Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 53

practicality of this transmission mode. In this example and in Example 2-1, we
emphasize that the medium is uniform and nondissipative.

2.1.4 Example 2-3. Asymptotic stationary phase expansion.
Although Equation 2.3 is exact, it is nearly always a complicated function
of x and t, which does not render the main features and physical consequences
immediately obvious. Thus, we often obtain asymptotic expansions of Equation
2.3 for large x and t (e.g., see Carrier, Krook, and Pearson (1966)) to search for
wave-like solutions, as suggested by the examples given early in this chapter.
The “method of stationary phase” or the “saddle point method” leads to
(x,t) F(k) (2 )/(t | "(k)|) cos(kx - t - sgn "/4) (2.4)

where k(x,t) is the solution of
x = '(k)t (2.5)
Note how, in this large-time expansion, the cos(kx - t - sgn "/4) term
conveys a sense of wave-like propagation along trajectories defined by Equation
2.5. In the above, primes denote derivatives of the frequency with respect to the
real wavenumber (e.g., '(k) = 2 (EI/ A) k in Example 2-1).
Since Equation 2.4 assumes that "(k) does not vanish, the results of this
chapter do not apply to axial or torsional waves, which satisfy the classical wave
equation; however, they will figure importantly later in lateral vibrations. We
emphasize that we have not restricted ourselves to any particular area of
engineering physics. Our results will apply generally because the most severe
restriction is the large time evaluation of the superposition integral. Equation
2.4 is an oscillatory wave; however, unlike the elementary wave cos( x- t) in
Equation 2.1, it is not uniform since k depends on x and t. But, when x and t are
large relative to typical wavelengths and periods, Equation 2.5 shows how the
change of k within a few wavelengths or periods is small, e.g.,
k x /k = { '(k)/(k "(k))}(1/x) = O(1/x) (2.6)
provided "(k) is nonzero. Thus, apart from a simple phase change that is not
relevant here, the solution given by Equation 2.3 ultimately takes the form
(x,t) = a cos(kx- t) (2.7)
where a, k and = (k) are slowly varying functions of x and t in the sense of
Equation 2.6. Importantly, the asymptotic slowly varying solution assumes a
form identical to that of the uniform wave satisfying Equation 2.1. It is also
interesting to observe that when Equation 2.7 is rewritten in the form

(x,t) = a cos (x,t) (2.8)

where the phase function (x,t) satisfies