Cross-waves. Part 2. Experiments

Abstract

In this paper experiments in which cross-waves were generated in front of a wave-maker at one end of a long channel are described. The primary field produced was a progressive wave train, but, at certain frequencies, a standing wave developed in front of the wavemaker. This wave, whose crests were at right angles to the wavemaker and which had frequency half that of the wavemaker, is known as a cross-wave.An instability mechanism for the formation of cross-waves in a long channel has been presented in Part 1 (Mahony 1972). In the present paper we describe some measurements concerning the formation of the waves. The initial growth rate of the waves has been measured and the curves of neutral stability determined for two of the possible modes. The results are in good agreement with Mahony's theory. After the early stages in the development of the waves, there is an increase in their growth rate. This new rate was found to be about twice the initial growth rate. Also reported are some measurements of the amplitude of the cross-wave field along the channel.

The Sloshing Motion of Cross‐Waves

“…Barnard and Pritchard's [7] experiments demonstrated the generation of cross-waves in a long water channel. They also observed that "the cross-waves never reach a true state of equilibrium, and after the cross-wave amplitude has passed through a maximum, a wave detaches itself from the wavemaker, propagates along the channel," and eventually decays.…”

“…Experiments on cross-waves have been conducted by Barnard and Pritchard [7], Lichter and Shemer [8], Ayanle et al [6], and Underhill et al [9]. Barnard and Pritchard's [7] experiments demonstrated the generation of cross-waves in a long water channel.…”

“…Miles and Becker [5], through analytical approximations and numerical integration, determined those stationary envelopes that are evanescent at large distances from the wave maker. They compared their envelopes with Barnard and Pritchard's [7] experiments and Lichter and Chen's [10] numerical calculations, and suggested that stationary envelopes with either no or one maximum are stable for sufficiently small amplitudes and evolve into limit cycles for somewhat larger amplitudes. Ayanle et al [6] observed a mixedmode state.…”

Pattern formation of parametrically driven surface waves induced by vibration

“…The subharmonic instability is a robust transition that is observed easily in vertically forced Faraday waves, wavemaker experiments (Schuler 1933;Lin & Howard 1960;Barnard & Pritchard 1972;Lichter & Underhill 1987;Shemer & Lichter 1990;Underhill, Lichter & Bernoff 1991) and horizontally vibrated systems (Porter et al 2012). These waves arise from parametric excitation (Benjamin andUrsell 1954, Garret 1970) and can exhibit complex behaviour such as modulations, mode interactions, and chaotic dynamics.…”

“…Cross waves (Barnard & Pritchard 1972) are strictly (2:1) subharmonic waves produced by a wavemaker consisting in a partially submerged horizontally vibrating plate. They were already identified by Faraday (1831) as a series of apparently permanent ridges projecting outward 'like the teeth of a coarse comb' (Miles and Henderson 1990).…”