Geophysical Ray Tracing - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 307

6.1 Classical Wave Modeling- Eikonal Methods
and Ray Tracing
A suitable starting point for an elementary discussion of eikonal methods
and ray tracing is the classical three-dimensional wave equation introduced
earlier in this topic,
2 P/ t 2 - c 2 ( 2 P/ x 2 + 2 P/ y 2 + 2 P/ z 2 ) = 0 (6.1)
describing P waves in an isotropic, homogeneous, nondissipative earth. In
Equation 6.1, P(x,y,z,t) represents pressure or normal stress, with the effects of
shear ignored in the model. The speed of sound c is determined from the
formula c = (E/ ), where is the mass density, and E is Young' s modulus.
Note that Equation 6.1 is identical to the pressure equation used to model
acoustic waves in liquids in Chapter 3. There, we considered waveguide
applications leading to eigenvalue problems, but in this section, we will
specifically consider wave propagation in unbounded domains.

6.1.1 The plane wave.
In Chapters 1 and 2, we examined the uniform plane wave “sin (kx- t)”
propagating in a single dimension. Its generalization in three-dimensions is the
monochromatic wave
P(x,y,z,t) = P 0 sin (k x x + k y y + k z z - t) (6.2)
where P 0 is a constant amplitude. Substitution in Equation 6.1 leads to the exact
dispersion relation
2 = c 2 (k x 2 + k y 2 + k z 2 )

(6.3)

connecting the wave frequency to each of the wavenumbers k x , k y , and k z ,

and the speed of sound c.

6.1.2 High frequency limit.
Equation 6.2 is a “plane wave,” since P(x,y,z,t) is constant whenever the
argument k x x + k y y + k z z - t is constant, that is, P does not vary on the plane
k x x + k y y + k z z = t + constant. But Equation 6.2 is not the only substitution
formally possible, for it is common to consider expansions of the form

) -k P k

P(x,y,z,t) = exp[i (t- )] (i (6.4)

k = 0
whenever periodic solutions having a specified frequency are desired. Note
that the leading k = 0 term in the summation is simply P 0 (x,y,x). Here, and the
functions P k are independent of and t, depending only on x, y, and z. Also,
the exponential in Equation 6.4 is analogous to our previous “sin (kx -

t),” but