Wave Propagation Isoelastic
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Wave propagation in isotropic linear elastic solids
As linear elasticity is a theory of small displacements, it is natural to consider the equation of motion in
isotropic linear elastic solids,
Dv
Dt= (+ )( u) + 2u+ b,
at this limit. In the righthand side
Dv
Dt= 0(detF )
1D2u
Dt2,
0(1 + u)1h t
+Du
Dt ih t
+Du
Dt iu(x, t) 0
2u(x, t)
t2
where we have neglected terms of higher than linear order in u. Thus, in the absence of body forces, we
obtain the equation
02u
t2= (+ )( u) + 2u.
This equation is very useful in studying various vibrations and wave propagation in linear solids. We shall
start by considering wave propagation in an infinite isotropic material.
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Shear waves
Assume, first, that the deformation in the solid is isovoluminous, u = 0. Thus,
2u
t2=
02u.
This is a standard wave equation describing the propagation of isovoluminous or shear waves (S waves).
The velocity of propagation is
Ct =
r
0.
The same result can be obtained by considering the infinitesimal rotation vector, = 12 u, andtaking the curl of the linearized equation of motion,
02u
t2= (+ )( u) + 2u
12
02
t2= 12(+ ) [( u)] {z }
0+2 = 2
indicating that satisfies the shearwave equation for an arbitrary (small) displacement field.
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Dilatational waves
Consider, then, a displacement field not accompanied with rotation, i.e., = 0. Using the vectoridentity,
2u ( u) ( u) =0= ( u)
we get 02u
t2= (+ )( u) + 2u = (+ 2)2u,
which is the wave equation for irrotational or dilatational waves (P waves). The propagation velocity is
Cl =
s+ 2
0.
The same result can be obtained by taking the divergence of the linearized equation of motion,
02u
t2= (+ )( u) + 2u
02
t2= (+ )  {z }
2+2 = (+ 2)2,
so the dilatation u fulfills the equation of dilatational waves for a general deformation field.
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Properties of dilatational and shear waves
Let us study the general wave equation
02u
t2= (+ )( u) + 2u
in an infinite medium by using the Fourier transform pair
u(x, t) =1
2pi
Zd
ZZZd3k u(k, ) exp{i(t k x)}
u(k, ) =1
(2pi)3
Zdt
ZZZd3xu(x, t) exp{i(t k x)}
Thus, the transformed wave equation becomes
02u = (+ )k(k u) + k2u.
Write u = ut + ul, where k ut = 0 and k ul = 0. Thus, k(k u) = k2ul and
02ut = k
2ut
02ul = (+ 2)k
2ul.
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Transforming these equations back to the real space gives
02ut
t2= 2ut
02ul
t2= (+ 2)2ul,
where nowut = 0 andul = 0, showing that the propagation of shear waves (with amplitudeut) is decoupled from the propagation of dilatational waves (with amplitude ul) in a infinite medium.
Sufficiently far from the source, waves can be treated as plane waves. Thus, we can take
u(x, t) = ue exp{i(t k x)},
where the physical displacement corresponds to the real part of the complex quantity. Here e is a unit
vector in the direction of displacement and k is the wave vector giving the propagation direction of the
wave. The density perturbation related to the wave is = 0(1+ u)1 0 ik u0. Wehave
shear waves: phase speed
k= Ct =
r
0, e k, = 0
dilatat. waves: phase speed
k= Cl =
s+ 2
0; e k, = iku0 ei(tkx).
Note that Cl/Ct = [(+ 2)/]1/2 = [(2 2)/(1 2)]1/2.
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Wave propagation in a linear anisotropic elastic medium
Consistent with the assumed material isotropy, we obtained wave propagation speeds that are independent
of the propagation direction of the waves for both wave modes. In an anisotropic medium, however, the
situation is more complex: waves may have different propagation velocities in different directions as well
as mixed states of polarization (i.e., neither u k nor u k).Example. Consider wave propagation a simple transversely isotropic medium that has the following
constitutive equation:
Tij = (Ekk + E33)ij + 2Eij.
Thus,
T = (+ ) u+ u3
x3
+ 2u
and the wave equation in Fourier space becomes
02u = (+ )k(k u+ k3u3) + k2u.
