Progressing waves in an infinite nonlinear string
| Autor: | B. A. Fleishman |
|---|---|
| Rok vydání: | 1959 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 10:329-334 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1959-0105903-x |
| Popis: | We wish to determine the types of waves which propagate in this medium at constant speed, and their propagation speeds, for various values of a and P. The problem reduces to that of interpreting the solutions of the nonlinear ordinary differential equation (5); the properties of these solutions, and the method of obtaining them, are discussed in §2. We proceed, in §§3, 4 and 5, to illustrate the noteworthy features of the wave solutions of (1) by considering particular combinations of a and p. Waves which travel at speeds greater (less) than one will be called supercritical (subcritical). In §3 it is shown that, for given positive a and P, all continuous wave solutions are of the form (13), where the amplitude a and (supercritical) velocity c may be chosen arbitrarily. In contrast, wave solutions of the simple wave equation (2) uu — uxx = 0 may have arbitrary shapes, but travel only with velocities +1. In §4 we note a duality principle between sub- and supercritical waves when the signs of a and P are changed. Finally, in §5, we see that when aP |
| Databáze: | OpenAIRE |
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