Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 76

(kx- t), for which we have u/ t = - a cos (kx- t) and u/ x = a k cos (kx- t).
If we substitute these in L, and take an average over the rapidly varying phase
= kx- t , we obtain an “average Lagrangian” L = 1/4 a 2 ( l 2 - Tk 2 ).
Taking variations in a leads to the Euler equation L a = 0, which is just the
phase relation of the slowly varying wave,

2 - Tk 2 = 0

(2.128)

l

but taking variations in leads to L / t - L k / x = 0 (since = - / t and k =
/ x, noting that disappears because averages over phase are taken), where
L = 1/2 a 2 l and L k = - 1/2 a 2 Tk. Thus,
( a 2 l )/ t + ( a 2 Tk)/ x = 0 (2.129)
Equation 2.129 reduces to our amplitude equation for a 2 when the properties l
and T are constant (see Equation 2.11).
But importantly, for variable properties, algebraic manipulation shows that
Equation 2.129 can be rewritten as either Equations 2.104 or 2.105 for wave
energy or momentum density! Equation 2.129 is clearly the more “universal”
conservation law, and the quantity L is known as the “wave action density.”
Equally significant is the applicability of KWT to fully nonlinear waves, where
sine solutions and Fourier superposition no longer apply; here, average
Lagrangian formulations provide elegant solutions to otherwise intractable
problems. Whitham (1974) gives an exposition of the early ideas; high-order
linear and nonlinear effects are discussed by Chin (1980).

2.6 Three-Dimensional Kinematic Wave Theory
Here we develop KWT in three dimensions. Our derivation will be
restricted to the low-order terms; high-order diffusive and dispersive effects,
such as those in Equations 2.94 and 2.95, are not modeled, but the required
modifications will be apparent to the skilled reader. We will avoid index
notation in order to eliminate any unnecessary confusion. Following our
approach for one-dimensional waves, we begin with the complex dispersion
relation for the uniform wave,
=

r (k x ,k y ,k z ,x,y,z,t) + i i (k x ,k y ,k z ,x,y,z,t)

(2.130)

Wave crest conservation requires that
k / t +

r = 0

(2.131)

or, in component Cartesian form,
k x / t +

r / x = 0

(2.132)

k y / t +

r / y = 0

(2.133)