AD-AIOO 805 AIR-FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOO--ETC F/6 8/3PLANE TIDAL WAVES GENERATED BY AN ARRAY OF SIMULTANEOUS UNDERWA-ETC(U)MAR 81 H J WEBER

UNCLASSIFIED AFIT/GNE/PH/81M-11 NLEhEE/I/EEE//EEIIIEEEEIIIIIEEIIIIwIIIII

AXFIT/GNE:/1111/8 II- I I

Acces sion For

NTIS GRA&IDTIC TAB r]Unannounced fjustification

ByDILtribution/_

Availability Codes~Avail and/or

Dist [Special

AN ARRAY OF

SIMIJLTANLOUS UJNDERWATER EiXPLOS ION'S

Henry -J.. Weber/ AFIT/GNE/P[I/81M- 11 CAPT USAF

J S L 1 981

D

Approved for Publ ic Re lase; Di st ribut ion linIirni ted

AFI T/GNl/IPII/8 II- I I

PLANI: TIDAL lIVAV I : liNEl .Vl'l1) I Y

AN ARRAY OP

S IMULTANEOIS JNL)ER'. ATFiR IiXPI.OSIONS

TIlESIS

Presented to the Faculty of the School of E~ngineering

of the Air Force Institute of Technology

Air University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

by

Henry J. Weber, B.S.

Captain USAF

Graduate Nuclear Engineering

March 1981

Approved for Public Release; Distribution Unlimited

Acknow edgem ent

I thank George 1-. Nickel for his previous work on this

problem and his guidance as my thesis advisor. Thanks to

Mike Stamm for his assistance and the use of his printer-plot

program which provided fast contour printer-plots of the

surface waves. I acknowledge C.. Ilirt's contribution of

providing research information from the Los Alamos Scientific

Laboratory. In addition, I am indebted to my wife for her

assistance and understanding.

Henry J. Weber

(This thesis was typed by Sharon A. Gabriel)

ii

Con Ien ts

Page

Acknowledgement----------------------------------------- ii

List of Figures----------------------------------------- iv

Abstract------------------------------------------------- v

T. Introduction--------------------------------------1I

Background--------------------------------------1IProblem and Scope------------------------------- 1IAssumptions------------------------------------- 'Approach and Sequence-------------------------- 2

IT. Analysis of Problem------------------------------ 4

Rationale for Shallow Water Theory -------------- 4Parameters Re ference Values-------------------- 5Initial Condition------------------------------- 7

Ill. Computer Prograin Development-------------------- 10

Derivation of Equations------------------------ 10Difference Scheme------------------------------15sBoundary Condition----------------------------- 18Verification----------------------------------- 18

IV. Discussion and Results--------------------------- 20

Yield Sensitivity------------------------------ 22Spacing Sensitivity---------------------------- 22

Alignment~~~~ ~ ~~~ Sestvt9----------- -Almigmensensvitivit-----------------------------7

Bottom Slope Sensitivity----------------------- 27Non-Linear Effect------------------------------ 30

V. Conclusion---------------------------------------- 33

VT. Recommendation----------------------------------- 34

Bibliography------------------------------------------

Vita--------------------------------------------------- 3

List of Figures

Figure Page

I Illustration of Coordinate System

and Variables -------------------------------- 11

2 Illustration of Plane Tidal Wave ------------ 21

3 Yield Sensitivity--------------------------- 3

4 Yield Combination Sensitivity---------------- 24

S Spacing Sensitivity------------------------- 2

6 Alignment Sensitivity----------------------- 26

7 Timing Sensitivity-------------------------- 28

8 Bottom Slope Sensitivity-------------------- 29

9 Non-Linear Contour Printer-Plot ------------- -1

10 Linear Contour Printer-Plot ----------------- 32

iv

AFI'r/(;Nli/PI/8 1 \,- 11

Abst rnut

This report considers an array of simultaneous under-

water nuclear explosions generating plane tidal waves on

the continental shelf. Computer simulations are used to

study this wave generation process. They are based on non-

linear shallow water theory with an initial condition of a

stationary raised cone of water. The sensitivity of the

wave height to different yields, combination of different

yields, spacing, alignment, timing, and bottom slope is

determined. The reference values used for this study are a

yield of five kilotons TNT, one kilometer spacing, water

depth of 100 meters, and no bottom slope. In addition to

the sensitivities, the amount of non-linear effect is illus-

trated by comparison of wave contour printer-plots generated

by linear and non-linear shallow water theory. The computer

results indicate the wave generation is relatively insensitive

to realistic variations in the parameters.

v

PI.:ANI ''IiI):I. WAVVS IINI;RATIIID BY

AN ARRAY OF

SIMULTANIOUS UNDIIWATIR EXPIOS[ONS

I. Tntroduction

Background

This thesis originated from the AI'l GNl-80M class

design project which analyzed a shallow underwater mobile

(SUM) basing alternative to the land based NIX intercontinental

ballistic missile weapon system (Ref 1). The SUMI system would

be deployed over the continental shelf of the United States.

