Perturbation Theory of Wave Propagation Based on the Method of Characteristics

Fox, Phyllis A.

doi:10.1002/sapm1955341133

Cited by 34 publications

(11 citation statements)

References 7 publications

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“…It is seen that (19) has the same form as the shallow water results (11) and (12); the implications are therefore parallel to those discussed in the sequel to (11) and (12).…”

mentioning

confidence: 51%

“…We have remarked that (11) and (12) are also valid for the case of an initially isolated disturbance. It is interesting to compare the predictions of (11) and (12) with those of (1) in a "simple wave" case in which-an exact solution of (1) is available. Let c denote the local signal speed, given by x/1 + r/ 1 + 1/2ft.…”

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confidence: 91%

“…The latter approach lends itself more readily to systematic algebraic formulation (especially for higher order terms) but it lacks the simplicity of interpretation associated with (11) and (12). The problems discussed here are of course ones in which the distortion of the characteristic curves is important.…”

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confidence: 96%

“…But within our first approximation, rt + u e(rt () + u (1)) 2e[ (from(7)), so that r/+ u is constant along any straight line in the (x, t) plane for which dx/dt 1 + 1/4(rt + u). Thus, treating rt and u now as functions of x and only, we have(11) (r/+ u), +(r/+ u)x[1 +](rt + u)] 0, and similarly, from the second of (10),(12) (n u),-(n u) [a + k(n u)] 0.Since the characteristics of (11) and(12) are straight lines, these equations are easily solved. For example, r/(x, 0) e cos 2zrx and u (x, 0) 0 in the periodic problem of 2, so that rt + u is constant along the (x, t) line passing through (xx, 0) and defined by x xx t{1 + ]e cos 2rxx}, and similarly rt u is constant along the line x-x2Note that a systematic consideration of higher order terms might make the use of basic variables like t(1 + e2tl + ), "r et, useful; see Lick[6] and[7].Downloaded 11/19/14 to 129.49.23.145.…”

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confidence: 96%

“…Of course, it is also possible to obtain a perturbation expansion for the characteristics of (1) themselves, and this we consider in 4. A consequence of (11) and (12) is that the time at which a discontinuity (shock wave) occurs may be computed. In fact, the characteristics of (11) will intersect one another at tx =-4/(3eM), where M is the most negative value of F'(x)+ G'(x); the characteristics of (12) will intersect one another at t2 4/(3eM2), where M2 is the most positive value of F'(x)-G'(x).…”

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confidence: 99%

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Dual Time Scales in a Wave Problem Governed by Coupled Nonlinear Equations

Pearson¹

1981

SIAM Rev.

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The method of multiple time scales is used to obtain a uniformly valid perturbation solution for initial value problems (periodic or isolated initial disturbance) arising in shallow water motion or in compressible gas flow. The results are used to modify the original equations so as to lead to a particularly convenient problem formulation. The time at which discontinuities first appear is found to be accurately predictable.Although the specific problem considered here does not seem to have been previously treated by this method, the basic ideas are familiar in related contexts. The paper is in part expository, and comparisons are made with other methods, including conventional perturbations, perturbation of characteristics, and direct numerical solutions.1. Introduction. The method of multiple time scales has proven effective in obtaining useful approximations for the solutions of appropriate nonlinear ordinary differential equations. Applications to partial differential equations are less common, although there is an increasing interest in its use for wave-type problems. Actually,

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“…It is seen that (19) has the same form as the shallow water results (11) and (12); the implications are therefore parallel to those discussed in the sequel to (11) and (12).…”

mentioning

confidence: 51%

mentioning

confidence: 91%

mentioning

confidence: 96%

mentioning

confidence: 96%

mentioning

confidence: 99%

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