On cross-waves

TLDR: In this paper, a second-order theory of the modes of oscillation of water in a tank with a free surface and wave-makers at each end leads to a form of Mathieu's equation for the amplitude of the cross-waves, which are thus an example of parametric resonance and may be excited at half the wave-maker frequency if this is within a narrow band.

Abstract: Cross-waves are standing waves with crests at right angles to a wave-maker. They generally have half the frequency of the wave-maker and reach a steady state at some finite amplitude. A second-order theory of the modes of oscillation of water in a tank with a free surface and wave-makers at each end leads to a form of Mathieu's equation for the amplitude of the cross-waves, which are thus an example of parametric resonance and may be excited at half the wave-maker frequency if this is within a narrow band. The excitation depends on the amplitude of the wave-maker at the surface and the integral over depth of its amplitude. Cross-waves may be excited even if the mean free surface is stationary. The effects of finite amplitude are that the cross-waves approach a steady state such that a given amplitude is achieved at a frequency greater than that for free waves by an amount proportional to the amplitude of the wave-maker. The theory agrees reasonably well with the experimental results of Lin & Howard (1960). The amplification of the cross-waves may be understood in terms of the rate of working of the wave-maker against transverse stresses associated with the cross-waves, one located at the surface and the other equal to Miche's (1944) depth-independent second-order pressure. The theory applies to the situation where the primary motion consists of standing waves and the cross-waves are constant in amplitude away from the wave-maker, but certain generalizations may be made to the situation where the primary waves are progressive and the cross-waves decay away from the wave-maker.