The solitary wave (John Scott Russell’s “Great Wave of Translation” ) was discovered in 1834 by Russell during investigations of the Edinburgh Union Canal. In the shallow water wave problem the solitary wave is a single hump, of large length- scale, travelling without change of form. Its existence was predicted theoretically by Rayleigh and Boussinesq in the 1870’s and then definitively by Korteweg and de Vries in 1895 in the paper in which the now-famous KdV equation was first derived. The wave of permanent form can exist only because of a delicate balance between the linear mechanism of dispersion (produced partly because of finite-depth effects and partly because of surface tension) and the nonlinear one by which the higher parts of a water wave travel more quickly than the lower. Dispersion allows the different frequency or wavelength components of a disturbance to travel away at different speeds, so that a concentrated elevation would in time disperse into an oscillatory wave train along which the wavelength changes slowly and continuously. The nonlinear mechanism in isolation leads to overturning of the wave, as the higher parts overtake the lower. The dispersive and nonlinear mechanisms precisely balance in the solitary wave, and the amplitude, lengthscale and propagation speed are all related; in a certain nondimensional set of coordinates moving with the speed of propagation of waves of infinitesimal amplitude the solitary wave solution of KdV is so that if the lengthscale a −l is given, the amplitude is 2a 2 and the speed V = 4a 2 — narrow pulses are high and travel fast.
KeywordsDepression Soliton Crest Acoustics
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