Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 67

x = k - k 0 = K(X,T)

(2.84)

and
- t = ( - r 0 ) = (X,T) (2.85)
The amplitude a(X,T) in Equation 2.81, following Equations 2.4 and 2.5, is
likewise assumed to vary slowly with x and t, changing significantly only over
X and T scales of O(1).
Interestingly, partial differentiation of Equation 2.84 with respect to time,
of Equation 2.85 with respect to space, and use of the identity

x / t = t / x

lead to the law for wave crest conservation

K/ T + (X,T)/ X = 0 (2.86)

previously taken as axiomatic.
Now, let us digress and consider any function u(x,t) that possesses slowly
varying wavelike solutions of the form
u(x,t) = U( ,X,T) (2.87)

Using the chain rule of calculus, we deduce the property
u(x,t)/ x = U X X x + U x = U X + K U (2.88)
In Equation 2.88 and in the subsequent analysis, subscripts are used to
represent partial derivatives. With some effort, we can show by mathematical
induction (Chin, 1976) that the n th partial derivative of U( ,X,T) with respect to
x satisfies
u nx (x,t) = K n U n (2.89)

+ [nK n-1 U (n-1) ,X + 1/2 n(n-1)K n-2 K X U (n-1) ]

+ 2 [ 1/2 n(n-1)K n-2 U (n-2)

XX

+ 1/2 n(n-1)(n-2)K n-3 K X U (n-2) ,X

+ 1/8 n(n-1)(n-2)(n-3)K n-4 K X 2 U (n-2)

+ 1/6 n(n-1)(n-2)K n-3 K XX U (n-2) ]

+ 3 [1/12 n(n-1)(n-2)(n-3)(n-4)K n-5 K X K XX U (n-3)
+ 1/6 n(n-1)(n-2)(n-3)K n-4 K XX U (n-3) ,X

+ 1/4 n(n-1)(n-2)(n-3)K n-4 K X U (n-3) ,XX

1/6 n(n-1)(n-2)K n-3 U (n-3) ,XXX

+

+ 1/24 n(n-1)(n-2)(n-3)K n-4 K XXX U (n-3)

+ 1/8 n(n-1)(n-2)(n-3)(n-4)K n-5 K X 2 U (n-3) ,X

+ 1/48 n(n-1)(n-2)(n-3)(n-4)(n-5)K n-6 K X 3 U (n-3) ]

n U nX