Choose the coordinate system so that k = k(e1 sin + e3 cos ). Thus,
02u1 = (+ )k
2sin [u1 sin + (1 + )u3 cos ] + k
2u1
02u2 = k
2u2
02u3 = (+ )k
2cos [u1 sin + (1 + )u3 cos ] + k
2u3 + k
2cos
2u3
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Clearly, one solution is obtained by taking u2 6= 0 = u2 = u3 and /k = p/0. This is a
shear wave (k u) with isotropic velocity.The other two solutions are obtained by taking u2 = 0 and solving for nontrivial solutions of the
remaining pair of equations:
[02 (+ )k2 sin2 k2]u1 (+ )(1 + )k2 sin cos u3 = 0
(+ )k2 cos sin u1 + [02 (+ )(1 + )k2 cos2 k2]u3 = 0.
These are obtained from
0 =
k
2 +
0sin
2
0+
0(1 + ) sin cos
+ 0
sin cos
k
2 +
0(1 + ) cos
2
0
=
"
k
2 +
0sin
2
0
#"
k
2 +
0(1 + ) cos
2
0
#
+
0
2(1 + ) sin
2 cos
2
=
k
4 2
+
20(1 + cos
2) +
0
k
2+
0
2+
0
+
0(1 + cos
2).
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Thus,
k
2=+
20(1 + cos
2) +
0
s
+
20(1 + cos2 ) +
0
2
0
2 0
+
0(1 + cos2 )
=+
20(1 + cos
2) +
0 +
20(1 + cos
2)
i.e.,
k
2=
0and
k
2+
=+
0(1 + cos
2) +
0.
Substituting the first solution, 02 = k2, with isotropic propagation velocity back to the wave
equation gives
u1 sin = (1 + )u3 cos yielding k u = k(u1 sin + u3 cos ) = ku3 cos . Thus, the wave is in general not a pureshear wave nor a pure dilatational wave, but reduces to a shear wave at 0 or 0 or pi/2.The other solution, 0
2 = k2[+ (+ )(1 + cos2 )], gives
u1 cos = u3 sin
implying that k u = e2(k3u1 k1u3) = e2k(u1 cos u3 sin ) = 0 so this is a purelydilatational wave. However, it has a propagation velocity that depends on the propagation direction.
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Elastic waves at material boundaries
When an elastic bulk wave meets a boundary surface of two
media it is reflected and refracted. In general, a longitudinal
wave is reflected (or refracted) not only as a longitudinal but
also a transverse wave. This is known as mode conversion.
Consider, first, the reflection of longitudinal waves from the in
terface between a linear elastic isotropic medium and vacuum.
Choose the coordinate system so that the planar interface be
tween the medium and the vacuum is the plane x1 = 0 and
that the incident wave is propagating in the x1x2 plane at an
angle wrt. to the interface normal. Thus, x1 = 0
inciden
t lwav
e
reflected swave
reflected lwave
uincl = eA1 exp{i(t kl x)} = eA1 exp{i[t kl(x1 cos+ x2 sin)]} = e1
where e = e1 cos + e2 sin and 1 = A1 exp{i[t kl(x1 cos + x2 sin)]} andkl = Cl/ =
p(2+ )/02. The boundary condition is
e1 T (x1 = 0) = e1 T (x1 = 0+) = 0 T11 = T12 = T13 = 0 at x1 = 0
They can be only satisfied if, in general, both longitudinal and shear waves are reflected.
For the reflected waves,
urefl = e
2; u
reft = e
3
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with
2 = A2 exp{i[t kl(x1 cos + x2 sin)]}, e = e1 cos + e2 sin
3 = A3 exp{i[t kt(x1 cos + x2 sin )]}, e = e1 sin + e2 cos
Here, kt = Ct/ =p/02. Note that vibrations of the shear wave tranverse to the x1x2 plane
are absent, as the incident wave has no such fluctuations.
The total displacement is
u = uincl + u
refl + u
reft = e1 + e
2 + e
3
and the strain components are
E11 =u1
x1= e1
1
x1+ e
1
2
x1+ e
1
3
x1
= i(kl1 cos2 + kl2 cos2 kt3 sin cos )
E22 =u2
x2= i(kl1 sin2 + kl2 sin2 kt3 sin cos )
E12 =1
2
u1
x2+u2
x1
= 12i(kl1 sin 2 kl2 sin 2 kt3 cos 2)
and all others zero.
