Wave Equation in Fluids
Transcript of Wave Equation in Fluids
-
7/29/2019 Wave Equation in Fluids
1/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
1. The wave equation in a source-free medium
Acoustic disturbances in fluids, i.e., gases and liquids, which cannotsupport shear stresses, propagate as longitudinal waves.
Plane wave propagation means that the acoustic variables, such as soundpressurep, have a constant instantaneous magnitude throughout any given
plane perpendicular to the direction of wave propagation
Direction of propagation
Pressurep/ p0 WavelengthRarefied Compressed
Direction of propagationp0
Figure 1 Symbolic depiction of plane longitudinal wave propagation for a sinusoidal
disturbance. The fluid particles are sketched in the compressed and in the rarefied regions.
1/48
-
7/29/2019 Wave Equation in Fluids
2/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Simplifying assumptions are
(i) The medium is homogenous and isotropic, i.e., it has the same
properties at all points and in all directions.
(ii) The medium is linearly elastic, i.e., Hookes law applies.
(iii) Viscous losses are negligible.
(vi) Heat transfer in the medium can be ignored, i.e., changes of state canbe assumed to be adiabatic.
(v) Gravitational effects can be ignored, i.e., pressure and density are
assumed to be constant in the undisturbed medium.
(vi) The acoustic disturbances are small, which permits linearization of the
relations used.
2/48
-
7/29/2019 Wave Equation in Fluids
3/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The following quantities are considered:
Pressure: ),(),( 0 trpptrptvv
+= (1)
where is the total pressure as a function of position ( ) and time (t),
p0 the pressure in the undisturbed medium, andthe pressure disturbance in the medium, the sound pressure.
),( trptv
rr
),( trp v
Particle Velocity zzyyxx eueueutrurrrrr
++=),( (2)
where is the particle velocity vector
ux, uy, uz are the corresponding velocity components,
is the unit vector
),( tru rr
zyx eeerrr
,,
Density:),(),(
0trtr
t
rr +=
(3)
where is the total density,
0 is the density in the undisturbed medium,
is the density disturbance in the medium.
),( trtr
),( tr
r
3/48
-
7/29/2019 Wave Equation in Fluids
4/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Absolute Temperature:
A fundamental assumption is that we considersmall disturbances, i.e., small
variations in pressure. As a rule of thumb, for air at normal temperature and
pressure, the sound pressure level should not exceed 140 dB
Equations of continuity
one-dimensional case
x
yz
+x
tux
)(tuxtux x
y
z
x
Figure 2 Mass flow in thex-direction through a volume element fixed in space.
4/48
-
7/29/2019 Wave Equation in Fluids
5/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
According to figure 2, the mass in the volume element is , the mass
flow into the element is , and the mass flow out is . The
net flow in the element is therefore , and must
equal the mass change , so that a mass balance is received
zyxt
xxt zyu )( xxxt zyu + )(
xxxtxxt zyuzyu + )()( )( zyx
t
t
xxxtxxtt zyuzyuzyx +=
)()()( (4)
( )
+=
xzyux
zyuzyuzyxt x
xtxxtxxtt )()()( (5)
The second term on the right-hand side can be expanded into a series, for small
variations about the undisturbed equilibrium state, and if higher-order terms canbe neglected, then
This can be simplified to
( ) 0=
+
xt
t uxt
(6)
5/48
-
7/29/2019 Wave Equation in Fluids
6/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Generalized to the three dimensional case,
( ) ( ) ( ) 0=
+
+
+
ztytxt
t uz
uy
uxt
(7)
Defining the del operator, as
+
+
=z
ey
ex
e zyxrrr
(8)
permits a simplified expression of the continuity equation, as
0=)( ut
tt r
+
(9)
Putting the total density (3) into the equation and taking advantage of the factthat the undisturbed density 0 is independent of time and position, and ignoring
second order terms, that consist of the product of two acoustic disturbances,
gives the linearized wave equation
6/48
-
7/29/2019 Wave Equation in Fluids
7/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
00 =+ ut
r
(10)
In one dimension
00 =
+
x
u
tx
(11)
Equation of motion
Consider a specific fluid particle, with a fixed mass Dm and a fixed volume
V = xyz, as in figure 6-3, that moves with the medium.
