Nonlinear gravity–capillary standing waves in water of arbitrary uniform depth

Abstract

Gravity–capillary standing waves in water of arbitrary uniform depth are considered. The classical perturbation calculation yields unbounded coefficients for some critical values of the depth. In the present paper solutions valid near the first critical value of the depth are derived. A problem of non-uniqueness is discovered and discussed. It is shown that two solutions exist, one with higher frequency and one with lower frequency than the zeroth-order solution. They are found analytically at the critical value of the depth and numerically in an interval around it. Graphs of the results are included.