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Returning to the boundary conditions, we use the constitutive equation
Tij = Ekkij + 2Eij
and notice that T13 = 0 is automatically satisfied. The other T1i at x1 = 0
0 = T12 = 2E12
0 = T11 = (+ 2)E11 + E22 = 0C2l E11 + 0(C
2l 2C2t )E22
From the first relation,
kl1 sin 2 kl2 sin 2 kt3 cos 2 = 0 at x1 = 0 t, x2 (1)
which can be satisfied if the exponents in 1, 2, 3 are all equal at x1 = 0. Thus,
kl sin = kl sin= kt sin or
= and
sin
Ct=
sin
Cl
Substituting this back to (1) gives
kl(A1 A2) sin 2 ktA3 cos 2 = 0. (2)
The second boundary condition gives
kl(A1 + A2)(C2l 2C2t sin2 ) ktA3C2t sin 2 = 0, (3)
and (2) and (3) together give the amplitudes A2 and A3.
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The amplitude relations,
kl(A1 A2) sin 2 ktA3 cos 2 = 0kl(A1 + A2)(C
2l 2C2t sin2 ) ktA3C2t sin 2 = 0
can be immediately used to see that no shear waves are reflected (A3 = 0) if
(A1 A2) sin 2 = 0(A1 + A2)(C
2l 2C2t sin2 ) = 0
These equations can only be satisfied iff = 0 or = pi/2, whence A2 = A1. In all other cases,the reflected waves consist of both transverse and longitudinal waves. The amplitudes can be solved as
A2
A1=
sin 2 sin 2 (Cl/Ct)2 cos2 2sin 2 sin 2 + (Cl/Ct)2 cos2 2
(may vanish!)
A3
A1=
2(Cl/Ct) sin 2 sin 2
sin 2 sin 2 + (Cl/Ct)2 cos2 2,
A 2/A
2
1
+1
90
[o]
where Cl/Ct = [(2 2)/(1 2)]1/2. The relations can also be used to show that conservationof energy applies, i.e.,
A21Cl cos = A
22Cl cos+ A
23Ct cos .
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If the interface is between two elastic media, re
fracted waves have to be considered. The boundary
conditions at the interface (x1 = 0) are
e1 T (1) = e1 T (2) and u(1) = u(2).
For the angles, a law analogous to the first case is
obtained by setting the complex exponents equal at
x1 = 0:
sin
C(1)l
=sin1
C(1)l
=sin 1
C(1)t
=sin2
C(2)l
=sin 2
C(2)t
x1 = 0
112
2
inciden
t lwav
e
reflected swave
reflected lwave
(1) (2)
refracted swave
refracted
lwave
The amplitude relations can also be formed in similar manner as before, but of course now they are more
complicated.
Finally, we note that the any material impurities in form of grains of different density or elastic constants
embedded in the medium causes scattering of waves at the surfaces of the impurities. This scattering, of
course, leads to mode conversion. Scattering and reflection of waves can be used to probe the properties
of materials.
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Elastic waves in a finite medium. Longitudinal wave
Waves in infinite media are different from those in bounded media. As an illustration, let us consider
longitudinal waves propagating in a thin rod made of linear elastic material.
Consider a thin rod along the x1axis with the radius of crosssection much shorter than the length,
r0 l. Let one of its ends be subjected to a periodic longitudinal stress T11. Since the rod is verythin, we may assume that other components of stress are zero.
Thus, T11 = EE11 = E u1/x1 and
02u1
t2=T11
x1= E
2u1
x21
i.e., the perturbation propagates at speed
k=
sE
0=
s
0
3+ 2
+
along the rod. The speed is obviously different from both Ct and
Cl. The material in the rod moves in the axial direction and (for
6= 0) in the radial direction.
x1
x1
r
r
For the limit of negligible thickness of the rod to be valid, one needs to assume that the wavelength
2pi/k of the wave is much larger than the thickness of the rod (i.e., kr0 pi).
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Torsional and flexural waves
The propagation speed of a longitudinal wave is given by
Youngs modulus. If, instead, the applied stress is a shearing
stress, the speed is determined by the shear modulus
k=
sG
0=
r
0.
In this case, the material in the rod moves in a torsional manner,
i.e., only u 6= 0. The mode is called the torsional wave.Longitudinal and torsional waves are of symmetric type.
x1
x1
x2
x2
The most complicated case is that of a flexural wave, where all components of the displacement are
nonzero and one takes u sin and ur, u1 cos. The phase speed of the flexural wavebecomes proportional to k, i.e., the wave mode is dispersive. In a flexural wave, elements of the cylinder
axis are in a lateral motion, and this type of wave is called antisymmetric.