( )( )
x
y
z
+0p+p [ ] x0p+p( ) 0
p+p
y
z
x
y yz zx-
Figure 3 Force in thex-direction on a
particular fluid particle moving with the
medium.
7/48
-
7/29/2019 Wave Equation in Fluids
8/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The force in thex-direction is
zyxppx
zyxppx
ppzyppFx +
=
+
+++= )())(()( 0000
(12)wherep0 is constant, so that
zyxx
pFx
= (13)
In three dimensions, the force vector becomes
zyxz
pe
y
pe
x
peF zyx
+
+
=
rrrr(14)
Putting in the del operator, as
+
+
=z
ey
ex
e zyxrrr
(15)
8/48
-
7/29/2019 Wave Equation in Fluids
9/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
and using the relation xyz= V, then equation (14) reduces to
VpF = r
(16)
For a given fluid particle, the velocity is a function of position and time.At time t and position (x, y, z), the velocity is (x, y, z, t). At a later instant t+t,
the position is (x+ x, y+ y, z+ z) and the velocity is .
The differential change in position (x, y, z) can be written x=uxt, y= uytand z= uzt, so that the acceleration can be written
),( tru
rr
( tzzyyxxu ++++ ,,,r
t
tzyxutttuztuytuxua
zyx
t
++++=
),,,(),,,(lim
0
rrr
(17)
The first term is reformulated with the help of a Taylor series, so that equation
(17) can be rewritten
t
tzyxutt
utu
z
utu
y
utu
x
utzyxu
azyx
t
+
+
+
+
+=
),,,(),,,(
lim0
rK
rrrrr
r(18)
9/48
-
7/29/2019 Wave Equation in Fluids
10/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The acceleration of the fluid particle becomes
t
u
z
uu
y
uu
x
uua zyx
+
+
+
=rrrr
r(19)
With simplifying notation this is
uut
ua
rrr
r)( +
= (20)
In one dimension,
x
uu
t
ua xx
xx
+
= (21)
For acoustic fields with small disturbances, the second term in equation(20)can be neglected, so that
t
ua
=r
r
(22)
10/48
-
7/29/2019 Wave Equation in Fluids
11/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
In one dimension,
t
ua xx
=
(23)
Making use of equations (16) and (22), as well as m = (0 + ) V, theequation of motion can be formulated as
t
uVVp
+=r
)( 0 (24)
If second order terms can be ignored, then the linear, inviscid equation of
motion is
00 =+
pt
ur
(25)
In one dimension,
00 =+
xp
tux (26)
The equation of motion, gives a relation between pressure and particle velocity
in a sound field.
11/48
-
7/29/2019 Wave Equation in Fluids
12/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The thermodynamic equation of state
For an ideal gas, the ideal gas lawapplies,
MRTpp )()( 00 +=+ (27)
where (p0 +p) [Pa] is the total pressure,
(0 + ) [kg/m3] is the total density,
R= 8.315 [J/(mol K)] is the ideal gas constant,M[kg] is the mass of a mole of gas, and
T[K] is the absolute temperature.
Two idealizations can be considered: an isothermal process, implying such
good heat conduction in the medium that the temperature is constant
throughout; or, an adiabatic process, in which no heat conduction occurs.
12/48
-
7/29/2019 Wave Equation in Fluids
13/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
where = cp/cvis the ratio of the specific heatof the gas at constant
pressure, to that at constant volume. For the sake of simplicity,pt= (p0 +p)denotes total pressure, and t= (0 + ) denotes total density, in the series
expansion. The total pressure can be expanded as
...2
1
00
2
22
00 +
+
+=+===
tt
t
t
t
tt
pp
pppp
where the partial derivatives are constants that remain to be determined for
adiabatic disturbances about 0, i.e., the density in the undisturbed medium.
