An alternative approach to study irrotational periodic gravity water waves

Biswajit Basu,Calin I Martin

doi:10.1007/s00033-021-01578-8

Abstract

We are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

Highlights

The paper by Benjamin and Lighthill [3], back in 1958, highlighted a quantity called flow force–defined as the rate of momentum of fluid flow–and emphasized its role as the driver of the flow
A novel flow force formulation of the free surface irrotational nonlinear water wave problem has been proposed by Basu [1] by way of exploiting the invariance property of the flow force on the fluid’s bed and on the free surface
Flow force has been used to uniquely parameterize the subset of waves with crests located on a fixed vertical line and to verify the Benjamin-Lighthill conjecture with values of Bernoulli’s constant close to a critical value by Kozlov et al [24]

Summary

Introduction

The paper by Benjamin and Lighthill [3], back in 1958, highlighted a quantity called flow force–defined as the rate of momentum of fluid flow (corrected for pressure and normalized with respect to density)–and emphasized its role as the driver of the flow. A novel flow force formulation of the free surface irrotational nonlinear water wave problem has been proposed by Basu [1] by way of exploiting the invariance property of the flow force on the fluid’s bed and on the free surface. We avail of bifurcation-type results and prove the existence of a solution curve consisting of exact periodic solutions to the irrotational gravity water wave problem in water of finite depth by a method different from the one in [1]. The main result showing the existence of a local curve of solutions to the water wave problem in the context of a fixed flow force is presented in Sect. The last section of the paper lists several properties of the Hilbert transform and of the Dirichlet–Neumann operator used throughout the paper

The water wave problem and several reformulations

Existence of exact solutions