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Surface waves
Other types of waves in bounded media include, e.g.,
surface waves. Examples of these are seismic waves
called Rayleigh waves and Love waves. In these
waves, the amplitude of the disturbance decays ex
ponentially with depth from the surface.
Figure source: Wikipedia
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Wave propagation in fluids
We will start the study of wave propagation in fluids by considering Newtonian fluids. As for the elastic
waves, we start by introducing the equation of motion. This is, of course, the NavierStokes equation:
Dv
Dt= bp+ (+ )( v) + 2v,
where and are now the viscosity coefficients (i.e., not the Lames constants).
The NS equation alone does not form a closed system of equations. In addition, we need the continuity
equation,
t+ (v) = 0
and an equation of state. In case of wave propagation, we will use the equation of state,
p = P ()
In addition, in wave propagation problems we will set the body force to zero, b = 0.
The next step is to derive the dispersion relation for linear waves in Newtonian fluids.
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Linearized wave equation for Newtonian fluids
Start by linearizing the NS equation, term by term. For this purpose, write
= 0 + 1(x, t); v = v1(x, t); p = P ()
and consider the terms of the NS equation:
Dv
Dt= (0 + 1)
v1
t+ v1 v1
0
v1
t
p = P1
P
(0)1
(+ )( v) = (+ )( v1); 2v = 2v1
Thus,
0v1
t= P
(0)1 + (+ )( v1) + 2v1.
Together with the linearized continuity equation,
1
t+ 0 v1 = 0,
we now have a closed set of equations.
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Taking a time derivative of the linearized NS eq. and using the continuity equation we get
02v1
t2=
P
0
1t
+ (+ )( v1t
) + 2v1t
2v1
t2=
P
0
( v1) ++
0( v1
t) +
02v1
t
This is the linearized equation of wavepropagation for Newtonian fluids. For inviscid fluids ( = = 0)
we get2v1
t2=
P
0
( v1)
By taking a divergence of this equation and denoting G = v1 = 10 (1/t) we get
2G
t2=
P
0
2G
This is the wave equation for dilatational waves, which have a propagation velocity
C0 =
sP
0
=
sbulk modulus
0sound speed.
By taking the curl of the wave equation, we see immediately that shear waves do not propagate in inviscid
fluids.
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Properties of sound waves in inviscid fluids
Wave equation for inviscid fluids:2v1
t2= C
20( v1).
In infinite media,
2v1 = C
20k(k v1).
Thus, it is immediately clear that sound waves are longitudinal (v1 k) waves. Fourier transform ofthe continuity equation gives
i1 = i0k v1Thus, the condition of linearity can be obtained by requiring that 1/0 1, i.e.,
v1
k= C0 or v1/C0 1
The intensity (= flux of energy F = 120v21C0) of plane sound waves produced in laboratory variesbetween 0.1 and 0.3 W cm2. In air (0 = 1.2 103 g cm3; C0 = 340 m s1), this gives (forF = 0.1 W cm2)
v1/C0 7 103and in water (0 = 1.0 g cm
3; C0 = 1500m s1)
v1/C0 2.4 105,
both clearly in the linear regime.
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Effect of viscosity
The full equation for wave propagation in viscous fluids reads
2v1
t2= C
20( v1) +
+
0( v1
t) +
02v1
t
We are analyzing the propagation of a longitudinal wave, so assume v1 = v1(x1, t) e1. Thus,
2v1
t2= C
20
2v1
x21++ 2
0
2
x21
v1
t= C
20
2v1
x21+C20v
t
2v1
x21,
where v 0C20/( + 2) is called the viscosity relaxation frequency. For a plane wave, v1 =v1 exp{i(t kx1)}, we get
2= C
20k
2(1 + i/v) or k
2=
2
C20(1 + i/v),
which gives the dispersion relation of waves in a viscous fluid. Writing k = kr + iki we get
v1 = v1 exp(kix1) exp{i(t krx1)} and
k2r k2i =
2
C20(1 2/2v); 2krki =
3/v
C20(1 2/2v)= (k2r k2i )
v
which may be solved for the phase speed C = /kr and the attenuation length k1i .
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At low frequencies, v and ki kr, we find
C C0 and ki 2
2vC0.
