Contributions to the Theory of Waves in Non-linear Dispersive Systems

This paper makes contributions to the general theory of wave propagation in conservative systems under conditions when the proportional change in amplitude or wavenumber over a distance of one wavelength is very small. For linear systems, such propagation is governed by the well-known theory of group velocity; there is “frequency dispersion”, in the sense that energy in components of different frequency is propagated at different group velocities. For non-linear systems without frequency dispersion, e.g. acoustic systems, a different, but also well-known, modification of the waveform occurs. It may be called “amplitude dispersion”, in that different values of an amplitude variable like the pressure are propagated at different speeds. A much more general theory of non-linear systems, where frequency dispersion and amplitude dispersion would be expected to be in competition, has been given by Whitham (1965b). Energy does not play a key role in the theory, because it is easily transferred between components of different frequencies. The fundamental equation follows from Hamilton's principle in an averaged form. In examples given by Whitham, changes in, for example, wavenumber (or amplitude) are propagated at two different velocities, because the fundamental equation is hyperbolic. However, in the limiting case of infinitesimal amplitude, the equation is parabolic and only one velocity of propagation (the group velocity) occurs. Thus, Whitham showed that non-linearity can “split” the group velocity. This paper is concerned with the inference of detailed conclusions from Whitham's theory, to enable comparisons with experiment that will show the range of applicability of the theory. It attempts to obtain these in the simplest case, namely, that of one-dimensional propagation when Whitham's “pseudo-frequencies” are absent. If the relationship between frequency ω and wavenumber k for infinitesimal amplitude is ω = f(k), then for finite amplitude the equation is shown to be hyperbolic or elliptic respectively, according as ω−f(k)f*(k) takes positive or negative values. For gravity waves on deep water this product is negative and these, it is inferred, may be good for comparison of theory with experiment in the elliptic case. A new non-linear non-perturbational theory of waves under the combined action of gravity and surface tension is used to indicate that waves at 9.6 c/s on mercury may be suitable for comparison with experiment in the hyperbolic case. When non-linear effects are only moderate, approximate transformations of Whitham's equation to the axisymmetric Laplace and wave equations respectively, in the elliptic and hyperbolic cases, are used to obtain particular solutions for comparison with experiment. A feature of these solutions is the appearance of discontinuities in wavelength. For example, when a wavemaker creates gravity waves of fixed frequency whose amplitude first increases and then decreases, the theory predicts that the length of waves in the group decreases ahead of the point of maximum amplitude and increases behind it. This produces in turn a concentration of energy towards the centre of the group, which continues during the whole period before a discontinuity in wavelength actually forms. This solution in the elliptic case is obtained with the aid of the theory of imaginary characteristics.