions are exactly the

same. For constant speed of sound and density, the unknown constants are

given by (327), (335), (342), (348), (354), (360), (366), and (373).

Having completed two successful tests of the programming technique,

we now return our attention to the general waveguide problem. As in the

simpler examples, the compact vector-matrix system equation (198) applies.

For the general waveguide prob!em, the vector matrix quantities involved

are as follows:

A is the 28 by 17 matrix of coefficients,

A - [10 al, 2 aI, 3 a1 ,4 0 0 0 0 0 0 0 0 0 0 0 0 01[a2,, 0 0 0 a2,5 a2,6 0 0 0 0 0 0 0 0 0 0 0110 0 a3,3 a3,4 0 0 0 a3,8 a3,9 0 0 0 0 0 0 0 0110 0 0 0 a4,5 a4.60 0 0 a4,10 a4 ,11 0 0 0 0 0 0110 0 0 0 0 0 0 a5,8 a5,9 0 0 a5,12 a5,13 0 0 0 0110 0 0 0 0 0 0 0 0 a6 ,10 a 6, 11 0 0 a6,14 a 6,15 0 0110 0 0 00 0 0 0 0 0 0 a7,12 a-7,13 0 0 a7,16 0110 0 0 0 0 0 0 0 0 0 0 0 0 as,14 a8,15 0 a8 ,17110 ag,2 a9,3 a9,4 0 0 0 0 0 0 0 0 0 0 0 0 01Ialo,, 000 a10,5 a10,6 00000 000000110 a1i ,2 al 1 ,3 a11 ,4 0 0 0 00 0 0 0 0 0 0 0 01

198

[a1 2,1 0 0 0 a12.5 a12.6 0 0 0 0 0 0 0 0 0 0 01[a13,1 0 0 0 a13.5 a13,6 0 0 0 0 0 0 0 0 0 0 0110 a14.2 a14,3 a14.4 0 0 0 0 0 0 0 0 0 0 0 0 0]10000 000 a15,8 a15,9 0 0 a15,12 a15,13 0 0 0 0110 0 0 0 0 0 0 0 0 a16,10 a16 ,11 0 0 a16,14 a16,15 0 0110 0 0 0 0 0 0 a17,8 a17,9 0 0 a1-, 12 a17.13 0 0 0 01to 0 0 0 0 0 0 0 0 a18,10 a18,11 0 0 a18,14 a18,15 0 01[0 0 0 0 0 0 0 0 0 a1g,1o a1g, 1 0 0 a19,14 a19,15 00110 0 0 0 0 0 0 a20 ,8 a20 ,9 0 0 a2 0,12 a2 0,13 0 C ) 01100 00000000 0 a21 ,1 2 a 21, 13 0 0 a2 1,16 01

10 0 0 0 0 0 0 0 0 0 0 0 0 a22,14 a22,15 0 a22,17110 0 0 0 0 0 0 0 0 0 0 a23,12 a23,13 0 0 a23,16 0110 0 0 0 0 0 0 0 0 0 0 0 0 a24 , 14 a24 ,1 5 0 a24 ,171[0 0 0 0 0 0 0 0 0 0 0 0 0 a25 ,14 a2 5,15 0 a25 ,17 110 0 0 0 0 0 0 0 0 0 0 a 2 6 ,1 2 a 2 6 ,1 3 0 0 a 2 6 , 16 01

10 0 a27 .3 a27 ,4 0 0 0 a27 ,8 a27 ,9 0 0 0 0 0 0 0 0110 0 0 0 a28.5 a28,6 -1 0 0 a28,10 a28,11 0 0 00 0 011

where

a 12 =- PI(ys) Jn(kr r) YI(Ys) (377)

a1.3 P2(YS) Jn(kf 2r) Y2a(Ys) (378)

a1.4 = P2(Ys) Jn(kr2 r) Y2a(Ys) (379)

a2,1 = - pi(Ys) Jn(kr r) Yi(Ys) (380)

a2,5 - P2(YS) Jn(kr 2r) Ya(Ys) (381)

a2,6 - P2(Ys) Jn(kr 2r) Y2a(Ys) (382)

a3,3 - Y2(yo) (383)

199

a3.4 Y-~o (384)

a3,8 - -~(O (385)

a3,9 -- Yb(yo) (386)

a4 .5 - Y-2a(Yo) (387)

a4.6 - y;a(yo,) (388)

a4, 10 Y 2b(yo) (389)

a, 1=Y~b(yo) (390)

a5.8 -P2(YBI) Jn,(kr1 r) Y2b(YB1) (391)

a5,9 - P2(YB91) Jn(k 12 rH Y2b(YB1) (392)

a5,12 =-P3(Y 91) Ju(k' 3 r) Y+(y, 1 ) (393)

a5,13 =-P3(YB 81) J,(kr13r H 3(yB1)(34

a6,10 =P2(YB,) JnAkQr Ylb(YB1) (395)

a6,11 -P 2(YB,) Jn(k f2r) Y~b(YBI) (396)

a6,14 -P3(YB 1) Jn(k13 r) Y3;(YB 1) (397)

a16,15 -- P3(Y91) Jn(kr3 r) Y3(YB1) (398)

a7,12 -P3(Y 92) Jn(kr 3 r) Y;3(YB2)(39

200

a7.13 P 3(YB 2 ) Jn(kr 3r) Y3(YB 2 ) (400)

a7,16 =- P4(Y B2) Jn(krr)Y4 (yB2) (401)

a8,14 - 3(YB2 ) Jn(kr 3 r) Y;(YB2) (402)

a89,15 - 3(YB2 ) J,(k, 3r) Y3(YB 2 ) (403)

aS.17 =-P4 (YB2) Jn(krfAr) Y4"(YB2) (404)

dY I(YS)a9,2 - - akrr dy (403)

a9,3 - jn(k r,)r) -Y~~ (406)

- ~dY2a(YS)(47a9,4 -in(kr r) dy(07

*dY (ys)a101 = -J,(k r') -(408)rI dy

dY*2a(YS)a 10,5 = Jn(kr 2 r) dy (409)

dY2a(YS)a, 0 ,6 - J(k r r) dy(410)

djfl(kr1 I r)

at 12 - kr - , y I YS)(411)d(kr, r)

