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A family of exact solutions approximating the gravitational waves of maximum amplitude. (English. Russian original) Zbl 0888.76009
Phys.-Dokl. 40, No. 10, 534-537 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 344, No. 5, 623-626 (1995).
The paper presents a particular family of exact solutions to equations describing finite-amplitude plane periodic waves in an ideal liquid layer of a finite depth over a flat bottom, neglecting the surface tension. The flow is assumed to be irrotational. By means of the appropriately chosen conformal map, the problem is reduced to a nonlinear boundary value problem. In the special case corresponding to the surface waves of the maximum amplitude (with cusp singularities on the free surface), the problem can be reduced to a system of three coupled ODEs that admits a solution in quadratures, in the form of a one-parametric family of periodic waves. The analytical findings are corroborated by comparison with numerical results.

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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