The amplitude attenuation per wavelength, thus, becomes
= ki = 2piki/kr = pi/vNotes:
(i) The observed attenuation of ultrasonic waves can be used to obtain the value of
v =0C
20
+ 2=
0C20
b+ 43,
where b = + 23 is the bulk viscosity. As the shear viscosity is, in general, easy to measure,
the ultrasonic attenuation experiment provides a means to determine b. The ratio of bulktoshear
viscosity measured this way for some liquids is given in the table below.
Liquid Glycerol Water Methyl alcohol Toluene Benzene
b/ 1.1 2.5 3.2 13 100
(ii) Attenuation length for plane waves is proportional to 2 (or 2). Thus, to probe the propertiesof matter, one should use as low a frequency as possible. (However, the wavelength needs to be
much smaller than the typical scale sizes under investigation.)
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(iii) Highly viscous fluids. For large enough values of b+43,
v becomes comparable to . Thus, the condition
w is no longer satisfied and the values of krand ki have to be solved exactly from the dispersion
relation. Thus (exercise),
C2=2
k2r= 2C
20
1 + 2/2w1 + (1 + 2/2w)
1/2
ki =C
2C20
/w
1 + 2/2w
As an example, the phase speed of the wave is plotted in the figure for Glycerol (5% of water).
Clearly, the theory does not well describe the observations. This means that the NavierStokes
equation is not valid for highly viscous systems. This is to be expected, as in the constitutive
equation for Newtonian fluids, all higherthanfirstorder derivatives of the velocity were neglected.
By applying the same reasoning in the present case, one expectsC20v t
2v1
x2i
C202v1x2i
,
but this for plane harmonic waves implies /w 1. Thus, this limitation is builtin limitationfor Newtonian fluids.
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Diffusion of vorticity. Viscous waves
Consider the rotational motion, i.e., vorticity in a fluid. Recall the linearized NS equation (omit the
subscript 1)2v
t2= C
20( v) +
t
+
0( v) +
02v
Taking the curl of both sides gives
2w
t2=
t
02w
w
t=
02w,
which is a diffusion equation for vorticity. Consider a transverse wave v = v2 exp{i(t kx1)}e2.Obviously, v = 0 and, thus,
2v2
t2=
0
t
2v2
x21so
2= ik
2/0 or k
2= i0/
Setting, again, k = kr + iki gives
k2r k2i = 0 and krki = 0/2, i.e., kr = ki = (0/2)1/2
The attenuation per wavelength, = 2pi, so the waves, called viscous waves, are very rapidly damped.
Thus, transverse fluctuations can penetrate a viscous fluid by only about one wave length thickness.
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Energy equation and heat flow.
So far we have applied an equation of state for the pressure. Next, instead, recall the energy equation
derived in Ch. 5:
De
Dt= TijDij q,
where e is the internal energy per unit mass, Tij = (p + Dkk)ij + 2Dij is the stress tensor,Dij =
12(vi/xj + vj/xi) the rateofstrain tensor, and q is the heat flux.
The internal energy in a fluid can be obtained from thermodynamics as e = cV , where cV is the
specific heat capacity in constant volume and is the thermodynamic temperature. Using this relation
and the Fouriers law of heat conduction, q = , where is the coefficient of heat conductivity,allows us to write the energy equation as
cV D
Dt= TijDij + 2.
Now, assuming that the pressure is given by p = P (, ) allows one to write
p =P
+P
Thus, the effect of heat flow on sound propagation can be studied by subtituting this into the Navier
Stokes equation and using it together with the energy equation and the equation of continuity after
linearising the equations.
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Effect of heat flow on wave propagation in inviscid fluids
For simplicity, consider an inviscid fluid, Tij = pij, with a thermodynamic pressure given by theideal gas law, P (, ) = R. Thus,
D
Dt= v
Dv
Dt= R R
cV D
Dt= R v + 2 = RD
Dt+ 2.
Linearize, i.e., substitute = 0 + 1(x, t), = 0 + 1(x, t) and v = v1(x, t) to these
equations:
1
t= 0 v1
0v1
t= R01 R01
cV 01
t= R0
1
t+ 21.
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Take a divergence of the second equation and use the first to eliminate v1 to get
21
t2= R021 + R021
cV 01
t= R0
1
t+ 21.