13/48
(29)
( ) ( )
+=+
0
0
0
0
p
pp(28)
For small disturbances below a certain frequency limit, it can be shown that
the process can, to a good approximation, be regarded as adiabatic. For an
adiabatic change of state,
-
7/29/2019 Wave Equation in Fluids
14/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
where
0
0
=
=
tt
tp(32)
is called the adiabatic bulk modulus.
The homogenous linearized wave equation
14/48
For small disturbances, second and higher order terms can be neglected,
and a linear relation is obtained as
0
=
=
t
t
tpp
0=por
(30)
(31)
-
7/29/2019 Wave Equation in Fluids
15/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
,
One dimension Three dimensions
Time derivative of continuity eq. (11), Time derivative of continuity equation (10),
Using
Spatial derivative of eq. of motion (26) Divergence of eq. of motion (25),
0
2
02
2
=
+
tx
u
t
x
(33)
02
22
0 =
+
x
p
tx
ux (35)
( ) 00
=+
p
t
ur
002
2
=
+
t
u
t
r
(34)
( ) 00 =+
pt
ur
(36)
In abbreviated notation,
020 =+
pt
ur
(37)
15/48
-
7/29/2019 Wave Equation in Fluids
16/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
One dimension Three dimensions
Subtraction of (33) from (35) gives
The wave equation in one dimension
(38)
Subtraction of (34) from (37) gives
02
22
=
tp
(39)
Equation of state (31) eliminates
(40)02
2
02
2
=
t
p
x
p
02
2
2
2
=
tx
p
Equation of state (31) eliminates
02
2
02 =
t
pp
(41)
The wave equation in three dimensions
01
2
2
22
2
=
p
cx
p(42) 0
12
2
22 =
t
p
cp (43)
16/48
-
7/29/2019 Wave Equation in Fluids
17/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The constant cis defined as
0/=c (44)
and is the propagation speed of a disturbance in the medium, the speed ofsound. According to equation (32)
0
0/
=
==
tt
tpc (45)
For an ideal gas, equation (28) implies
)ln(lnlnln 0000
=
= tttt ppp
p(46)
17/48
-
7/29/2019 Wave Equation in Fluids
18/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
so that
t
t
t
t pp
=
(47)
Put into the expression for the speed of sound (45), this yields
0
=
=t
ttpc
i.e.,
00 pc =
(48)
(49)
The temperature dependence of the speed of sound is obtained by putting the
ideal gas law (27) into (48)
MRTc = (50)
If the speed of sound at 0 C (273 K) is denoted c0, then for other temperatures
2730 Tcc = (51)
18/48
-
7/29/2019 Wave Equation in Fluids
19/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The speed of sound increases with temperature, a relationship with great
significance for sound propagation outside, where the temperature often varies
with distance to the ground.
isothermal bulk modulus T . the relation between these is b = T , after
which the sound speed in liquids is found from
0 Tc=(52)
Solutions to the wave equation
General solution for free plane one-dimensional wave propagation
Forplane wave propagation in thex-direction, the wave equation (42) applies.
Assume a general solution form of
( ) ( )cxtgcxtftxp ++=),( (53)
19/48
-
7/29/2019 Wave Equation in Fluids
20/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
where fand gare arbitrary functions and (tx / c) and (t+x / c) are theirrespective arguments. That assumed solution is known as dAlembert s
solution. Derivation of equation (53) with respect toxgives
( ) ( )cxtgc
cxtfcx
p++=
11
(54)
( ) ( )cxtgc
cxtfcx
p++=
222
2 11 (55)
and similarly with respect to tgives
( ) ( )cxtgcxtft
p++=
(56)
( ) ( )cxtgcxtft
p ++=
2
2(57)
Putting (55) and (57) into (42) shows that the solution fulfills the wave equation.
