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dy/dt =
r
ky
= 1/2 c (k
x
2
+ k
y
2
+ k
z
2
)
-1/2
2k
y
= c k
y
(k
x
2
+ k
y
2
+ k
z
2
)
-1/2
(2.163)
dz/dt =
r
kz
= 1/2 c (k
x
2
+ k
y
2
+ k
z
2
)
-1/2
2k
z
= c k
z
(k
x
2
+ k
y
2
+ k
z
2
)
-1/2
(2.164)
The right sides of Equations 2.162 to 2.164 are proportional to the components
in Equation 2.160b; thus, the rays corresponding to the isotropic dispersion
relation in Equation 2.161 are parallel to the normals to the wave front.
Now, consider the
anisotropic
problem, with different sound speeds c
x
,
c
y
,
and c
z
in the x, y and z directions, that is
=
r
= ( c
x
2
k
x
2
+ c
y
2
k
y
2
+ c
z
2
k
z
2
)
1/2
(2.165)
The new velocity components are
dx/dt =
r
kx
= c
x
2
k
x
(c
x
2
k
x
2
+ c
y
2
k
y
2
+ c
z
2
k
z
2
)
-1/2
(2.166)
dy/dt =
r
ky
= c
y
2
k
y
(c
x
2
k
x
2
+ c
y
2
k
y
2
+ c
z
2
k
z
2
)
-1/2
(2.167)
dz/dt =
r
kz
= c
z
2
k
z
(c
x
2
k
x
2
+ c
y
2
k
y
2
+ c
z
2
k
z
2
)
-1/2
(2.168)
The vector associated with Equations 2.166 to 2.168 is clearly
not
parallel to the
normal
n
= ( k
x
i
+ k
y
j
+ k
z
k
)/ (k
x
2
+ k
y
2
+ k
z
2
) in Equation 2.160b. Straightness
and wave front orthogonality are only true of motions satisfying the three-
dimensional wave equation in homogeneous isotropic media. The normal in
Equations 2.160a,b is perpendicular to the wave front, but it is related to the
phase velocity, and not the ray velocity associated with energy propagation.
2.7 References
Bourbie, T., Coussy, O. and Zinszner, B.,
Acoustics of Porous Media
, Gulf
Publishing Company, Houston, 1987.
Chin, W.C.,
Physics of Slowly Varying Wavetrains in Continuum Systems, Ph.D.
Th e s i s
, Massachusetts Institute of Technology, Cambridge, Massachusetts,
1976.
Chin, W.C., “Effect of Dissipation and Dispersion on Slowly Varying
Wavetrains,”
AIAA Jo u r n a l
, Feb. 1980, pp. 149-158.
Chin, W.C.,
Modern Reservoir Flow and Well Transient Analysis
, Gulf
Publishing, Houston, 1993.
Cole, J.D.,
Perturbation Methods in Applied Mathematics
, Blaisdell Publishing,
Waltham, Massachusetts, 1968.
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