This large area of deployment with the location of the SUI, s

unknown would require a large number of warheads to defeat the

weapon system. However, an easy defeat of the SUM based weapon

system has been proposed which uses a plane tidal wave generated

by an array of simultaneous underwater explosions. The tidal

wave would travel across the continental shelf, disabling the

SUMs. Computer simulations based on shallow water theory verify

the formation of a plane tidal wave. The wave energy resulting

from the underwater explosion was approximated with an initial

condition of a stationary raised cone of water. Further study

of this plane tidal wave generation is the topic of this thesis.

Problem and Scope

The primary purpose or this study is to determine the

sensitivity of the plane tidal wave ,eneration to the explosions'

yield, vield combinations, sp1acing between explosions, align-

ment of explosions, timing of explosions, and slope of the

continental shelf. The reference values used are five kilotons

TNT for yield, one kilometer spacing between explosions, and

a water depth of 100 meters. The yield is varied from 2 to 8

kilotons. The combination of two yields is varied as a ratio

of one yield to the other yield from .25 to 1 with an average

yield of S kilotons. The spacing is varied ±409 from the

reference value of one kilometer. The misalignment is varied

from 0.0 to 0.S kilometer and the timing is varied from 0.0 to

5 seconds, The bottom slope is varied from 0.0 to 0.002.

In addition to the sensitivities, the amount that the non-

linear effects contribute to the generation of the plane tidal

wave is illustrated.

Assumpt ions

There are two ma ill assumpt ions. The first is that an

explosion on the ocean floor whica produces a large enough gas

bubble for blowout will generate shallow water waves. The

second assumption is part of shallow water theory. It assumes

the water's vertical acceleration doesn't affect pressure and

the water's vertical velocity doesn't affect the conservation

of momentum equation.

Approach and Sequence

The general approach starts by analyzing the problem

physically to determine the type of wave generated, realistic

parameter reference values, and modeling of initial cond it ion

Next , the di f ferenc ing scheme i s deve 1 oped by der iv int the

shallow water equations and d i fferenc i ng the equat ions. The

scheme is used to simulate the wave generation. The wave

heights are evaluated to determine sensitivity to several of

the parameters. Finally, the non-linear effects are illus-

trated.

. . .. . . . i, I H .. . . .. . . . .3

SI. Ana I v s is of Iroh I C

The analysis of this problem Focu ses on thrce areas.

First is the rationale and basis for using shal low water

theory. Next is the determinat ion of the reference values

for the independent variables and their range of variation.

The last area is thle initial condit ion and its relationship

to the explosions' yield.

Rat ion" le for Shal low Water Theory

The rationale for using shallow water theory is primarily

based on the type of cavity formed. Thaillow water theory can

he used if the total depth of the water moves horizontally

together. The explosions used for this problem generate a

gas bubble with a radiLIS greater than the depth of water.

Since the expanding bubble wili horizontally excite the total

depth of the water, shallow water waves are expected. On this

basis, shallow water theory is used.

Research at Los Alamos Scientific Laboratory (LASL) found

that shallow water theory was not applicable for a near surface

explosion (Ref o). However, the difference in burst location

can account for this. The near surface burst used by LASL

formed a hemispherical surface cavity with a radius equal to

1/6 the depth of the water. The surface cavity would not hori-

zontally excite the total depth of the water at the same time.

Therefore, shallow water waves are not expected, which is

consistent with shallow water theory not accurately duplicating

the experimental data.

Parameters Reference \alues

TI rtf erence Values of 1hle iJ)dC])UI1dfl-1t )araitle't rs and

range of variation were selectel hased on many factors. The

water depth and bottom slope are based on the average depth

and slope oF the continental shelf. The yield was selected

based on wave generation efficiency and the minimum yield

to ensure blowout. The spacing was I imi ted by the accuracy

of the bomb placement. The alinment and timing are assumed

perfect for the reference value.

Depth and Slope. The reference water depth is about the

average of the continental shelf's depth (0 - 000 ft), I he

water depth reference value is 100 meters with no variation

except when bottom slope is used. The average bottom slope

of the continental shelf is .0002 to .0004 based on a width

of 300 to 600 miles. The reference value for bottom slope is

zero with a variation of 0.0 to .002.

Timing and Alignment. The reference value for timing is

zero seconds with a range of variation of 0 to S seconds.

The reference value for alignment is also zero with a range

of variation of 0 to 500 meters. Misalignment distance is

the perpendicular distance from the array centerline to the

misaligned explosion.