201

dJa(kr 2 r)

at 1.3 - kr Y2a(Ys) (412)d(kf2 r)

dJn(kr 2 r)a, l,4 - kr2 Y2.(Ys) (413)d(kr r)

a12, 1 - Jn(kr r) YI(ys) (414)

a1 2,5 = Jn(kr 2 r) Y2(Ys) (415)

a1 2,6 - Jn(kr 2 r) Y2a(YS) (416)

djn(kr, r)a, 3,! I - I (ys) (417)

d(krI r)

dJ0(k, 2 r)a1!3 .5 = kra Ya(Ys) (418)

d(kr a r)

dJn(kr 2 r)a 13.6 - kr2 Y2a(ys) (419)

a,4,2 - - Jn(kr I r) Y1(ys) (420)

a14,3 Jn(kr2 r) Ya(ys) (421)

202

214,4 - Jn(kr 2 0) Y(YS) (422)

dY24b(YB,)

a nk20 dy (423)

dY2b(YBI)

a,5,9 jn~(k r2 r) dy (424)

dY -(y,3)

a,51 f~ 30 dy (425)

dY3(YB1 )

a1,3--j~ r3 0 dy (426)

dYb4B 1

a61- nkQ0 dy (427)

dYib(YBI)

a a16.11 -Jn(kr2 0) dy (428)

dY3(yB,)

a 16,14 - - Jn(k t30r)(49

dY3(YBI)

a 1615 --jn(k r3) dy (430)

djn(k r2 r)*

d(kr 2 r)(41

203

2 179 - kr2 djl(k 2 r) Y2b(YBI)(42

a171 - -k~ 3dJ(k 2 r)Yiy 1

d(k,, r)

a 17.12 - - kr3 -J (k 3 r) (434)d(kr3 r)

21&lO -J(k, r) bY 1

a11--k3 dJ( kr r) Y3(y1 ) (436)

a IS.1I - Jn(kr r) Yb(YB 1) (437)

a18 15 - - J,(kr3 r) Y3(y, 1 ) (438)

dJ,(kr2 r)a19,10 - k2 kf2 Y~b(YB1)(49

a1911 - k jir2 r -(B (440)2d(kr 2r) 2

204

djn(kr 3 r)a19,14 - - kr3 dYk(3 r) (441)

d(kr 3 r)

a19,15 - -kr3 Y3(YBI) (442)d(kr, 0

a20,8 - jnkr) Y2b(YB 1) (443)

a20,9 -Jf(kr 2 r) Y2b(YB1 )(4 )

a20,12 - -J(k r3 r) Y 3(B1 ) (445)

a201 3 - -Jf(kF3 0)Y3Y) (446)

dYB 2)

* a21 ,12 - J,(k r) d- (447)

dY 3(yB2)

aF,3- nk 3 r dy (448)

dY4y 2

a2 ,16 - - Jn(k r4r) dy (449)

dY;(YB2)a22,14 - Jn(k r) - (450)r3 dy

a22,15 -Jfl(kr~)d 3 y 2 (451)

205

dYB(Y2)

22217 - - Jn(kr4r) dy (452)

dJn(kr 3 r)a23,12 - Y(Y) 2) (453)

d(kr3 r)

dJn(kr r30)a23.13 - kr3 Y3(YB2 ) (454)

d(kr r)

dJa(kr 4 r)a23,16 - - kv - Y4(I2 (455)

4 d(kr 4 r) Y(yB2 )

a24,14 - Jn(kr 3 r) Y3(YB2 ) (456)

a24,15 - Jn(kr 3 r) Y3(y. 2) (457)

a24,17 = - Jn(kr4 r) Y*(yB2) (458)

dJ(k 3 r)

a25, 14 - k d(kr3 r) y (459)d(kr 3 r)

dJn(kr 3 r)

a25,15 - kr3 Y3(YB2 ) (460)d(kr 3 r)

206

dJn(kr 4 r)

a 25,17 - - kr4 Y4(YB2 (461)d(kr4 r)

4

a26,12 - J,(kr3 r) Y;(YB2) (462)

a26,13 - Jn(kr 3 r) Y3(y.2 ) (463)

a26,16 - - Jn(kr 4 r) Y4(yB 2) (464)

a27.3 - dy (465)

dY2a(Yo)a27,4 - dy (466)

dY2b(Yo)

a27 9,8 - dy (467)

dY2b(Yo)a27,9 = - dy (468)

dY+ (Y.)a28,5 - dy (469)

dY2a(Yo)a28,6 - dy (470)

dY2b(YO)a28,10 - - dy (471)

dY2b(yo)a28,1 - - dy (472)

and

207

z x is the 17 by one vector of unknown constants,

S( A1 BI A2a B2aCaD 2aG2 A2b B2b Cab D2b A3 B3 C3 D3 A4 B4 )T, (473)

where the superscript T indicates the transpose matrix operator (which

means that z is a column vector).

The vector of known constants, b, is simply made up of the right-hand

sides of ( 149) through (176), or more specifically

b=( 00000000000000000000000000G0)T (474)

where here again the transpose operator is used to indicate that b is a

column vector.

One should note that the arbitrary constant G, appears in the known

constant vector while the arbitrary constant G2 appears in the unknown

constant vector. This occurs because G, represents the known amount of

discontinuity required to achieve the free-space Green's function solution

under the necessary conditions for that solution to exist (i.e., constant speed

of sound, no boundaries). G2, on the other hand, is really an artifact of the

method used to derive these boundary condition equations, and, as such,

should be treated as an unknown quantity in the general case.

In the general waveguide problem, the form of matrix A leads to some

complications. First, the fact that the number of equations (i.e., 28) is greater

than the number of unknowns (i.e., 17) implies that the solution to this

problem will not be unique. Secondly, the fact that the matrix A is not a

208

square matrix implies that the simple matrix inversion technique of (222)

cannot be utilized. Thus, an alternate solution technique must be employed

to set up the problem so that Mathematica may be used in the solution.

Gelb (1974) describes this situation (more equations than unknowns) as

an overdetermined case. Gelb and Haykin (1986) both suggest the use of a

pseudoinverse matrix in obtaining a least squares estimate for the vector x.

The pseudoinverse matrix is defined by Gelb as follows for real matrices A:

A*=(AT A) I AT. (475)

where As is the 17 by 28 pseudoinverse matrix.

Haykin and Menke (1984, p. 253) both define a similar pseudoinverse

matrix for complex matrices A as follows:

A- - (AH A) I AH. (476)

where the superscript H indicates the Hermetian or complex conjugate

transpose matrix operator.