20/48
-
7/29/2019 Wave Equation in Fluids
21/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
To interpret (53), consider a special point (x1,t1) in the wave, in figure 4. It
represents a certain sound pressurep1(x1,t1). To have the same sound pressur
at another point (x1 + x), at a later instant in time (t1 + t), then the arguments
have to be the same, i.e.,
( ) ( )cxxttcxt )( 1111 ++= (58)
which gives the condition
tcx = (59)
Thus, the solution f(tx / c) implies wave propagation in the positive
direction along thex-axis, with speed c.
21/48
-
7/29/2019 Wave Equation in Fluids
22/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Figure 4 Instantaneous picture of the wave propagation in the positivex-direction at time
instants t1 and (t1 + t). The propagation speed of the disturbance is c.
Similarly, g(t+x / c) implies propagation in the negativex- direction, so that
tcx = (60)The propagation speed cof a disturbance is called the speed of sound(or
sound speed), wave speedorphase velocity.
22/48
-
7/29/2019 Wave Equation in Fluids
23/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Harmonic solution for free, plane, one-dimensional wave propagation
From Fourier analysis, it is known that everyperiodic process can be built up
of the summation of harmonic, sinusoidal processes with different
frequencies, the set of which is called a Fourier series.
The harmonic solution we seek for the angular frequency = 2f must, for
a certain x-value, say x1 + x, give the same sound pressure at time tas it
does one period later at time t + T, where T is called the period. For a
harmonic function, that implies that the argument, i.e., the angle, increases
by 2. The solution must also, for a certain time value, say t1
= t, give the
same sound pressure at points separated along thex-axis a distance equal
to the wavelength . Even in that case, the argument must increase by 2,
see figures 5a and 5b. Thus, we attempt a solution of the form
)(cos),( cxtptxp = + (61)
where is the amplitude, i.e., the highest value of the sound pressure. The
argument (t - x/c) = t kxis called thephase and+p
23/48
-
7/29/2019 Wave Equation in Fluids
24/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
k= /c (62)
is called the wave number. The first condition above gives
))((cos)(cos 11 kxTtpkxtp += ++ (63)
i.e.,
T = 2p , = 2 /T (64)
The second condition gives
( )( ) += ++ xktpkxtp 11 cos)(cos (65)
i.e.,
2,2 == kk (66)From (62), (64) and (66), one obtains the relation
fc= (67)
24/48
f S
-
7/29/2019 Wave Equation in Fluids
25/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
which applies to all types of wave propagation, and in which the frequency is
f= 1 / T (68)
p(x1,t) $p+
Period T
t
Figure 5a The variation of sound pressure with time at a fixed positionx1.
25/48
F d t l f S d d Vib ti
-
7/29/2019 Wave Equation in Fluids
26/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
p(x,t1)
Wavelength
$p+x
Figure 5b The variation of sound pressure with position at a fixed time t1.
complex notation
)()( ),( kxtikxti epeptx +
+ +=p (69)
in which bold print means that the variable concerned is complex. The
first term on the right-hand side refers to propagation in the positive x-
direction, and the second term to propagation in the negativex-direction.