Spacing. The spacing reference value is limited at the

lower end by the accuracy of the warhead placement. According

to Scientific American, the 1985 estimated circular error

- i .. . - -, " ' ' . . . . . . ... . .. . .[] 1 .. . . . .. .. . . . '.. .. . - ..... I il m

p)rohb It, (CFP) f-or 11uss i ;ininmi ss i I es i 22 k i 1onie r (Ref 2:5S4)

The reference -spac i no ..;Is I im i te d to a d ist allcc of V*CFP.

Based onl this ,tile re ference va Inc of' one k i 1 orieter i s used

wi th a variat i on of- . to 1.14 k i Iloijet er.

Yiel Id. T[le reCfereceI 'i(cld ( Y) is based on the efficiency

of wave format ion and thle mii lium yield requiired for a blowout

case. [Tie efficiency as a funjction of' yield is- indicated by

thle wave he ight. T[he wave he i lit 1I) isPrp tinait ,I/6b

(Ref 5:133). Therefore, tile lowest possible yield will be used

to obtain the hig~hest efficiecncy.

The in i imum1 y i eld re~U i red to produILce a maximum hem 1 5per -

ical bubble with a raldiuis equial to t he wateri depth of' 100 met ers

Is 17kiloons. This is based onl thle fol lowing ajiproxi mt ion-

and calculations. The energy reu 'rd to ior thle hemi spheri cal

bubble i s approxiniat ed by the energy requ ired to form an eql

voIlme spherical bubble with coinciding cent roids. Approximately

half of the explosion's energy is absorbed by thle ocean floor

(Ref 5:126) . U~sing these approx imiat ions and FH' (1I) for thle max i-

mum bubble radiuIs (Ref S:113), thle minimuili yield is determined

as fol lows. 1, is thle maUx bubble rad ins, Y is thle yield, 1)

is thle depth of burst , R s is tile ra1dius Of thle sphe)Crical

bubble, and R1 is thle radius Of tile 1be1isphr ica bubble.

I~~(ft) 13. hs)) 1/3(1

l.Lm ~ ~ n)) k-- +- I- o2

Y(kt) ([)(m) + 10JJ .-09 (3)

Rh - 100 meters

R I- R ) = 79.4 meters

To obtain the minimum yield, use Eq (3) with L, Rs

D equal to the depth of the heinisphere's centroid, and1

Y = Y to account for the energy absorbed by the bottom.

Y(kt) (62.5 + l0)(79.))

Y = 1.7 ktons

Therefore, the yield should be equal to or greater than the

minimum yield of 1.7 kilotons. A reference yield of 5 kilotons

was selected since it is small and yet more than twice the

minimum yield to ensure shallow water waves.

Initial Condition

The initial condition used to represent the wave energy

imparted by the explosion is very simplistic. It consists of

a raised stationary cone of water. The cone surface has a

constant slope or 0. 2 %ith it . vol tile dependent on the

yield.

The relationship between the yiild and the potential

energy of the cone is based on the scaling laws for the

blowout case (Ref 5:132-133). In the blowout case, the

wave height is scaled by Y Since the potential energy

of the wave is proportional to the wave height squared, it

is scaled by Y Therefore, the potential energy (Pl

of the cone is equal to some constant times Y1/3

pio = Const*Y 1/3 (4)c o11 e

The constant is obtained by analyzing the potential

energy of the cavity. Starting with the minimuiii yield of

1.7 kilotons, the maximum bubble radius (L) is 100 meters.

Assume the water displaced by the bubble is now located at

the surface. The amount of increase in potential energy is

determined as follows: where o is water density (1000 kg/m 3

and g is the acceleration of gravity (9.8 m/s 2 )

1 4 3 12APE = (62.5)(- .- T L ) pg 1.28*10 Joules (S)

As the bubble collapses, much of the potential energly is

lost in turbulence. One half of the wave energy is assumed

to be lost in turbulence. Based on this assumption, the constant

is determined as follows:

2 I-A I 2(1. 28*10 1 1

ro n s t - 2 -. 8 1 >J ) = . * 1 1 1 (6

C¥ st -l3 i- - -- . 4 *10 (6)

PE= 5.4*1 (11 Y1/3(kt) (7)

The size of the cone is determined by the amount of wave

energy imparted by the explosion. The potential energy of

the cone of water is set equal to the energy obtained from

Eq (7). Solving for the height (h) of the cone's peak

determines the cone's size with a constant slope (s) of 0.2.

h4 ()

12 s(8)

1 (9)h~m) [(S.4*1o 11 )y1/3 12 sj]1/4(9

gP

h(m) = 54 Y1/12 (kt) (10)

9

Ji

IlI. Computer Ilrograi fevelop lcnt

The computer program developmJent includes the derivations

and differencing of the equations and treatment of the boundary

conditions. Both the non-linear and linear equations will be

derived and differenced. fn addition, a check of the differ-

encing scheme is accomplished.