Consider the propagation of plane waves, i.e., t i and ik, to get
(2 R0k2)1 R0k21 = 0
i0R1 + (k2 + cV 0i)1 = 0,
so
(2 R0k2)(ik2 cV 0) + R200k2 = 0
gives the dispersion relation. For = 0 we get
cV2= R(cV + R)0k
2
and using the thermodynamic relation cp = cV +R, where cp is the specific heat capacity in constant
pressure, we get
C20 =
cp
cVR0 = R0
consistent with the earlier result for an adiabatic equation of state, p = const.
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For > 0, we obtain again a complex wavenumber for a real frequency. Denoting = /0 we
have
(2 R0k2)(ik2 cV ) + R20k2 = 0
and solving for gives
2= R0k
2 R20k
2
ik2 cV= R0k
2 ik2 cV R
ik2 cV= R0k
2 cp ik2cV ik2
Taking, again k2 k2r +2ikrki and approximating that ik2 ik2r i/C200 = icp/with 0C20cp/, we get
k2r + 2ikrki
2
R0
cV icp/cp icp/
=2
R0
1 i/1 i/
and separating the real and imaginary parts gives
C2= C
20
1 + 2/21 + 2/2
= R01 + 2/21 + 2/2
ki =C( 1)2R0
2/
1 + 2/2= kr
12
/
1 + 2/2
Thus, thermal conductivity produces dispersion at frequencies & and at the highest frequenciesC2 R0, i.e., isothermal sound propagation. The attenuation has a similar dependence at lowfrequencies on as attenuation by viscosity.
The value of is 1013 rad s1 for water and 1010 rad s1 for air.
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The viscous attenuation and the attenuation due to heat conduction are additive, i.e.,
kclassi = kvisi kthiand the obtained attenuation of waves is called classical absorption. For monoatomic fluids, classical
values are found to be in good agreement with experiments. For polyatomic fluids, the observed at
tenuation is significantly greater than classical values. This occurs, e.g., because sound waves disturb
the equipartition of energy between the internal and external degrees of freedom of the molecules. The
phenomenon is called thermal relaxation.
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Shallowwater waves
Also in fluids, we can consider surface waves. Consider a layer of incompressible, inviscid fluid of thickness
d on top of a solid horizontal floor. Let h = d+(x, t) be the height of the surface from the bottom.
Consider a shallow layer, so that the velocity component along z can be taken as small. Conservation of
mass implies
t(h(x, t) dx) = [h(x+ dx, t)v(x+ dx, t) h(x, t)v(x, t)] dx
x(hv)
h
t+
x(hv) = 0
Similarly, conservation of momentum implies (exercise)
(hv)
t+
x
hv
2+ 12gh
2= 0
Linearising
t+ d
v
x= 0;
v
t+ g
x= 0
giving2
t2= gd
2
x2
i.e., a wave propagating at speed C =gd is found. The solution is valid for d and d.
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Solitary waves and solitons
If the nonlinearities are not neglected but taken into account to the lowest order, one can derive after
transforming to frame moving with the linear waves (x = xCt) and rescaling of the dependent andindependent variables the Kortewegde Vries equation for the propagation of the wave (see, e.g., G.L.
Lamb, Elements of Soliton Theory, Wiley, 1980). In dimensionless form
t + 6x + xxx = 0,
where the subscript denotes differentiation. Seek for wavelike solutions of the equation, i.e., take
= f(x ct)
and substitute to the equation:
cf + 6ff + f = 0 cf + 3f2 + f = A  f
12cf2 + f3 + 12f 2 = Af + BB = 12f 2 + f3 12cf2 Af = 12f 2 + V (f)
where V (f) = f3 12cf2Af andA andB are integration constants. Thus, f(x) behaves like theposition z(t) of a particle with total energy of B in a cubic potential V (z) [i.e., 12z
2 + V (z) = B].
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As V (f) = 3f2 cf A, there are two extrema of thepotential at
f =cc2 + 12A
6; A > 112c2,
where f is a maximum and f+ is a minimum. [Note: forA = 112c2 only one extremum and for A < 112c2 none.]If B < V (f), solutions oscillating between two roots ofV (f) = B are obtained.
V
ff+
f
Choosing A = B = 0, i.e.,
V (f) = f2(f 12c)
gives a potential with local maximum V = 0 at f = 0. This admits a solution with f, f 0 asx and with a maximum value of f = 12c somewhere in between.Its a easy exercise to show that this solution is given by
(x, t) = f(x ct) = c2 cosh2{
c2 (x ct a)}
,
where a is an arbitrary constant. This describes a solitary wave, i.e., a nonperiodic, localized propagating
wave with a permanent form.
