The semi-analytic theory and computation of finite-depth standing water waves

Abstract

We propose a Stokes expansion ansatz for finite-depth standing water waves in two dimensions and devise a recursive algorithm to compute the expansion coefficients. We implement the algorithm on a supercomputer using arbitrary-precision arithmetic. The Stokes expansion introduces hyperbolic terms that require exponentiation of power series, which we handle efficiently using Bell polynomials. Although exact resonances occur at a countable dense set of fluid depths, we prove that for almost every depth, the divisors that arise in the recurrence are bounded away from zero by a slowly decaying function of the wave number. A direct connection between small divisors and imperfect bifurcations is observed. They are found to activate secondary standing waves that oscillate non-uniformly in space and time on top of the primary wave, with different amplitudes and phases on each bifurcation branch. We compute new families of standing waves using a shooting method and find that Pad\'e approximants of the Stokes expansion continue to converge to the shooting method solutions at large amplitudes as new small divisors enter the recurrence. Closely spaced poles and zeros of the Pad\'e approximants are observed, which suggests that the bifurcation branches are separated by branch cuts.