Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 79

which shows that frequency varies along rays only when the medium varies in
time (e.g., as tension increases, the pitch on a guitar string increases).

2.6.4 Energy variation.
Equation 2.130 assumes that the uniform wave dispersion relation =
r (k x ,k y ,k z ,x,y,z,t) + i i (k x ,k y ,k z ,x,y,z,t) is available. The three-dimensional
energy law corresponding to Equation 2.104 is

E/ t + ( r kx (k x ,k y ,k z ,,x,y,z,t)E)/ x
+ ( r ky (k x ,k y ,k z ,,x,y,z,t)E)/ y + ( r kz (k x ,k y ,k z ,,x,y,z,t)E)/ z
= E{ 2 i (k x ,k y ,k z ,,x,y,z,t) + r t (k x ,k y ,k z ,,x,y,z,t)/ r } (2.158)
Equation 2.158 can be written along the rays of Equations 2.150 to 2.152, by
extending the steps taken in Equations 2.108 to 2.112. Changes to energy
density arise from three effects: time-dependent heterogeneities, dissipation and
geometric spreading, all of which are included in Equation 2.158.

2.6.5 Ray topology.
Earlier we showed how rays remain straight in homogeneous media, while
in heterogeneous media, they are typically curved. Now we demonstrate that
they need not in general remain orthogonal to their wave fronts. Again, wave
fronts are surfaces are defined by the constancy of the phase function ,
(x,y,z,t) = (x,y,z,t) - constant = 0 (2.159)

From vector calculus, the unit normal to this surface is given by
n = ( x i + y j + z k )/ ( x 2 + y 2 + z 2 ) (2.160a)
where the subscripts represent partial derivatives, and i , j , and k are unit vectors
in the x, y, and z directions. Since the wavenumber components satisfy k x =
/ x, k y = / y and k z = / z, we can write Equation 2.160a as
n = ( k x i + k y j + k z k )/ (k x 2 + k y 2 + k z 2 ) (2.160b)
In general, rays are obtained as characteristics of the “eikonal equation,” that is,
the trajectories in dx/dt =

r kx , dy/dt = r ky , and dz/dt = r kz (see Equations

2.150 to 2.152).

2.6.6 Example 2-9. Acoustics application.
Consider the dispersion relation in Equation 3.46, describing three-
dimensional acoustic waves in isotropic homogeneous media,
=

r = c ( k x 2 + k y 2 + k z 2 ) 1/2

(2.161)

We easily see that the velocities in Equations 2.150 to 2.152 satisfy
dx/dt = r kx = 1/2 c (k x 2 + k y 2 + k z 2 ) -1/2 2k x
= c k x (k x 2 + k y 2 + k z 2 ) -1/2

(2.162)