The solitarywave solution of the KdV equation is an example of a soliton. Solitons are solitary waves
which can interact with other solitons emerging from the collision unchanged, except for a phase shift.
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Burgers equation. Shock waves
The KdV equation t + 6x + xxx = 0 without the nonlinear term represents a dispersive wave,
i.e., the linear dispersion relation becomes
+ k3= 0 or
k= k2
showing that highfrequency fluctuations are lagging behind the lowfrequency fluctuations. The soliton
solutions describe a system, which attains a balance between nonlinear and dispersive effects.
Another nonlinear equation of considerable importance in fluids is the Burgers equation,
vt + vvx = vxx,
where is viscosity. It represents a system that can attain a balance between nonlinearity and dissipation.
A propagating wave solution can be obtained by v = f(x ct)
cf + ff = f 12A cf + 12f2 = f df
f2 2cf + A =dx
2
x x02
=
Zdf
c2 A (f c)2A

Therefore, at x , f c c2 A = v, soc2 A = 12(v+ v) = 12v and
c = 12(v+ + v) = v. Thus,
v(x, t) = v 12v tanhx vt x04/v
This represents a solution propagating from left to right at speed v, where the speed of the fluid increases
from right to left (i.e., a fast fluid is overtaking a slower one). The transition layer from the slower to
the faster speed has a thickness of 4/v. This kind of a wave is a shock wave.
Actually, as Burgers equation contains no pressure gradient term, it is not a completely valid description
of shocks. By using p = P () we get as a onedimensional momentum equation
vt + vvx = P ()x/+ vxx,where the equation of continuity, t + (v)x = 0, has to be used to eliminate the extra term. Write
P () = P ()/ and seek for a propagating solution in form v = v(x ct), = (x ct).
c + v + v = 0 (c v) =M = const. = Mc v
c v + vv = P () + v
v = P() cv + 12v2 = P Mc v
cv + 12v2
which shows that the speed cannot cross v = c without yielding an infinite density and that the pressure
gradient term keeps the flow speed from crossing the speed of propagation of the shock wave.
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For a more physical description of shock waves in hydro
dynamics, one should use an energy equation instead of
an equation of state.
For 0, the thickness of the shock becomes verysmall, and the solution can be treated as a discontinuity.
The conservation laws for mass, momentum and energy
then read
v =+v
+
v2+ P =+v
+
2+ P+
12v
3+ (P + U)v
=
12+v
+
3+ (P+ + U+)v
+,
v+
v+
v
4/v
v
x
v
x
v
c
Burgers equation
Real shock wave
~/v
c
where v = c v is the velocity measured in the shock frame and U is the internal energy per unitvolume. For an ideal gas it is given as
U =P
1where = cp/cV . One can solve the conservations laws for the values behind the shock (subscript )for known values in the ambient medium (subscript +).
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Wavepropagation in inhomogeneous media
So far, we have considered wave propagation in homogeneous media, infinite, semiinfinite and finite.
We briefly mentioned scattering of elasic wave in granular material, but otherwise inhomogenieties have
not been considered. Next, focus on wave propagation in an inhomogeneous medium. We will restrict
to the weakly inhomogeneous case, where the scale length of the medium is large compared to the wave
length.
Consider a wave propagating in such a medium. Let f be the quantity that fluctuates, and write it as
f = Aei.
The quantity is called the eikonal. For a monochromatic plane wave in a homogeneous medium, we
may write
= k x t,where k and are constants, but in an inhomoneous medium, a more general treatment is needed.
In a small region, we may expand the eikonal as
= 0 + x + tt,
where the origin of the coordinate system lies within the region. Thus,
k = and = t,
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which corresponds to the fact that in each small region, the wave can be regarded as planar, but now
the frequency and wavenumber may vary from one region to the other.
The wavelength and the frequency are related to each other by a dispersion relation of form
= (k;x, t),
where now the slow dependence of the propagation characteristics (like phase speed) on the properties
of the medium (i.e., x and t) is explicitly shown. Thus, we obtain
t + (,x, t) = 0, ( ? )
which shows a direct analogy to the HamiltonJacobi theory: identifying
eikonal Hamiltons principal function S(q, t;),xi generalized coordinate qik = xi canonical momentum pi = qiS, and(k,x, t) Hamiltonian H(q, p, t)of the system, the equation (?) is just the HamiltonJacobi equation H(q, qS, t) + tS = 0 for a
particle,which now corresponds to a wave packet propagating in the medium.