Derivation of Equations

The shallow water wave equations are derived from the

conservation of mass and conservation of momentum equations

using the 3-dimensional rectangular coordinate system. The

viscosity and compressibility of the water are ignored. The

coordinate system is as shown in Figure I with the free-standing

water surface at z = 0 The actual water surface position

is described by li(x,y,t) and the bottom is described by

b(x,y) The averaged velocity components u , v , and w

are in the x , y , and z directions, respect ively, with

representing the velocity vector (u,v,w)

The derivation starts with the conservation of mass equat ion

(Ref 4:2,24) and its integrat ion with respect to z

3 + p V. 0 (1)

u + v + I - (12)5x yz

Ortc i ir nc

Eigtrc .IIrus ra ionof oor in~~ Sste

I, (x t

LI u d3 ,v d ,(13)f a - d - + I d -z + ,,, Cb b b

The boundary conditions are used to determine w for

z H and z b The values of w at these points are

used to expand the integral of the conservation of mass

Eq (13).

all + u all AllWz=H a + U 5X + V (14)

b+ v 1bWz= b =U Y'x" + 7 V(15

f u dz + f ' dz + l +b b

+ v---y uV - v - = 0 (16)

3 11 + f u '-6 l!b5-- ( dz + ' d: + uat b b - a x

+ y - 0 (17)

Now the first approximation must be made for the shallow

water theory. The vertical acceleration of the water is

assumed not to affect the pressure II1) (Ref 8:23-24). This

results in hydrostatic pressure, P) = gp(lI-z) where

b - z _ I! .I Hydrostatic pressure provides a horizontal

12

pressure grad ient whi i i s i nide pCndetI I of z Therefore,

the horizontai accel erajt iokIs ad yelo 1c it ies are independent

of z IVith this alp proximat ion, the integration of the

conservat ion of mass equat ion ( 17) i completed.

I_ I u V -

+ Il-b) + (l-hi 3 - 0

+ x + ,o (18)

+ [u(Il-b) + [ [v(1I-h) J = 0 (19)

The conservat ion of moiicntun equat ion (Ref 4:3,24) is

used to derive the velocity equations. Th1e previous approxi-

mation is expanded to accomplish this. The vertical velocity

w is assumed to have a negligible effect on the conservation

of momentum. Therefore, the velocity vector S becomes (u,v)

p u + p V-u S + (20)

,31 itu 2 Duv l+ + - 0 (21)at 3y f lx

'T'he equ( at ion For v is obtained simi larly.

v )X' 2v II+ + , -(22)

13

Equation. (19), (21) and (22) arc the non-linear

shallow water equations. 'he 1 inear shal low water equations

are obtained by asSuming u , v , If , and their derivatives

are small enough so their squares and products can be neg-

lected (Ref 8:24). This results in linear equations, where

-b is simply the depth of the water.

all1 + (-b) + v(-b) = (23)

+ - 0 (24)

D V+ g = 0(25)

If the water depth is assumed constant, the variables

u and v can be eliminated as follows to obtain the linear

wave Eq (30) (Ref 8:25).

- b u 1) : v 0 (26)

D 211 ) 2- U b) 5- v V 0 (27)D -t 2 j .) t a y a 5 t

23 u 1 l-- (28)

- U

D3 x2 = 12211 (8

T 292"v 3 2(2)

1 4

+ + = (30)

This shows that the characteri St ic 1JvC velocity is 'g(-bT

Difference Scheme

The second order explicit difference scheme is three

levels in t ime and space, conservat ive , and has an ampl i f i -

cation factor of unity. Correct space and time centering

provide an amplification factor of unity if the Courant-

Friedrich-Lewy criterion is satisfied (Ref 7:288). This

criterion is uy-t- + vt < I Using the charac-

teristic velocity equally distributed between u and v

the criterion value is 0.44 for At = .5 sec and Ax = Ay

= 50 meters . The difference equations are based on the

integral conservation of mass Fq (19) and the conservation

of momentum Eqs (21) and (22)

S[u(1L-h)j - f Vt(l-b) 1 (31)t -x 1) ,3u -)u ;wv - H1 (32

)v :v2 J 11l- y - ---- - g , (33)

Kinematic viscosity can be added to the last two

equations to account for viscosity or to help prevent

instability. The kinematic viscosity is incorporated by

is

add ing to thc i Lht ide of I ts ( 2) and (33) a k iI)n at ic

viscosity coefficient, mull ipl icd h, t1e I pac ian of u

and v , respect ivel. A slight diffut sion couplingi might

also be added to keep the independent conse rvat ive Flows

in step. This is done by adding the product of a coupling

coeffic ient and the Laplacian of If to the right side of

Fq (31).