In the solution for the general waveguide case, we will use a

combination of these pseudoinverse matrix techniques. This combination

incorporates a 28 by 28 weighting matrix, W, which allows us to obtain a

weighted least squares estimate for the vector x. Therefore, our approach

will be to use the weighted pseudoinverse formulation suggested by Gelb

and Menke (1984, p. 54) with the complex conjugate transpose operators

209

suggested by Haykin in lieu of the standard (and less general) transpose

operator. Thus, the pseudoinverse matrix to be used in this thesis is defined

as follows:

A=O-(AH VA)-'AR V. (477)

Using this pseudoinverse matrix, a weighted least squares estimate for

the vector z may be calculated as follows:

z -A" b . (478)

The following AMbIema/ica code was developed to solve the general

waveguide problem using this weighted least squares technique:

b -(0, 0,0, 0, 0,0,,,0,0, 0, 0,0,.0, 0.0,0, 0, 0, 0, 0, 0, 0, 0, 0, GI1, 0);

w -((wlcl, wlc2, wc3, wlc4, wlc5, wlc6, wlc7, wlc8,wlc9, wlclO, wlcl 1, wlcl2, wlcl3, wlcl4, wiciS5,wicI 6, wlc]7, wic] 8, wlcl9, wlc2O. wlcZ 1. wlc22,w Ic23, w Ic24, wlIc25, w Ic26, w Ic27, w Ic28),(w2c L w2c2, w2c3, w2c4, w2c5, w2c6, w2c7, w2c8,w2c9, w2c 10, w2c 11, w2c 12, w2c 13, w2c 14, w2c 15,w2c 16, w 2c 17, w2c 18. w2c 19, w2c2 0, w2c2 1, w2c2 2,w2c23, w2c24, w2c25, w2c26, w2c27, w2c28),

(w3cl1, w3c2, w3c3, w3c4, w3c5, w3c6, w3c7, w3c8,w3c9, w3cl0, w3cI 1, w3cl 2, w3cl 3, w3cl 4, w.7c1 5,w3cl 6, w3cI 7, w3cl 8, w3cl 9, w3c20, w3c2 1. w3c22,w3c23, w3c24, w3c25, w3c26, w3c27, w3c28),(w4cl, w4c2, w4c3, w4c4, w4c5, w4c6, w4c7, w4c8.w4c9, w4cIO0, w4cI 1, w4clZ, w4c13, w4c1 4, w4c15,w4cl 6, w4c17, w4ci 8, w4cI 9, w4c20, w4c2 1, w4c22,w4c23, w4c24, w4c25, w4c26, w4c27, w4c28),

(w5clI, w~c2, w5c3, w5c4, w5c5, w5c6, w5c7, w5c8,w5c9, w5clO, w5cl1, w5cl 2, w5c1 3, w5cl 4, w~cl5,w5c1 6, w5c17, w5c1 8, w5c1 9, w5c20, w5c2 1, w5c22,

210

w5c23, w5c24, w5c25, w5c26, w5c27, w5c28},(w6cI, w6c2, w6c3, w6c4, w6c5, w6c6, w6c7, w6c8,w6c9, w6cIO, w6cl 1, w6c12, w6c13, w6cl 4, w6c15,w6cl 6, w6c17, w6cI 8, w6cl 9, w6c20, w6c2 1, w6c22,w6c23, w6c24, w6c25, w6c26, w6c27, w6c28),(w7c1, w7c2, w7c3, w7c4, w7c5, w7c6, w7c7, w7c8,w7c9, w7c 1O, w7c I1, w7c 12, w7c 13, w7c 14, w7c 15,w7c1 6, w7c17, w7c 18, w7c 19, w7c20, w7c2 1, w7c22,w7c23, w7c24, w7c25, w7c26, w7c27, w7c28),(w8cI, w8c2, w8c3, w8c4, w8c5, w8c6, w8c7, w8c8,w8c9, w8cl O, w8cl I, w8cl 2, w8c13, w8cl 4, w8cl 5,w8cl 6, w8c17, w8cl 8, w8cl 9, w8c20, w8c2 1. w8c22,w8c23, w8c24, w8c25, w8c26, w8c27, w8c28),(w9cI, w9c2, w9c3, w9c4, w9c5, w9c6, w9c7, w9c8,w9c9, w9clO, w9cl 1, w9c 12, w9c13, w9c14, w9c 15,w9cI 6, w9c 17, w9cl 8, w9cl 9, w9c20, w9c2 1. w9c22,w9c23, w9c24, w9c25, w9c26, w9c27, w9c28),(w I Oc I, w I c2, w I Oc3, w IOc4, w I Oc5, w I Oc6, w 0c7, w I c8,wi Oc9, wIOclO, wI Oci 1, wIOc 12, wIOcl 3, wlOcl4, wi Oc 15,wI Oc16, w IOc17, wIOcl 8, wlOcl9, wi Oc20, w IOc2 1, wI0c22,wiOc23, wI0c24, wI0c25, wI0c26, wl 0c27, wI0c28),(wl Icl,wl 1c2, wl 1c3, wl 1c4, wl Ic5, wl 1c6, wl 1c7, wl Ic8,wl 1c9, wl IclO, wl Icl , wl Ic12, wl IcI3, wl Ic1. wl Ic 5.wl Ic16, wl Ic17, wl Ic18, wl Ic19, wl 1c20, wl Ic2I, wl Ic22,wl Ic23, wl 1c24, wl 1c25, wl Ic26, wl 1c27, wl 1c28),(wI 2ci, wI 2c2, w1 2c3, w1 2c4, w 12C5, w 12c6, w 2c7, w12c8,w12c9, wl2clO, wl2cl I, wl 2c12, w12c3, w12c14, w12cI5,w12c16, wI2c17. wi2c18, wI2c19, wI2c20, wI2c2 1, wi2c22,w12c23, wl2c24, w12c25, wl2c26, wl2c27, wl2c28},

(w I3c I, w I3c2, w 13c3, w 13c4, w 13c5, w 13c6, w 13c7, w 13c8,w13c9, wl3clO, wl3c1 I w13c12, w13c13, w13c14, wI3ci 5,wI 3c16, wI 3c17, wI 3ci 8, wI 3c19, wI 3c20, w13c2 1, wI 3c22,w 13c23, w 13c24, w 13c25, w 13C26, w 13c27, w 13c28),

(w I4c i, w 14c2, w 14c3, w 14c4, w 14c5, w 14c6, w 14c7, w 14c8,wl4c9, wl4c10, wI4cI w14Ic12, wI4c3, w14c14, w14c15,wI 4c16, wI 4c17, wI 4c 18, wI 4c19, wl 4c20, wI 4c2 1, wI 4c22,w 14c23, w I4c24, w1 4c25, w 14c26, wl 4c27, w I4c28),(w I5c I, w 15c2, w 15c3, w 15c4, w 15c5, w 15c6, w 15c7, w 15c8,wI 5c9, wI 5c 0, wl 5c1 I, w1 5c12, wi 5c13, wi 5c1 4, w I5c15,

