Abstract
The linear instability of small but finite amplitude cnoidal waves on shallow water is examined. A new set of equations valid under the shallow water approximation is used to derive periodic wavetrains and to study the behavior of small perturbations. A resonance condition k2 = k1 + 2k, similar to that used by other authors, is required to hold between the side-band perturbations with wavenumbers k1 and k2, and the main wave with wavenumber k. It is also necessary for the perturbations to have a specific nonzero frequency Ω, measured with respect to the main wave which is stationary in a wave frame. In the region 0 < F2 < 1, where F is a Froude number of the flow underlying the cnoidal wave, all wavetrains are unstable to these side-band perturbations. Like all other parametric instabilities, the instability is due to the resonant transfer of energy from the main wave to the perturbing waves.
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