For the sake of physical interpretation, the real part of (69) is needed, i.e.,
26/48
F d t l f S d d Vib ti
-
7/29/2019 Wave Equation in Fluids
27/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
)cos()cos()Re(),()()(
kxtpkxtpepeptxpkxtikxti ++=+= +
++
(70)The equation of motion (26)
00 =+ xp
tux (71)
relates the particle velocity to the sound pressure; rearranged, it gives
dtx
pux
1
0
=
(72)
Next, putting in (69) gives the particle velocity
( )
+
= ++ )()(
0
1, kxtikxtix epi
ikepi
iktx u (73)
Since k/ = 1/c, the particle can be written
27/48
F d t l f S d d Vib ti
-
7/29/2019 Wave Equation in Fluids
28/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
)(
0
)(
0
),( kxtikxtix e
c
pe
c
ptx ++ =
u (74)
The two terms refer to wave propagation in the positive and negative x-
directions, respectively. The ratio of pressure to particle velocity is called thespecific impedance Z,
xupZ = (75)
and, for the free plane wave case, is therefore
cZ 00 =+
(76)
and
cZ 00 = (77)
28/48
for propagation in the positive and negative directions, respectively. The
quantity 0cis called the wave impedance.
F d t l f S d d Vib ti
-
7/29/2019 Wave Equation in Fluids
29/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Sound intensity for free, plane, one-dimensional wave propagation
The sound intensity is defined as the sound energy per unit time that
passes through a unit area perpendicular to the propagation direction. the
instantaneous power can be written
)()()W( tutFtrr
= (78)
A general expression for sound intensity is therefore that
),(),(),( trutrptrIrrrrr = (79)
),(),(),(x txutxptxI x=
For propagation in thex-direction,
(80)
The time-averaged sound intensity is
29/48
F d t l f S d d Vib ti
-
7/29/2019 Wave Equation in Fluids
30/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
=T
xx dttxutxpT
xI
0
),(),(1
)( (81)
Making use of the expression for pressure, i.e., the real part of (69), and the
particle velocity, i.e., the real part of (74), in (81), gives the intensity in theform
cppI x 022
2)( + = (82)
The first term refers to a wave moving in the positive x-direction, and thesecond term to a wave moving in the negative x-direction. For harmonic
waves, the relation between the rms amplitude and the peak value
is , so that (82) can be written2pp=
c
p
c
pI x
0
2
0
2 ~
~
+ = (83)
30/48
F ndamentals of So nd and Vibration
-
7/29/2019 Wave Equation in Fluids
31/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Energy and energy density in free, plane, one-dimensional
wave propagation
The energy Eassociated with a sound wave consists of two parts, the kinetic
and the potential energy. The kinetic energy Ek
can be related to the velocity of
a fluid particle. The potenti-al energy Ep is due to the compression, i.e., the
elasticity. Considera part icular mass of gas that has density 0 and volume V0in the undisturbed medium, so that its mass can be written
VV t =00 (85)For wave propagation in thex-direction, its kinetic energy is, from elementary
mechanics,
2
),(),(
200 txuVtxE xk
= (86)
The potential energy of the fluid mass comes from the work that expended to
compress it. Consider, now, the fluid volume shown in figure 7. The forceapplied to the piston is equal to the product of the pistons area S and the
pressurep against its inside surface, i.e., Fx=p(x,t)S. When the piston moves a
distance dx, the differential amount of work dEp = Fx dx = p(x,t) S dx is
performed.
31/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
32/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Figure 7 Work performed to compress a fluid volume.
Since Sdx= dV, dEp can, with complete generality, be written
dVtxpdEp ),(= (87)
where the minus sign implies that a positive sound pressure p(x,t), which
gives a negative volume change (i.e., a volume reduction or compression),
corresponds to a positive potential energy. We choose to express dVas a
function of the sound pressure. Differentiating (85),
32/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
33/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
tt
t
tt
dV
dV
dVVV
== 2
0000
1(88)
Since equation (46) states that
= 00
tt
p
p
tt
tt
t
t
t
tdppdp
dpd
==1
then
(89)
equations (88) and (89) give
tt
dpp
VdV
= (90)
33/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
34/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
For small pressure and volume disturbances, the undisturbed pressure andthe undisturbed volume can be used
dpp
VdV
0
0
= (91)
The work (87) can now be written as
dppp
VdEp
0
0
= (92)
The potential energy is, finally, obtained by integrating from 0 to the soundpressurep; thus,
2
0
0
2p
p
VEp
=(93)
According to (49), ; thus, for plane wave propagation in thex-
direction, a general expression for the energy is002 pc =
( ) ),(2
, 22
0
0 txpc
VtxEp
= (94)
34/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
35/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The concept ofenergy density [J/m3] refers to the energy per unit volume. Ofcourse this quantity, just as the total energy does, consists of two parts: kinetic
and potential energy densities, i.e.,
),(),(),( trtrtr pkrrr
+= (95)
Using expressions for the kinetic energy (86) and the potential energy (94), the
energy density for propagation in thex-direction is then
2
0
220
2
),(
2
),(),(
c
txptxutx x
+= (96)
According to (76) and (77), applies to plane waves.