As the wavepacket can be treated as a particle with a canonical momentum of k and a Hamiltonian ,
the equations of motion of a linear wave become
x =
k; k =
k; =
t
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Sound propagation in a nonisothermal atmosphere
Consider a polytropic atmosphere with absolute temperature = (z). Thus,
= (k, z) C(z)k with C(z) 1/2
gives the dispersion relation of sound waves in the atmosphere. The equations of motion for sound waves
with C(z)/C (z) become
x =
k=C(z)k
k; k =
x= C (z)ez; =
t= 0
Thus, the canonical momenta kx and ky, corresponding to the cyclic coordinates x and y, and the fre
quency corresponding to the energy of the wave packet are constants of motion. Consider propagation
in the xz plane. We have
x =Ckx
k=kx
C
2(z); z =
Ckz
k=kz(z)
C
2(z); k
2z(z) =
2
C2(z) k2x
dx
dz=kx
kz= kxC(z)p
2 k2xC2(z),
dt
dz=
C2kz=
C(z)p2 k2xC2(z)
These equations can be integrated to give the ray path, x(z), and the propagation time, t(z), of the
sound waves from a source in the atmosphere.
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In the simplest case of C = C0z/h we get (for upward propagating waves, kz > 0)
x x0 =Z zz0
C0kxz/h dzq2 k2xC20z2/h2
= z0C(z0)kz(z0) C(z)kz(z)
C(z0)kx
t =
Z zz0
h/C0 dz
zq2 k2xC20z2/h2
=h
C0ln
/kx +
p2/k2x C2(z0)
/kx +p2/k2x C2(z)
z
z0
!
It is interesting to note that nonvertically propagating waves (kx 6= 0) will be reflected from upperlayers of the atmosphere if C(z) is a function increasing without a limit. The reflection height, zref , is
determined by
0 = k2z(zref) =
2
C2(zref) k2x i.e., C2(zref) =
2
k2x,
which in the case of linearly increasing sound speed, C = C0z/h, gives
zref =h
kxC0.
If there is a maximum value of the sound speed in the atmosphere, only waves with k2x > 2/C2max =
[k2x + k2z(z0)]C
2(z0)/C2max k2x/k2z(z0) > C2(z0)/[C2max C2(z0)] will be reflected.
Wave propagation is a powerful tool to analyze sound speed in inhomogeneous fluids. An example of this
is provided by helioseismology, where the internal structure of the Sun can be probed by analysing sound
waves propagating below the surface and observed as fluctuations of different quantities at the surface.
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Evolution of wave intensity
So far we have analyzed only the propagation of individual wave packets, but we still need informa
tion about the evolution of the intensity of the waves. To obtain this, we will take the waveparticle
analogy one step further. We will consider the wave field as being composed of quanta with energy
~ and momentum ~k. The wave field can be seen as a gas of such quanta. We will, however, useclassical mechanics to describe the motion of these quanta in phase space (x, ~k). This is called thesemiclassical approximation. For simplicity, we will consider a timeindependent medium.
The energy density of the wave field isU(x, t) =Rd3kUk
Rd3kA2k, whereAk is the amplitude
of the waves in the wavenumber range from k to k + dk. In the semiclassical approximation, the
energy density can be given as
Uk = ~Nk,whereNk is the number density of the wave quanta in the 6D phase space and
is the wave frequencyin the restframe of the medium. According to statistical mechanics, the phase space density of the wave
quanta follows a continuity equation in phase space, i.e.,
Nk
t+ (xNk) +
k (kNk) = 0,
where x = /k and k = /x. Thus,
Nk
t+
k Nk
x Nkk
= 0.
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In a static medium (v0 = 0, = = const.), we can multiply the equation with the constant
wave frequency and writeUk
t+
k Uk
x Ukk
= 0.
However, if the medium is not at rest, we have to use the Dopplershift formula
= v0 k
to calculate the frequency in the rest frame to relate the number density of the wave quanta to the
observable wave intensity Uk = ~Nk.Weakly nonlinear wave propagation in inhomogeneous media can be treated by adding terms describing
collisions of the wave packets to the righthand side of the continuity equation, i.e., using Boltzmanns
equation for the wavequantum gas. These wavewave interactions are the basis of the weakturbulence
theory of often used, e.g., in plasma physics.
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