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The superharmonic normal mode instabilities of nonlinear deep-water capillary waves

We present results of the calculation of superharmonic normal mode perturbations to the exact nonlinear deep-water capilary wave solution of Crapper (1957). By using the method of Longuet-Higgins (1978a), we are able for the first time to consider all waveheights up to and including the maximum for two-dimensional perturbations. We find agreement with the recent asymptotic analysis of Hogan, Gruman & Stiassnie (1988). Superharmonic instabilities are found at various waveheights less than the maximum.