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- W1990274078 abstract "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<0, there exist both periodic and nonperiodic waves, of suband supercritical type. In this case, also, it is possible to have discontinuous waves which travel at unit speed. In [l]1 the author and F. A. Ficken have demonstrated the existence of time-periodic solutions, with period p, of a related equation with forcing and damping terms," @default.
- W1990274078 created "2016-06-24" @default.
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- W1990274078 date "1959-01-01" @default.
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- W1990274078 title "Progressing waves in an infinite nonlinear string" @default.
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- W1990274078 doi "https://doi.org/10.1090/s0002-9939-1959-0105903-x" @default.
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