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Parametrically excited, progressive cross-waves

The variational formulation of the nonlinear wavemaker problem, previously applied (Miles 1988) to cross-waves in a short tank, is extended to allow for slow spatial, as well as slow temporal, variation of cross-waves in a long tank. The resulting evolution equations for the envelope of the cross-waves are equivalent to those derived by Jones (1984) and may be combined to obtain a cubic Schrödinger equation in a semi-infinite domain. The corresponding criterion for the stability of plane waves (i.e. for the temporal decay of cross-waves) agrees with Jones but differs from Mahony (1972). Weak damping is incorporated, and those stationary envelopes that are evanescent at large distances from the wavemaker are determined through analytical approximations and numerical integration and compared with the experimental observations of Barnard & Pritchard (1972) and the numerical calculations of Lichter & Chen (1987). These comparisons suggest that stationary envelopes with either no or one maximum are stable for sufficiently small amplitudes (solutions with multiple maxima may be stable but more difficult to attain) and evolve into limit cycles for somewhat larger amplitudes, but the analytical question of stability remains open.