Using this in (96) yields the instantaneous value of the energy density
)(),(),( 0ctxptxux =
20
2
20
2
20
2 ),(
2
),(
2
),(),(
c
txp
c
txp
c
txptx
=+=
(97)The time average is obtained by integrating with respect to time
2
0
2
0
)(~),(
1)(
c
xpdttx
Tx
T
== (98)
35/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
36/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
For a plane wave however, since it doesnt decay with distance, the rmspressure is independent of positionx; that naturally applies to the time-averaged
energy density as well, i.e.,
20
2~
c
p
= (99)
A comparison to (83) gives the relation between the sound intensity and the
energy density of a plane wave, as
cI x = (100)
General solution for free spherical wave propagation
The spherical wave is a basic cornerstone in the study of acoustic fields.
More complex sound fields can be built up of combinations of spherical waves
36/48
-
7/29/2019 Wave Equation in Fluids
37/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
38/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
For spherical wave propagation, it is of course more convenient to make use ofthe latter; see figure 9.
x
y
z
r sinsin
r sincos
r cos
r
(r,, )
Figure 9 Relation between spherical and Cartesian coordinate systems.
38/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
39/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
In spherical coordinates, the wave equation (43) takes the form
2
2
22
2
2222
2
1
sin
1sin
sin
11
t
p
cp
rrrr
rr
=
+
+
(101)
For spherical symmetry, the sound pressure has no angular dependence, and
equation (101) reduces to
2
2
22
2
11
t
p
c
p
r
r
rr
=
(102)
or
2
2
22
2 12
t
p
cr
p
rr
p
=
+
(103)
Unlike plane wave propagation, the sound pressure amplitude decays with
increasing radius, since the sound power in the wave is divided over an
ever-expanding spherical surface of area 4 r2. In a plane wave, the sound
intensity is, according to (82),
39/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
40/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
i.e., proportional to the squared sound pressure amplitude. Assuming thatapplies to spherical waves as well, their amplitude would therefore have to
decay at a rate of 1/raccording to the energy principle; by analogy to the plane
wave case, an assumed solution might therefore take the form
( ) ( )crtgrcrtfrtrp ++=11
),( (105)
Here, the first term represents an outgoing diverging wave and the second term
an incident converging wave. The incoming wave seldom exists in connection
with acoustic radiation from machines. The solution is verified by inserting it intothe wave equation (103). On a term-by-term basis, we have
( ) ( ) ( )
( ) ( ) ( )crtgr
crtg
cr
crtg
rc
crtfr
crtfcr
crtfrcr
p
+++++
+++=
322
3222
2
221
221
(106)
( ) ( ) ( ) ( )crtgr
crtgcr
crtfr
crtfcrr
p
r+++=
3232
22222
(107)
40/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
41/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
( ) ( )crtgrc
crtfrct
p
c++=
222
2
2
111(108)
Putting all of these into (103) shows that the assumption does indeed fulfill
the wave equation. The equation of motion (25) relates the particle velocityto the sound pressure. In spherical coordinates, it takes the form
0sin
110 =
+
+
+
per
er
et
ur
rrrr
(109)
With spherical symmetry and , the equations of motion can be
reduced to
00 =
+
r
p
t
ur (110)
and the particle velocity expressed as
(111)dtr
pur
1
0
=
41/48
rr etruurr
),(=
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
42/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
The first term on the right-hand side refers to an outgoing, diverging, wave,
and the second to an incident, converging, wave. The sound pressure
amplitude in the outgoing wave isA+/ r, whereA+ is a constant. The amplitude
