Some special cases treated by the Whitham theory

Abstract

The theory of the preceding paper (by G. B. Whitham ) may be valuable in problems of nonlinear dispersive wave systems governed by equations too complicated for analytical or numerical treatment. It suggests, in fact, that the development of wave groups, whose parameters vary gradually enough, can be represented by much less complicated approximate equations, derived from the assumption that in each separate small region the waves are closely like plane periodic ones. Nevertheless, the effort demanded to solve even those approximate equations would normally be very substantial, and therefore it is important (see §1) to get a preliminary idea of whether the solutions would correspond well with reality or not, by calculating the implications of the theory in detail for some rather simple system or systems and comparing them with experiment. It is argued (§2) that unidirectional waves on deep water are a suitable system for this purpose. Their propagation, as an analysed by the Whitham theory, is governed by how the Lagrangian density L depends on the frequency ω and the wavenumber k . In fact, L k 2 / ρg (where ρ is the density of water) is a function of z = ω 2 / gk alone; and, by interpolating between known values for low waves ( z → 1) and for the wave of greatest height ( z = 1·20), a polynomial approximation L k 2 / ρg = ⅛ [( z - 1) 2 –( z - 1) 3 –( z - 1) 4 ], expected to be good throughout the interval 1 ≼ z ≼ 1·20, is found (§4). It is deduced from this (§5) that, when a wavemaker produces a slightly modulated train of waves of constant frequency on deep water, the exponential rate of increase of modulation with distance from the wavemaker, shown by Brooke Benjam in later in this Discussion to be proportional to amplitude for waves of moderate amplitude, attains a maximum for a wave of about half the greatest height, and then falls to zero, no increase of modulation being predicted in the case of waves of more than three-quarters of the greatest height, which should exhibit instead Whitham’s ‘splitting of the group velocity’ (figure 2). Brooke Benjamin & Feir’s agreement with experiment is improved (figure 3) by this modification, the remaining discrepancy being readily attributable to dissipation. However, additional experiments are needed for large values of the amplitude. The theory is worked out and compared with experiment also for groups within which the wave amplitude varies by a large fraction of itself, although only gradually on a scale of wavelengths and subject to an upper limit on magnitude (§7). The predicted change with time, previously worked out (Lighthill 1965 b ) for one special form of group (reciprocalquadratic; see §8), is calculated in §9 for a general group of uniform wavenumber whose amplitude varies smoothly and symmetrically about its maximum; and detailed results are given for a sinusoidal modulation with arbitrary depth of modulation. The theory predicts that the group remains symmetrical, and that the peak amplitude increases, first according to a hyperbolic-cosine law but then still faster, until a critical condition is reached when the distribution of wave amplitude has a cusp at the centre of the group, where also the wavenumber exhibits a discontinuity. Beyond this critical time the assumptions of the theory break down and a changed type of propagation is expected. Experiments by Feir (described below, p . 54), in which the spatial period of modulation was about 12 wavelengths, showed just such a catastrophic alteration at a time close to that predicted (see §10), and wavenumber changes close to those given by the theory. Dissipation of wave energy (e. g. by friction at the side walls) was again too great to permit an accurate check on the theory’s amplitude predictions, but the value of the critical time (after which the observed wave group becomes markedly asymmetric) is encouraging evidence that the theory is reliable so long as the variations it predicts remain smooth and slow enough to satisfy its own basic assumptions. Earlier, in §§3 and 6, the way (not essentially more complicated) in which the theory would be used to study a two dimensional pattern of stationary waves on a uniform stream of deep water is outlined.