The difference equations use the indices i ,j

and n for the variables x , y , aind t , 'espectively.

The equat ions are second order in t ime and space with no

kinematic viscosity or dirffusion coupling as shown below.

l,+ = ,1,-. " n 1}- , - b

Ax [',.+1Ax , 1 ,j 'i+l ~,)

n In {- uA1. Tj i'

-v1H~ i,j -1 i 1)]

n+l 11I" t 11 2 Fr i

+ i J I J A -L,.- , i,.j+l vi,j+l u. j 1

(35)

lo

i+ -I ,

[Lll V ( I Ij i(

The I i,.la d i terencc equat i i s based oil t he Ii na r

wave Eq (30). It is second order with space and t inie

centering.

--- H = - gb + (37)t" Lx 3 y-

21l 1)ii>. = 2 Ii) ,- I

, A [ 1111 + .

- gh )> E ''If - , II'! + I' i (8Ay)'- ~ lj l~ - (38)

The d i f fe renc e sc heme used a standard rectangular mesh

with all the point values calculated for each time step.

Symmetry i,,as used as much as possible wvith the aid of

reflecting boundaries.

17

k(o) ipdda CO 11 id i t 10on1S

Ref lect iig boundary coid it ions a rc UsLLd in t l1c progrFaIll

to take advant age of symmet rv (Re 7:288 -29)1 tit h

reflect ing bounda ries, t he lhounda ry cond it ion.s are set up

to ensure no mass fow ac l)ss the o 1Ilda ry .

Sirst , the veloc it)' coIponlIt s plerpend ic I ar to the

boundary are set to zero at the boundary and both elocity

comiponenlts are zero at the co'nie's. The perpendicular

velocity colmponents just outside the houndary are set equal

to the negative of the perpendicular components _just inside

the houndary. The parallel velocity components just outside

the boundary are set equal to tile parallel velocity components

just inside the boundary. The surface height Just outside

the boundary is set equal to the sunface height _just inside

the boundary. The corners arc treated similarly with the

corner point being the center of symmetry (e.g., ii corner

point is (2,2) , the V(l,2) 3-V(,2) , 1(1,2) = )

and L( I , I) = - I S , 3) 1

Ver I -icat ion

The program ba.Ned oni these di fference e(Luat ions and

boundary coiid it ions was veri ied by dup ica t ing the results

obtained by Mader using his SiWAN code (Ref I)l. 'the problem

solved by Mader consisted of a .5 meter rad ius hemispherical

surface cat i ty in water three meters deep. The cavity was

tIle result of a near su r face burst . This p rogran used the

18

same time step of 0.001 seconds and cul l size of .Amo meters

square. The restilts o1 this lrograw ageed e vvy tull with

the results oht ained by Mader.

19

.l~lijt I 0 1- 111 LI 1 t 1 n1>' ha I ,'lOil hVb d i f Ic CI VII cC e(lI i on l

iii it i J Ic :Oild I t ios OlSAnId 1,loNi.F Yv ComA i t i Ons we WI-C U to0

du2term ifi tl t li sksi i t v o0 t lie u i d' I w'avc rtie11cr~It il

1i t ir c 2 ilIIlus-;tI-a teCS t 1e iic i shp 1C AIS1 1)2Of- theI t idl IWavegenticra ted by the cOmtpu tcr -; illu 11 a t i onsl ' he wa v c* VC cocit

i s 5 3 (I/s , ) wi ch i s very- clIosec t o thli clia rac te r i t i c 1.ave

Ve IOC it)' 0F 31 . 3(111/) S.''l I L ill it ilt COndLit ionl used produced

a W aVeC'A with1 I p~er iod 0 ' bl)l t SO) secondls Vu i ch is !e1Ca r I y

thri ee t ime s tile ex pc cted per i od ol C 1 8 sec ond." sbased o)n t lie

relIa t ion sh i PIe r i odIl ' 1.I Y 0 14 ( Recf 3 : 22) . '1[his

i nd ic ates - t le wa ve I c ng t l o F 10 ) me tecrs p)roduiced hy th1is

in1 i t ialI c or[i it i onl s alIso jbult t h 1ce t lue s to0o Ilng. TIlie

he Ci g11t of1 the1 i it fl :1wave ab0oe thIe SurI'fa ce was SSelIect ed

as the jiid ica7tor- ol the wave v2eulrat ion seus. it ivitv . Thle

wavec lie i gh t i s me asuLired aIt a nea rlIy cons ta nt. d i s tanice, o f

8 kjIome tcr s f rom thle exp 1o s ionls . 'I'l ci avc highi t wa s

o bt a in ed friom tile p r, i n ter -p1 lo ts w lii ch ex prie ssed thle wave

lie iijilit Wi thlii i nfc remi It s; o f . 25 meCtOYis . T[he er-ror bars onl

the graphs indicate thle vir i at ion inl wave height a long the

1)l;a11C tidal1 Wave. I~otli the l inear and( non- Linecar di fference

Schemes Used the saule t flue Step of 0.5 Seconds and cellI

siz Of 0 50 mecters squa re.