211

w15c16, w15c17, w15c18, w15c19, w15c20, w15c21, w15c22,w15c23, w15c24, w15c25, w15c26, w1 5c27, w15c28),(wI6cI, w16c2, w16c3, w16c4, w16c5, w16c6, w16c7, wl6c8,wI6c9, wI6clO, wI6cl 1, wI6cl2, wI6c 13, wI 6c14, w16cl5,w16c16, w16c17, wI6ci 8, w16c19, w16c20, w16c21, w16c22,w16c23, w16c24, w16c25, w16c26, w16c27, w16c28),(w 17c 1, w 17c2, w 17c3, w 17c4 w 17c5, w 17c6, w 17c7, w 17c8,w17c9, wI7ci0, wl7cI 1, w17c12, w17c13, w17c14, wI7c15,w17c16, w17c17, w17c18, w17c19, w!7c20, w17c21, w17c22,w17c23, w17c24, w17c25, w17c26, w17c27, w17c28),(wI8cl, w18c2, w183, w 18c4, w 18c5, w18c6, w18c7, w18c8,w18c9, wI 8c10, wl8cl I, w I8c12, w18c13, w18c1 4, wl8cl 5,w 1 8c 16, w18c 17, w1 8c 18, wI 8c 19, w1 8c20, w 18c2 1, w18c22,w18c23, w18c24, w18c25, w18c26, w18c27, w18c28),(wl9cl, w19c2, w193, w19c4, w19c5, wI9c6, w 19c7, w19c8,w 9c9, wl9clO, wl9cl 1. w19c12, w19c13. w19c1 4, w19c15,w19c16, w19c17, w19c18, w19c19, w19c20, w19c21, w19c22,w19c23, w19c24, w19c25, w19c26, w19c27, w19c28),(w20cl, w20c2, w20c3, w20c4, w20c5, w20c6, w20c7, w20c8,w20c9, w20cl 0, w20cl 1, w20c12, w20c13, w20c 14, w20c15,w20c16, w20c17, w20c18, w20c19, w20c20, w20c21, w20c22,w20c23, w20c24, w20c25, w20c26, w20c27, w20c28),(w2cl, w21c2, w2 Ic3, w2 Ic4, w2 Ic5, w2 Ic6, w2 Ic7, w2 Ic8,w2 1c9, w21clO, w2 cI 1 w2 Ic 12. w2Ic13, w2 Ic 14. w2 Ic15,w2Ic 16, w21c17, w2Ic18, w21c19, w21c20, w21c2 1, w2 1c22,w21c23, w21c24, w21c25, w21c26, w21c27, w21c28),(w22ci, w22c2, w22c3, w22c4, w22c5, w22c6, w22c7, w22c8,w22c9, w22cI0, w22cI 1, w22c12, w22c13, w22c14, w22c1 5,w22c16, w22cl7, w22cl8, w22c1 9, w22c20, w22c21, w22c22,w22c23, w22c24, w22c25, w22c26, w22c27, w22c28},(w23cl, w23c2, w23c3, w23c4, w23c5, w23c6, w23c7, w23c8,w23c9, w23ci O, w23ci 1, w23ci 2, w23c1 3, w23c1 4, w23ci 5,w23ci 6, w2317, w23c18, w23c19, w23c20, w23c2 1, w23c22,w23c23, w23c24, w23c25, w23c26, w23c27, w23c28),(w24cI, w24c2, w24c3, w24c4, w24c5, w24c6, w24c7, w24c8,w24c9, w24ci0, w24c1 I, w24c12, w24c] 3, w24c14, w24c15,w24c16, w24c17, w24c1 8, w24ci9, w24c20, w24c2 1, w24c22,w24c23, w24c24, w24c25, w24c26, w24c27, w24c28),(w25ci, w25c2, w25c3, w25c4, w25c5, w25c6, w25c7, w25c8,

212

w25c9, w25c1 0, w25c 11, w25cl 2, w25c1 3, w25c14, w25c15,w25c16, w25c17, w25c18, w25c19, w25c20, w25c21, w25c22,w25c23, w25c24, w25c25, w25c26, w25c27, w25c28),(w26cI, w26c2, w26c3, w26c4, w26c5, w26c6, w26c7, w26c8,w26c9, w26c1O, w26cI 1, w26cI 2, w26ci 3, w26ci 4, w26ci 5,w26c16, w26c17, w26c18, w26c19, w26c20, w26c21, w26c22,w26c23, w26c24, w26c25, w26c26, w26c27, w26c28),(w27cI, w27c2, w27c3, w27c4, w27c5, w27c6, w27c7, w27c8,w27c9, w27c 10, w27c1 I, w27c12, w27c 13, w27c 14, w27c 15,w27c 16, w27c17, w27c18, w27c1 9, w27c20, w27c21, w27c22,w27c23, w27c24, w27c25, w27c26, w27c27, w27r28),(w28cI, w28c2, w28c3, w28c4, w28c5, w28c6, w28c7, w28c8,w28c9, w28ci 0, w28ci I, w28cI 2, w28c! 3, w28c1 4, w28cl 5,w28c16, w28c17, w28c18, w28c19, w28c20, w28c21, w28c22,w28c23, w28c24, w28c25, w28c26, w28c27, w28c28));

a = ((0, alc2, alc3, alc4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(a2cl, 0, 0, 0, a2c5, a2c6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, a3c3, a3c4, 0, 0, 0, a3c8, a3c9, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, 0, 0, a4c5, a4c6, 0, 0, 0, a4clO, a4cl 1, 0,, 0, 0, 0, 0),(0, 0, 0, 0 , 00 , 0, a5c8, a5c9, 0, 0, a5c12. a5c13, 0, 0, 0, 0),(0 0, 0, , 0, 0, 0, 0, 0, a6clO, a6cl 1, 0, 0, a6c14, a6c15, 0, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, a7c12, a7c13, 0, 0, a7c16, 0),(0, 0 0, 0, 00, 0, 0,0, 0, 0, 0, 0, a8c14, a8c15, 0, a8c17),(0, a9c2, a9c3, a9c4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(alOc1, 0, 0, 0, aIOc5, aIOc6, 0. 0, 0O , 0 , 0 , 0 , 0 , 0 , 0 , 0),(0, al 1c2, al 1c3, al 1c4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(al2c1, 0, 0, 0, a12c5, a12c6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(aI3cl, 0, 0, 0, a 13c5, a13c6, 0, 0, 0, , 0 , 0 , 0 , 0 , 0 , 0 , 0),(0, a14c2, a14c3, aI4c4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, 0, 0, , 0, 0, a15c8, a15c9, 0, 0, al5c12, al5cI3, 0, 0, 0, 0),(0, 0 0, 0, , 0, 0, 0, 0, al6clO, al6cll, 1, 0, a16c14, a16c15, 0, 0),(0, 0 0, 0, 0, 0, 0, a17c8, a17c9, 0, 0, a17c12, a17c13, 0, 0. 0, 0),(0, 0 O, 0, 0, 0, 0, 0, a8cIO, al8cl 1, 0, 0, a18c14, al8c15, 0, 0),(0, 0, 0, 0, , 0, 0, 0, 0, al9clO, al9cl 1, 0, 0, a19c14, a19c15, 0, 0),(0, 0, 0, 0, 0, 0, 0, a20c8, a20c9, 0, 0, a20c12, a20c13, 0, 0, 0, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a21c12, a21c13, 0, 0, a21c16, 0),(0, 0, 0, 0, 0, 0, , 0 , 0 , 0 , 0 , 0 , 0 , a22c14, a22c 15, 0, a22c17),(0, 0, 0, 0, , 0 , 0 , 0 , 0 , 0 , 0, a23c 12, a23c13, 0, 0, a23c16, 0),

