Resonant sloshing in shallow water

Abstract

The ordinary differential equation \[ {\textstyle\frac{1}{3}}\kappa^2(g^{\prime\prime}+g) - \lambda g - {\textstyle\frac{3}{2}}g^2 + \frac{2}{\pi} \cos t = -\frac{3}{2}\int_{-\pi}^{\pi}g^2\,{\rm d}t, \] which represents forced water waves on shallow water near resonance, is considered when the dispersion κ is small. Asymptotic methods are used to show that there are multiple solutions with period 2π for a given value of the detuning parameter λ. The effects of dissipation are also considered.

Steady free-surface flow over spatially periodic topography

“…Ockendon et al 1986;Amundsen et al 2007). In the latter problem the fKdV equation arises in the limit of small amplitude oscillations and shallow water.…”

“…In the latter problem the fKdV equation arises in the limit of small amplitude oscillations and shallow water. When the driving frequency is close to resonance, Ockendon et al (1986) used an asymptotic analysis to obtain periodic solutions with very rapid and short-lived spiked bursts occurring at regular intervals in the time signal. Local solutions are constructed inside each of the rapid burst regions; these solutions have a solitary-type character which are not dissimilar to those we have presented in figure 2 (see their figure 4, for example).…”

“…Following a genuinely non-autonomous viewpoint helped to avoid such assumptions, which are often used in similar fKdV equations (e.g. Ockendon et al 1986;Amundsen et al 2007). …”

“…The equation also applies in other flow problems, notably the resonant sloshing of water in a horizontally oscillating tank (e.g. Ockendon & Ockendon 1973;Cox & Mortell 1986;Ockendon et al 1986;Amundsen et al 2007). Its popularity stems from its relative simplicity and utility-as a single equation flow model-and from the fact that it provides insight into the effects of weak nonlinearity and dispersion without solving the corresponding fully nonlinear problem.…”

Nonlinear resonant characteristics of shallow fluid layers

“…Following the procedure outlined by Ockendon et al (1986), it is pertinent to consider the case of two-dimensional motion in a tank of width pl where the depth of the undisturbed liquid layer is hl. The tank oscillates horizontally with frequency u and amplitude a.…”