2 0

0 (A

-,1 0

41 -4

m -4

u x]

Y -1 L-1 S ei it iv i t

'Ilewave 110ight Sens',it iVity' toV VIL is p~roportionalto ./ '[his does not agree with tile scal ing of wave

height as Y I/o Used for tile initijal condit ion (Ref 5:133).

However, it does come close to the scal ing of wave height

a s Y - or (Ice water arnd Y1/ 4 for sihallow water used

by (Hasstoie (Ref 3:2722173). [he sensit ivity to yield over

the range of 2 - 8 kilotons is +4 . S (i/ (kt ) I 3 )

Thbe wave sensitivity to combinations of two yields

whose average is S Kilotons; is small I. The sensitivity is

determined relat ive to tile ratio of the small yield over the

larger yield. As shown inl Figure 4, the wave height is not

very sens it ive to combinat ions of vie Ids over a practical

range o f 0.5S to 1 .

S~ac i en, I t iv ity

The spac infg sells itiv ity was determined ovel thle range

of . t o I .4 Koilme ter1,s. '[hle wave height d id not change

I inear 1y with spac ing as shown inl I iu ic S. Over thle range

of 0.8 - 1 .2 R~ i lometers Spa1c ing, the approx imate sensitivity

is -1.5mk)saig

'[he sens it iv it v to ill i '- I i gImenlt i a teni ine foi-

ill i sa 1 i gtlneit s uip t o .5 k i Ioic' I~ a, V m-, ld i ca t ed in 1 [igure 0.

Thle in i salI i gmint si tuit ion tA, - t ilp 1))' liat-ing every other

explosion o [['set a t iren d ki 't ;nce f rom tilie array. Thle

mYIELD

SENSITIVITY

w /

Il/

Ljjo /

wl /

1'.0 0 1. 25 1. 50 1. 75 2. 00 2. 25CUBE ROOT OF YIELD (KT** (1/3>)>

Figure 3. Yield Sensitivity

23

kL __

i .. . . . i l , . . . . , , , . . . .

YIELD COMBINATIONSENSITIVITY

c)

we

z

.-0

>a;-

YIELO#1/YIELD#2 RATIO

F igUre 4 ic wd comb) iat ion Sens it iv it),

. 2L

SPACINGSENSITIVITY

X-W4w

0I

W'.

0 0.70 o.90 1.10 1. . 1.50EXPLOSION SPACING (KM)

Figure 5. Spacing sensitivity

2 5)

w*4 ALIGNMIENTSENSITIVITY

In

in

w

x0 .002 03 .005MISLINMET KM

figur I. Algmn Snii

C26

it isa 1I gnillent l i st allk L I , ic I- I d I a II I oIl set J i Stance

The approx intate ev it iv It\' ') e t I Il l.' r I e U I () to .5

k il omet er is - 2 . 7 ( /kil mill Ia 1 llell t . 'he [ii sa I i .ynilent s

did increase the ,ave heigOht ia riat ion alonl thlc wave front

as ind icated by the e uror h rs.

'I i i Itli , S ellsi it i t

The wave he i.,,h1 sensitivity to til int delays of 0 - S

seconds is very, smal I , as: Thomn in Vigure 7. The t ime

delay situat ions are iode led by hvil' eve ry' other explosion

delayed the same amiount otf t imle. In the range of 0 - 2.5

seconds delay, the sensitivity i; onl] y . 2(lin/sec ) time

delay.

l;(t toiL Slope Sell. 1t ix it V

'Ie ae rage sl tyc of tit' CO it iliCltal -,hel f had a

negl igible effect on the t,ivc heighi in 8 kilometers of

trave I. The hottom was treated as being smooth and twith

constant slope. ihe slope ias varied from 0.) to 0.002

as indicated by I igure 8. The tave height sensitivity at

8 kilomieters fromu the explosio1s is +182(m.) per the slope.

Assuming the elfect was constant over the 8 kilometers,

the sensitivity was +23(rn/kmi ) per kilometer traveled by the

wave per the slope. For example, using the averaged con ti-

nental shelf slope of 0.0(003, the expected increase in wave

height in 100 kiloimeters is 0.7 meters.

TIMINGSENSITIVITY

I -4I-.

