Analytic derivation of the dependence of the frequency of standing surface waves on the amplitude in a liquid of finite depth

The dependence of the frequency of non-linear free standing surface waves on the amplitude in a heavy ideal incompressible liquid of finite depth contained in a rectangular vessel with vertical sides is derived analytically, assuming fairly shallow waves. Attention is focused on the case of the near-critical depth of the liquid, which has not been studied analytically before. The hydrodynamic equations are written in Lagrange variables, and an asymptotic procedure based on the method of Krylov and Bogolyubov is used. The relation obtained is compared with those obtained by other researchers, either numerically or analytically for far from critical depths.