213

(0. 0, 0, 00. 0, 0, 0, 0, 0, 0,00 0, a24c14, a24c15, 0, a24c17),(0, 0, 00, 0, 0,0,0,0,0,0,0, a25c14, a25c15, 0, a25c17),(0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, a26c12, a26c13, 0, 0, a26c16, 0),(0, 0, a27c3, a27c4, 0, 0, 0, a27c8, a27c9, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, 0, 0, a28c, a28c6, -1, 0, 0, a28c10, a28cl 1, 0, 0, 0, 0, 0, 0));

aherm - Transpose[Conjugatelall;

x = (Inverselaherm.w.a]).aherm.w.b

Again, the lower case w was required to represent the weighting matrix

W due to the Mathemaica variable naming convention mentioned earlier

(i.e., matrices have variable names which begin with lower case letters since

Mathemaica reserves variable names which begin with capital letters for

built-in functions). The variable "aherm" represents the complex conjugate

transpose of the matrix A (i.e., AH).

This code ran on a Macintosh II computer which is equipped with five

megabytes (MB) of random access memory (RAM) for about 25 minutes

before halting due to an "out of memory" error. This same code was run on a

different Macintosh II computer equipped with similar hardware and a

software package which allowed access of up to eight MB of hard disk space

for use as virtual memory. Thus, Mathematfca had 13 MB of RAM available

to it. On this 13 MB machine, the code ran for about 75 minutes before it

again halted on an "out of memory" error. Before attempting an alternate

approach, the code was run in steps to see which calculation was causing the

trouble. The multiplication of AH,W, and A ran successfully as did the

multiplication of AH, W, and b. The problem recurred when an attempt was

214

made to take the inverse of the product of AH,W, and A. We speculate that

the problem occurs because the program is required to store and operate on

a large number of string variables, thereby requiring large quantities of

memory to store intermediate results. To confirm this, we attempted to take

the inverse of a 17 by 17 symbolic matrix using only simple generic variable

names (such as jlcl) without success. We must conclude that Mathematica

requires too much memory to successfully run this type of symbolic problem

on a personal computer. The methodology should be validated when less

memory intensive or main frame based symbolic programs become

available.

In order to attempt the use of Mathematica's LinearSolve function, we

must first modify the matrix A so that it is a square matrix. This may be

accomplished while maintaining the integrity of our weighted least squares

formulation as follows:

Recall that

Ax -b. (198)

Premultiplying both sides of (198) by the matrix W yields

W A z - W b. (479)

Premultiplying both sides of (479) by the matrix AH yields

A W A z - All W b. (480)

If we let

215

jAHW A. (481)

and

d -AHW b, (482)

then we have reformulated the problem as desired to

jx-d , (483)

where j is the 17 by 17 matrix defined by (481) and d is the 17 by 1 column

vector defined by (482).

The following revised Mathem.uica code was developed for this

reformulated problem:

b - (0,0,0, 0, 0, 0,0, 0,0,0, 0,0, 0,0, 0,0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, GI1, 0);

w - U(wlcl, wlc2, wlc3, wlc4, wlc5, wlc6, wlc7, wlc8,wlc9, wiclO, wicI 1, wlcl2, wlcl3, wlcl4, wlcl5,wlcl6, wlcl7, wlcl8, wlcl9, wlc2O, wlc2l, wlc22,w1c23, wic24, wlc25, wlc26, wlc27, wlc28),(w2cl, w2c2, w2c3, w2c4, w2c5, w2c6, w2c7, w2c8,w2c9, w2cl10, w2cl 11, w2c 12, w2c 13, w2c 14, w2c 15,w2c 16, w2c 17, w2c 18, w2c 19, w2c20, w2c2 1, w2c22,w2c23. w2c24, w2c25. w2c26, w2c27. w2c28),(w3c I, w3c2, w3c3, w3c4, w3c5, w3c6, w3c7, w3c8,w3c9, w3c 10, w3cl 11, w3c 12, w3c 13, w3c 14, w3c 15,w3c 16, w3c 17, w3c 18, w3c 19, w3c20, w3c2 1, w3c2 2,w3c23, w3c24, w3c25, w3c26, w3c27, w3c28),(w4c I, w4c2, w4c3, w4c4, w4c5, w4c6, w4c7, w4c8,w4c9, w4cl10. w4c 11, w4c 12, w4c 13, w4c 14, w4c 15,w4c 16, w4c 17, w4c 18, w4c 19. w4c20, w4c2 1, w4c22,w4c23, w4c24, w4c25, w4c26, w4c27, w4c28),