-r

400 1.00 2.00 3.00 4.00 5.00TIME DELAY (SECONDS)

1:igure 7. Ti inin ! Seiis it ivi ty

28

5' BOTTOM SLOPESENSITIVITY

w

54

V-4

w

i.U 0.04 0.08 0.12 IL 16 I0.20BOTTOM SLOP&B1 o 2

Upgure S. (Rottoill slope sellsitivity

29

Noii-Lincar L fects

The non-linear effects are illustrated I), comparing

the wave formation o1f a linear and a non-linear computer

simulation. Wave contour printer-plots of Figures 9 and

10 show the non -linear and linear initial waves, respCC-

tively, at four kilometers from the explosions. Lach 1 ine

of the contour plots indicates a change in height of two

meters. The troughs or areas of the water surface below

the normal free surface are not represented by contour

lines, but lcft blank. The plots show how the non-linear

effects produce a slightly smoothe r and larger wave in the

same distance. However, the linear simulation provides a

very good approximation for the initial conditions of this

study.

3(1

I

- -- "-.- .--.--.----- -- --- - ----- ------- - - -------- -

, J

0

. . . . . . ... .. . . . . . . . ..-

t4--4

I T

.. .. .-- ------ ----... ... .. . ... ....... -- ---- :- - ---- ---- --

I"1I I

I.- ,, I'

o.i- IPt.c A.lon. I1)istance Along Array of Lxplosions (kin)

Figure 9. Non-Linear Contour Printer-Plot

31

.. .-... .. .. . ... -- o

o ...-..

/\ . ..

,, ., . - ' _ _ ! ' _ _

'..-

An ar'ray of S iIII I t ZaneOIus- 111de rwat c-e xp pI s ionis i s very

e ff ec t iv o in i ge cr a t i i~ ag z1 )1;iiic t i I wa ,,ziv e. This conclIus ion

is based on shallow water theor>, L\ithi the explosion represented

by a raised Cone Of wate2r Which prdcswavVe with a wavelength

of about 1000 meIter-s.

TIhie sensitivity of the wvave geneLrat ion process was deter-

mined. by the height of the initial wave. The wave height was

not very senisitiv e to miisal ignment s less than 200) ieters , time

delaYs less than S seconds, hot to slope) less t han 0.0004, and

rat ios of yields in the ranL'c, of 0.5 to 1.0. The wave heigh11t

s en s it iv ity t o sp)a c in gs o f 1 .0 2 Rki Iore te rss a )1) r ox ima te 1y

-11 I.25S met e rs pe r k ilIom et e rs s pa c in g. The yield sensitivity,

which depends onl the derived initial condition, is 1I.55 meters

per cube root of the yield in kilotons. 'Fhe non-I ineair effects

inc rease the size of thle tidal wave.

3.

V . URec oie cl lt i on

More research oI the in it i:il conJ it ion is reconlniiended.

The waves produced by the raised cone of water have a wave-

length about three t imes too loilv' anid the in iti 1wave is

not p receded by a t roiug h is norlna lIv expected (Ref 3.-5:273).

In Ldd it ion, the cone does not conserve the volume of water.

A new in it i, condition should be developed which conserves

the water voLtumVe and produces waIves with a shorter wavelength.

It is recommllended that a cvindrical cavity with its depth

equal to its radius ( IM)P metcrs) and with a lip of water

surrounding it ic ued As al new in it ia l condition.

54

1;)i) I i o'4 r1,)h

1. Dickey, Robert 1'. , lan W. Hian i fen , LIawrence R. 1'arch,Lawrence R. Mlart in, and (hr i s R. Nest brook . Sha I lowUnderwater Moobile: An Alternative to Land-Base-d NiX.KITClass Design Project. 'right-Patterson AFB 0-H:Air Force I nsti tut e of Technol oy , larch 1 98(.(AF 1T/(-NF/PII/8OM-lDesign).

2. FeId , Bernard T. and Kosta ls i pis . "Land-Based Inter-continental Ballistic issiles ' cientific American,241 (S): 54 (November 1979).

3. Glasstone, Samuel and Phil ip J. Dolan. The Effects ofNuclear Weapons (Third Idition). Washiington: Departmentof Defense and )epartment of I nergy, 1977.

4. Harlow, Francis II. and Anthony A. Amsden. Fluid I)ynamics(LA-4700, UC-34). Springfield VA: National Te inicalInformation Service, June 1971.

S. Hlartman, Gregory F. Wave Makimn By An (nderwaterlxplosion. Silver Springs MD: Naval Surface WeaponsCenter, White Oak Lab, September 1976. (AD A038 276).

6. 'IMader, Charles L. "Calculation of Wves Formed fromSurface Cavities," Proceed in s of the 15th Coastalngineering Conference. 1079-1092. Honolulu III:

American Society of Civil :ngineers, July 11-17, 1970.

