Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 55

The total differential dk is dk = k/ t dt + k/ x dx. If we divide by dt, we have
dk/dt = k/ t + dx/dt k/ x. Comparison with Equation 2.14 shows that

dk/dt = 0 (2.15)

provided that

dx/dt = C(k) = d (k)/dk (2.16)

Equation 2.15 states that the wavenumber k(x,t) does not change along the
characteristic or ray dx/dt = C(k) = d (k)/dk defined by Equation 2.16. In other
words, k(x,t) propagates with the group velocity d (k)/dk.
We turn our attention to the wave amplitude in Equation 2.12. The total
differential da = a/ t dt + a/ x dx, on division by dt, gives da/dt = a/ t +
dx/dt a/ x =

a/ t + C(k) a/ x. Thus, Equation 2.12 is equivalently

da/dt = -1/2 C'(k) k x a (2.17)

along the ray dx/dt = C(k). If an initial disturbance is limited to a finite interval
around x = 0, then for sufficiently large x and t, we have x/t = C(k); upon partial
differentiation with respect to x, we find that 1/t = C'(k) k x . Substitution in
Equation 2.17 shows that da/dt = -a/2t. The simplified amplitude law

da/dt = -a/(2t) (2.18)

easily integrates to

a = a 0 (k)/t 1/2 (2.19)

where a 0 is an integration constant related to the initial amplitude. Equation
2.19 shows that amplitude a(t) is inversely proportional to t, a fact consistent
with Equations 2.4 and 2.5 obtained using classical methods.
Also note that the group velocity d (k)/dk appears naturally in Equations
2.15 and 2.16 for k(x,t), and in the derivation of Equation 2.17 for a(x,t); it also
follows as a consequence of classical stationary phase theory, as is evident from
Equation 2.5. Now, if x is the distance between two neighboring
characteristics, differentiation shows that d(a 2 x)/dt = 2a(da/dt) x + a 2 C =
{2a(da/dt) + a 2 C'(k)k x } x + O( x) 2 . Since Equation 2.17 requires that da/dt =
-1/2 C'(k) k x a, it is clear that the curly bracketed { } quantity must vanish, so

d(a 2 x )/dt = 0 (2.20)

Thus, the total energy a 2 x between two consecutive rays defined by the group
velocity is constant in a homogeneous, nondissipative medium. All of our KWT
results so far are consistent with classical physical phenomena.