216

(w5clI, w5c2, w5c3, w5cl, w5c5, w5c6, w5c7, w5c8,w5c9, w5cI1O, w~cI 11, w5c 12, w5c13, w5ci4, w5c 15,w~c 16, w5c 17, w5c18, w5c 19, w5c20, w5c21, w5c22,w5c23, w5c24, w5c25, w5c26, w5c27, w5c2 8),(w6c I, w6c2, w6c3, w6c4, w6c5, w6c6, w6c7, w6c8,w6c9, w6cl10, w6cl 11, w6c 12, w6c 13, w6c 14, w6c 15,w6c 16, w6cI17, w6c 18, w6c 19, w6c20, w6c2 1, w6c22,w6c23, w6c24, w6c25, w6c26, w6c27, w6c28),(w7c I, w7c2, w7c3, w7c4, w7c5, w7c6, w7c7, w7c8,w7c9, w7cI10, w7cI 11, w7c 12, w7c 13, w7c 14, w7c 15,w7c 16, w7c 17, w7c 18, w7c 19, w7c2 0, w7c2 1, w7c2 2,w7c23, w7c24, w7c25, w7c26, w7c27, w7c28),(w8c I, w8c2, w8c3, w8c4, w8c5, w8c6, w8c7, w8c8,w8c9, w8ci10, w8cI 11, w8c 12, w8c 13, w8c 14, w8c 15,w8c 16, w8c 17, w8c 18, w8c 19, w8c20, w8c2 1, w8c2 2,w8c23, w8c24, w8c25, w8c26, w8c27, w8c28),(w9cl, w9c2, w9c3, w9c4, w9c5, w9c6, w9c7, w9c8,w9c9, w9cIO0, w9cl 1, w9c1 2, w9c1 3, w9cI 4, w9c15,,w9cI 6, w9c17, w9c1 8, w9c19, w9c20, w9c2 1, w9c22,w9c23, w9c24, w9c25, w9c26, w9c27, w9c28),(w IOci1, w IOc2, wlIOc3, wlIOc4, w IOc5, w IOc6, wl1c0, wlIc8,wI cM, w IOcl10, w IOcI 11, wlIOcI12, w IOcI 3, w IOcI14, w IOcl15,w I Oc 16, w I 0c 17, w IOc 18, w IOc 19, w I0c20, w I 0c2 1, w 1 0-22,w I0c23, w1I0c24, w I0c25, w1I0c26, w I0c27, w IOc28).(wi Icd, wi 1c2, WI 1c3, wI 1c4, wi Ic0, wi 1c6, wi 107, wlIM8wI 1c9, wi IclO0, wi IcI 1, wlIc 1c2, wi 1c13, wlId 4, w Ic 1c5,wi 1c16, wi IcI7, wi 1c18, wi 1c19, wi lc20, wi 1c21, wi 1c22,wi 1c23, wI 1c24, wI 1c25, wi 1c26, wi 1c27, wi 1c28),(wI 2cl, w12c2, w12c3', w12c4, w12c5, w12c6, w12c7, w12cS,w1 2c9, wI2c1O, wi2cI 1, w12c12, w12c13, w12c1 4, w12c1 5,w 1 2c1 6, Wi 2c 17, Wi 2c1 8, wi 2c1 9, wI 2c20, wl 1Wc 1, wi 2c22,w1 2c23, w12c24, w12c25, w12c26, w12c27, w12c28),(wl3cI, w13c2, w13c3, wI3c4, w13c5. wI3c6, w1 30, w13C8,wI 3c9, wI 3d 0, wI 3d 1, WI 3d 2, WI 3d 3, wi 3d 4, WI 3d 5,WI 3R16, wi 3c17, WI 3d 8, WI 3d 9, wI 3c20, wi 3c2 1, wi 3c22,w13c23, w13c24, w13c25, w13c26, w13R27, wl3c28),(wi I4c 1, w I 4c2, wI 146, w I 4c4, wI 140, w I 4c6, w 1407, w 1 4c8,w I4c9, w I4cl10, wlI4cl 11, w1I4c 12, w1I4c 13, w1I4c 14, w1I4c 15,w I 4c 16, w I 4c 17, w I 4c 18, w I 4c 19, w I 4c20, w I 4c2 1, w1I4c2 2,

217

w14c23, w 14c24, w 14c25, w 14c26, w 14c27, w 14c28),(wI 5cI, w1 5c2, w15c3, wl 5c4, w 5c5, w 15c6, w 15c7, w15c8,wI5c9, wI5cIO, wl5cl 1, w15c12, w15c13, w15c14, w15c15,w15c16, w15c17, w15c18, w1 5c 19, w 15c20, w15c21, w1 5c22,w 1 5c23, w I5c24, w I5c25, w I5c26, w 1 5c27, w 15c28),(wl6cl w16c2, w16c3, w16c4, w16c5, w16c6, w16c7, w16c8,wl 6c9, wl 6c10, wI 6c 11, w16c12, w16c13, w16c14, w16c15,w16c16, w16c17, w16c18, w16c19, w1 6c20, w16c2 1, w1 6c22,w1 6c23, w1 6c24, w I6c25, w1 6c26, w1 6c27, w1 6c28),(wl7cl, w17c2, w17c3, w17c4 w17c5, w17c6, w17c7, w17c8,w17c9, wl7clO, wI7cI I, w 17c 12, wl7cl 3, wl7cl 4, wI7ci 5,w17c16, w17c17, wI7c18, w17c19, w17c20, w17c21, w17c22,w 17c23, w17c24, w 17c25, w 17c26, w 17c27, w 17c28),(w 8c1, w18c2, w18c3, w18c4, w18c5, w18c6, w18c7, w18c8,w18c9, wl8clO, wI8c 11, w18c12, w18c13, wl8cI 4, wl8cl 5,w18c16, wI 8c17, w I8c18, w18c19, w18c20, w18c2 1. wI 8c22,w1 8c23, w 8c24, w1 8c25, w1 8c26, w18c27, w1 8c28),(wl9cI, w19c2, w19c3, w19c4, w19c5, w19c6, w19c7, w19c8,w19c9, wI9cIO, wl9cI 1, wI9c12, w19c13, wI9c14, w19c15,wI 9c16, w19c17, w19c18, w19c19, w19c20, w19c2 1, w19c22,wl 9c23, w 19c24, w 19c25, w19c26, w 19c27, w1 9c28),(w20cl, w20c2, w20c3, w20c4, w20c5, w20c6, w20c7, w20c8,w20c9, w20c 1O, w20cl 1, w20cl 2, w20cl 3, w20c1 4, w2Ocl 5,w20c16, w20c 17, w20c18, w20cI9, w20c20, w20c21, w20c22,w20c23, w20c24, w20c25, w20c26, w20c27, w20c28),(w2 Icl w21c2, w21c3, w21c4, w21c5, w21c6, w21c7, w2 Ic8,w2 1c9, w2Ici0, w2Icl I, w21c12, w21c13, w21c14, w21c15,w21c16, w21c17, w21c18, w21c19. w21c20, w21c21, w21c22,w2 Ic23, w2 Ic24, w2 Ic25, w2 1c26, w2 Ic27, w2 1c28),(w22ci, w22c2, w22c3, w22c4, w22c5, w22c6, w22c7, w22c8,w22c9, w22c1O, w22c1 1, w22cI 2, w22cI 3, w22ci4, w22c1 5,w22c1 6, w22ci 7, w22c1 8, w22ci 9, w22c20, w22c2 1, w22c22,w22c23, w22c24, w22c25, w22c26, w22c27, w22c28),(w23cI, w23c2, w23c3, w23c4, w23c5, w23c6, w23c7, w23c8,w23c9, w23c1 0, w23c 11, w23c 12, w23c1 3, w23ci 4, w23c1 5,w23c16, w23c17, w23c18, w23c19, w23c20, w23c21, w23c22,w23c23, w23c24, w23c25, w23c26, w23c27, w23c28),(w24ci, w24c2, w24c3, w24c4, w24c5, w24c6, w24c7, w24c8,w24c9, w24ci0, w24cl 1, w24c1 2, w24ci 3, w24c14, w24ci 5,