7. Roberts, K.V. and N. 0. W eiss. Mathematics of Computation,20 (94): 272-299 (April 1966).

8. Stoker, J.J. Water Waves. New York: lntersciencePublishers, Inc., I-f T.

35

Henry Jauics Weber was horn oi 4 Januajry 19S2 in

,Jamestown, North Ofakot a. Hie gritduaitcd from North Cent ral

Ihigh SchooL of Rogers, N\orth lzakotai ini 11)7(), and then

at tended North [Dakot a State Univers ity~. Upon graduation

in 1974, he received it Bachelor of Science degree in

Mechianical ling ineering and a tin ited States Air For-ce

commission through the Reserve Officer Training Corps.

Start ing in September 1974, hie served in the 6S96th

Miss il1e Test Group, Vandenberg AEI- , Californiia, as a

requirements officer anid 1launch controller uint il eite ring

the School of Flnginecring, Air Force Inist i(Ite of' fechlnolog)y

in Aiiuist 1979.

Peirmanent Add re ss: 1039 2nd Avenue N. 1%.

Vallecy City , North DiakotaS80712

3 o

__ NC LASS I jLj)l( UHIY T A y 11 t A 1 ,,iV 4 9- T I I, I.'A It Kh,, ,. -, f

hiAll IN'," *I, IJ

REPORT DOCUMENTATION PACE I~ij IN' ., I-,I RIPO CT N'M lt - I All I N I I . . . T. A. 1, ,Mb i

4 TITLE -J I- bra &

PLAN. IDAI. WAVES tNI I,11 II -AN APRAY O: ,1 1 hes i s

SIILLI ANIO US 1'NDIW.1\' H ER I NI 1' 6' t). 1- WAN LPR?~~~~f ,-HN. NFA R ' ANT N,MdE HI 11:IN! I.. . . . ..I.I.I....

.....

ILNE.Y,' .1. |,LI|,.R{

CAPT ISAI[9 PE FL(IRM IN k, k C, A f.' I N:A 1 , A. .A,, , I Tl 1e M IM) , lSJ T T I

XEt Institute of 'llechnologv (AF II'-IN)Wright-Patterson At:B OI .15,1.33

It CON TI ,OLLIN(,Of FI L NAMl ANDI A[!NA ti . , t r t

M'ircl 981IH II- - - - - _

? N'MH I ' I AA a

I742 F I ,4 M N u~I~ A ;ENCY NAME &AIO? , -1 4a~a~, ' jH Y L, AI,, SiH rA .JN,'t

, A-,SIF,. AT )N f)OaWNIaRADIN

16 OISTRIBJTIN S 17 7r-M .NT ,,i '"a ' F.

Approved for Public Release; Distribution lInlinmitcd.

17. 01ST RIBUTION STATEMENT (., I he rt, t I - t f I. . 1 11- Ira- Rar,

1. SUPPLEMENTARY NOT ES A prived for iluhli - Release; lAW AFR 190-17

FR D FRI K C. 1, .11, ajor, USAIXl g7 ' - C/ 7 ;Jgy(ATC

Director of PutTic AffairsI9 KEY WORDS (CortIini.. - re"Orr e at-, 11 1~ r sta id AI.Jr lfv hv h IN n-ra'r

Simultaneous lnderwater xplosion Inderwater Explosions

Water Waves lane Water Waves

Shal low Water haves Underwater Vxplos ion Arrav

20 ABST i 0 TI ((oeri auii mIf rver..i e l.11 1 r f hi.. k n..r Ia,,r)

his report considers in array of Sililtmllt;rnteoliH 1inde0orwaternuclear explosions geiieratilli plane tidal waves On the continentalshelf . Computer simul1ations are used to studv this wave genera-tion process, The) are based on non-linear shallow water theorywith an initial condition of a stationarv raised cone of waterj.'rhe sensitivity of the wave lieicoht to dii'ferent yields, combina-tion of different yields, spacing, al!ignment, liming, and bot tomn

(:ot nue.on RWeerse )

DD I ,JAN 1473 EDITION OF INOV 6S IS OBSOL -TF TV" :',' m...... ... , . ... I n L. . . I l en .i ri.. . .. Jd

I ) ,rI.Ass I: IFI)SECURITY CL ASSIFICATION OF THI PAGE(Wh.,, )1. Pnt-.J)

BLOCLK 2(0: A B3lRA 'CT (tC ont 'd)

-Islop)c is deter i11ed. The rC frencC va I kie used for this studyare a yield of five kilotons TNT, onc ki loiteter spacing, waterdepth of 100 meters, and un hottom slope. III add ition to thesensitivities, the amount of non-linear effect is illustratedby comparison of wave contottr printer-plots ,.eneratte] by I inerand non-I inear sha I low twatcr theorv . lh c)mpl ter restl Itsindicate the wave generation is reltlt i vly ilscns itive to real-istic variations in the paraiietrs.

HI\(,4 A ,. I I I 11

kSfI T k6

DAT

DIC• 1