is therefore a function ofr. From (111), the particle velocity becomes
( ) ( ) dter
ik
r
Adt
r
r,ttr krtir
1
1, )(
200
+
+
=
=
pu
Harmonic solution for free, spherical wave propagation
A complex harmonic solution is obtained, as
)()(
),(
krtikrti
er
A
er
A
tr
++
+=
p
(112)
(113)
Integrating, and applying the relation (62), k= /c, gives the particle velocity
)(
0
11),( krtir e
rkicr
Atr +
+=
u (114)
42/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
43/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
By analogy to the definition (75) of the specific impedance of a free plane wave,we define here the complex quantity Z as the ratio of the complex sound
pressure to the complex radial particle velocity at a point in a sound field
ru
pZ = (115)
For an outgoing spherical wave, the specific impedance, using equations (112)
and (114), becomes
ikr
ikrc
ikr
c
r+
=
+
==11
1
100
u
pZ (116)
Multiplying the numerator and denominator by the complex conjugate of the
latter, gives
++
+=
+
+=2222
22
022
22
0111 rk
kri
rk
rkc
rk
ikrrkc Z (117)
43/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
44/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
and the specific impedance Z can be divided into a resistive part Rand areactive partX, i.e.,
iXR+=Z (118)Some observations that follow from the preceding are:
(i) Nearfield
In the acoustic near field (kr = 2r/ 1), i.e., when the radius is small in
comparison to the wavelength, both the resistance and the reactance approach
zero, but the resistance does so more quickly. The reactance therefore
dominates, and the impedance approaches
krcir )( 0Z (119)
That means that for a given sound pressure, the particle velocity becomes largeand its phase shift approaches 90 with respect to the sound pressure. Forkr=1, both the resistance and the reactance are equally large, 0c/ 2,
44/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
45/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
(ii) Far field
In the acoustic far-field(kr= 2r/ 1), i.e., where the radius is large with
respect to the wavelength, the resistance approaches 0cand the reactance
approaches zero as rgoes to infinity.
The resistance dominates and the impedance approaches the same
expression as for plane waves
cr 0)( Z (120)
That means that the phase difference between the sound pressure and the
particle velocity approaches zero, as is the case for plane waves.
The curvature of the spherical waves in the farfield, with increasing distance
to the source, becomes all the less significant and the situationasymptotically approaches that of plane waves.
45/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
46/48
Fundamentals of Sound and Vibration
Indian Institute of Technology Roorkee
Wave Equation in Fluids
kr 201510500
0,8
0,6
0,4
0,2
1,0
R/0c
X/0c
Figure 10 Normalized resistance R/0c, and normalized reactanceX/0c, for outgoing
spherical wave propagation.
46/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
47/48
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Sound intensity for spherical wave propagation
The time-averaged sound intensity of outgoing spherical waves can be
determined by the same methods as for plane waves. Write the intensity as
2)*Re(
2)*)(Re()( rrr rI upup == (121)
Putting equations (112) and (114) into equation (121) gives
20
2
20
2
211Re
21)(
crA
ikrcrArI r
++ =
+= (122)
The time-averaged energy flow through a closed, spherical control surface
of radius ris
cArrrrIW0
2
2 24)(
+== (123)
For a loss-free medium, the sound power is therefore independent of the
radius, which is in agreement with the energy principle.
47/48
Fundamentals of Sound and Vibration
-
7/29/2019 Wave Equation in Fluids
48/48
Indian Institute of Technology Roorkee
Wave Equation in Fluids
Thank you