218

w24c 16, w24c 17, w24c1IS, w24c 19, w24c20, w24c21, w24c22,w24c23, w24c2 4, w24c25, w24c26, w24c27, w24c28),(w25c1, w25c2, w25c3, w25c4, w25c5, w25c6, w25c7, w25c8,

4V w25c9, w25c10, w25c1 1, w25c1 2, w25c1 3, w25c1 4, w25c1 5,w25cl6, w25c17, w25c1 8, w25c19, w25c20, w25c2 1, w25c22,

.0 w25c23, w25c24, w25c25, w25c26, w25c27, w25c28),(w26c1, w26c2, w26c3, w26c4, w26c5, w26c6, w2607, w26c8,w26c9, w26c1 0, w26c1 1, w26c1 2, w26c1 3, w26c1 4, w26c1 5.w26c1 6, w260 7, w26c] 8, w26c1 9. w26c20, w26c2 1, w26c22,w26c23, w26c24, w26c25, w26c26, w26c27, w26c28),(w27c1, w27c2, w27c3, w27c4, w27c5, w27c6, w2707, w27c8,w27c9, w27c 10, w27c1 11, w27c 12, w27c 13, w27c 14, w27c 15,w27c 16, w27c 17, w27c 18, w27c 19, w27c20, w27c2 1, w27c2 2,w27c23, w27c24, w27c25, w27c26, w27c27, w27c28),(w28c1, w28c2, w28c3, w28c4, w28c5, w28c6, w28c7, w28c8,w28c9, w28c1 0. w28c1 1, w28c1 2, w28c] 3. w28c1 4, w28c1 5,w28c16, w28c17, w28c1 8, w28c1 9, w28c20, w28c2 1, w28c22,w28c23, w28c24, w28c25, w28c26, w28c27, -w28c28));

a - ((0, alc2, alc3. alc4, 0, 0, 0, 0,0, 0, 0, 0,0, 0, 0,0, 0),(a2cl, 0. 0, 0, a2c5, a2c6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, a3c. , a3c4, 0, 0,0, a3c8, a3c9, 0 0, 0, 0, 0,0, 0,0 J,(0, 0, 0, 0, a4c5, alc6, 0, 0,0, aicl10, a4cl 11, 0, 0, 0, 0, 0, O'l(0, 0, 0, 0, 0, 0, 0, a5c8, a~c9, 0, 0, a5c 12, a5c 13, 0, 0, 0. 0 ).(0, 0, 0, 0, 0, 0, 0,0, 0, a6c 10, a6cl 1,.0, 0, a6cl14, a6c 15, 0,0),(0, 0, 0,0, 0,0, 0,0, 0, 0,0, a7c 12, a7c 13, 0, 0, a7c 16, 0 ),(0, 0, 0,0, 0, 0,0,0, 0, 0,0, 0,0, a8cI14, a8c 15. 0, a8c 17),(0. a9c2, a9c3, a9c4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(al~ci, 0, 0, 0, albc5, alOc6, 0, 0,0, 0, 0,0,0, 0, 0,0, 0),(0, al 1c2, a] 1c3, al 1c4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(a I2cl1, 0, 0, 0, a 1205, a 12c6, 0, 0, 0,0, 0,0, 0, 0, 0,0,0),(a I3cl1, 0, 0,0, a 1305, a 13c6, 0, 0,0, 0, 0,0, 0,0, 0, 0,0),(0, a I c2, a 146, a I4c4, 0, 0,0, 0, 0,0,0, 0, 0,0, 0, 0, 0),(0, 0, 0, 0, 0, 0, 0, a15c8, a15c9, 0, 0, a15c]2, al5c13, 0, 0, 0, 0),(0,0, 0,0, 0, 0,0, 0,0, al6clO, al6cl 1, 0, 0, a16c14, al6c15, 0, 0),(0, 0, 0, 0, 0, 0,0, a17c8, a17c9, 0, 0, a17c12, a17c13, 0, 0. 0, 0),(0, 0, 0,0, 0,0, 0, 0,0, aI8cIO, aI8cI 1, 0, 0, al8c14, a18c15, 0, 0),(0, 0, 0,0, 0, 0,0, 0, 0, al9clO, al9cl 1, 0, 0, a19c14, a19c15, 0, 0),(0, 0, 0, 0, 0, 0, 0, a20c8, a20c9, 0, 0, a20c 12, a20cl13, 0, 0,0, 0),

219

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a21cl2, a21cl3, 0, 0, a2lcl6, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a22c14, a22c15, 0, a22c17),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a23c12, a23c13, 0, 0, a23c16, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a24c14, a24c15, 0, a24c17),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 00, 0, a25c14, a25c 15, 0, a25c17),(0, 0, 0, 0, 0,00,0,0,0,0, a26c12, a26c13, 0, 0, a26c16, 0),(0, 0, a27c3, a27c4, 0, 0, 0, a27c8, a27c9, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, 0, 0, a28c5, a28c6, -1, 0, 0, a28ci0, a28ci 1, 0, 0, 0, 0, 0, 0)):

aherm - Transpose[Conjugatefall;

j = aherm.w.a;

d - aherm.w.b;

LinearSolve[j,dl

This revised code ran on the five MB Macintosh for about 25 minutes

before halting on an "out of memory" error. It also ran on the 13 MB

Macintosh for about 110 minutes before halting on the memory error. These

failures have led us to conclude that the solution of the general problem is

possible with this technique but is not practical with currently available

hardware/software configurations.

One additional test was run using the code generated for the general

case employing Mathemalca 's LinearSolve function. In this test case, the

three media waveguide with plane, parallel boundaries was simulated by

setting the weighting matrix, W, equal to the identity matrix (using the

Mathematica command: w - IdentityMatrix[281;), setting appropriate values

of the A matix equal to zero, and directly assigning values to the AH matrix

as follows (in Mathematica code):

220

aherm - ((0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),

(alc2CC, 0, 0, 0, 0, 0, 0, 0, a9c2CC, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0),(a I c3CC, 0, a3c3CC, 0, 0, 0, 0, 0, a9c3CC, 0, 0, 0, 0, 0, 0,0, 0, 00, 0, 0, 0, 0, 0, 0, 0, a27c3CC, 0),(alc4CC, 0, a3c4CC, 0, 0, 0, 0, 0, a9c4CC, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a27c4CC, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. 0, 0,0, 0, 0, 0, 0, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),(0, 0, a3c8CC, 0, a5c8CC, 0, 0, 0, 0, 0, 0. 0, 0, 0, a l5c8CC,0. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a27c8CC, 0),(0, 0, a3c9CC, 0, a5c9CC, 0, 0, 0, 0, 0, 0, 0, 0, 0, a15c9CC,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a27c9CC, 0),(0, 0, 0, 0, 0, 0. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 00, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),(0, 0, 0, 0, a5c12CC, 0, 0, 0, 0, 0, 0, 0, 0, 0, al5cl2CC, 0, 0,0, 00, 0, 0, 0, 0, 0, 0, 0, 0),(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0),

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0)),

where the notation CC in each of the variable names indicates that the

element represents the complex conjugate of the appropriate element of the

A matrix.

221

This code ran on the five MB Macintosh for ten days without converging

to a solution. This additional failure confirms our suspicions that the general

problem is not practically solved using current technology. t

222

V. CONCLUSIONS AND RECOMMENDATIONS

The main purpose of this thesis was to obtain the symbolic solution of a

multilayer (four fluid media) ocean waveguide problem. The waveguide was

assumed to have depth-dependent ambient density and sound-speed

profiles in all fluid media, and arbitrarily shaped boundaries between all

fluid media. A system of 28 equations in 17 unknowns was generated by

satisfying all of the boundary conditions (including the boundary condition

at the source) in cylindrical coordinates. A weighted least squares estimation

technique was employed to formulate a symbolic solution to this

overdetermined (more equations than unknowns) case. A computer program

capable of performing symbolic algebra was sought to minimize the number

of assumptions required to be made, thereby maximizing the generality of

the solution obtained. Mathematica (version 1.2.1 f33 (enhanced)) for the

Macintosh computer was selected for this work due to its availability at the

Naval Postgraduate School and its symbolic algebra capabilities. Mathematica

code was developed which programmed the weighted least squares

estimation technique for the most general case. Unfortunately, this code was

unable to provide a solution to the most general case due to software and

hardware limitations (i.e., speed and random access memory problems).

By relaxing the arbitrary boundary shape assumption, Mathematica

code was developed which programmed a direct solution to the three media

waveguide problem for plane, parallel boundaries. This code ran

successfully, and provided results which could be verified by direct

223

comparison with the known solution to this classical (i.e., three fluid media

with plane, parallel boundaries and constant ambient density and sound

speed in each medium) waveguide problem. During this process, built-in

Mathematica functions were used in an attempt to simplify the resulting

symbolic expressions. This effort revealed that these built-in functions

lacked sufficient sophistication for applications of this complexity. This lack

of sophisication resulted in manual reduction of the program output so that

verification was possible. Mathemaica code was then developed to solve the

four media waveguide problem for plane, parallel boundaries. This code also

ran successfully and yielded results which could be verified using known

classical waveguide solutions when some judicious assumptions were made

to mathematically eliminate the fourth medium.

In addition to validating the symbolic solution technique, this thesis

provides a series of generic expressions for the unknown constants for each

of the three and four media waveguide problems with plane, parallel

boundaries. Each of the generic expressions is a combination of generic

variables whose definitions are provided in the text. Each of these generic

variables can be programmed in a high level language (i.e., FORTRAN) as a

unique subprogram or function. In this manner, the unknown constants can

be calculated by combinations of calls to appropriate subroutines. This

modular programming technique is enhanced by the fact that each of the

generic expressions has a common denominator, which can also be

programmed in a similar manner.

224

The generic results from the four media waveguide problem for plane,

parallel boundaries should be programmed in FORTRAN. When this

programming has been completed, the standard test cases (i.e., three fluid

media with plane, parallel boundaries and constant ambient density and

sound speed in each medium) should be run on the new code to verify the

results. This verification will provide additional credibility to Mathematica's

output.

It is recommended that the most general case (i.e., four fluid media

with arbitrarily shaped boundaries) be attempted again when one of the

following conditions are met:

• a Macintosh computer with more than 13MB of RAM becomes availableat the Naval Postgraduate School,

• an advanced version of Mathematica is released, or" another computer program with symbolic algebra capabilities becomes

available for use on a workstation or main frame computer.

225

LIST OF REFERENCES

Boas, M. L., Mathematical Methods" in the Physical Scinceff Second Edition,John Wiley & Sons, Inc., 1983.

Berkey, D. D., Cacuu, Second Edition, Saunders College Publishing, 1988.

Haykin, S. S., Adaptive Filter Theory Prentice-Hall, 1986.

Kinsler, L. E., et al., Fundamentals of Acoustics, Third Edition, John Wiley &Sons, Inc., 1982.

Leithold, L., The Clculu_7 With Analytic Geometry, Second Edition, Harper &Row, 1972.

Menke, W., Geophy.siczl Data Analysis': Discrete Inverse Theor, AcademicPress, Inc., 1984.

Officer, C. B., Introduction to the Theory of Sound Transmission, pp 124-127,McGraw-Hill Book Co., 1958.

Ziomek, L. J., Fundamentals of Acoustic Field Theory And Space-Time SignalProcessing, in progress, Aksen Associates, Publishers, 199 1.

226

INITIAL DISTRIBUTION LIST

1. Defense Technical Information Center 2Cameron StationAlexandria, VA. 22304-6145

2. Library, Code 52 2Naval Postgraduate SchoolMonterey, CA. 93943-5002

3. Professor Lawrence J. Ziomek, Code EC/Zm 2Department of Electrical and Computer EngineeringNaval Postgraduate SchoolMonterey, CA 93943-5000

4. Professor Hung-Mou Lee, Code EC/LhDepartment of Electrical and Computer EngineeringNaval Postgraduate SchoolMonterey, CA 93943-5000

5. Mr. Thomas MartinUndersea Warfare Program OfficePRC, Inc.1555 Wilson Blvd.Arlington, VA 22209

6. Dr. Richard SeesholtzIUndersea Warfare Program OfficePRC, Inc.1555 Wilson Blvd.Arlington, VA 22209

7. LCDR Charles J. Young, Jr.CommanderPortsmouth Naval ShipyardPortsmouth, NH. 03804-5000

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NAVAL POSTGRADUATE SCHOOL Monterey, California · Approved bN . -/Od # Aprvdb. _ Lawrence ijo ......

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