A Nonlinear Theory of Water Waves for Finite and Infinite Depths

This paper is concerned with the construction by a direct approach of a fairly general nonlinear theory of an incompressible inviscid fluid for application to water waves, which upon specialization yields (a) a theory suitable for deep waters, (b) one for waters of finite (non-shallow) depth and also reduces to (c) the theory of a directed fluid sheet given previously by Green & Naghdi (Proc. R. Soc. Lond. A 347, 447-473 (1976a); J. appl. Mech. 44, 523-528 (1977)). Whereas our development again is based on a model known as the directed- (or Cosserat-) surfaces model, our approach to the subject differs from the earlier one in two respects: (1) the basic conservation equations are recast here in an Eulerian form by means of a procedure utilized recently for viscous fluid flow in channels Green & Naghdi (Arch. ration. Mech. Analysis 86, 39-63 (1984)) and (2) a new procedure for identification of various quantities in the conservation equations which can be specified by constitutive equations. After development of the basic equations in the context of the purely mechanical theory (part A) and their reductions to the three above-mentioned categories for an incompressible, homogeneous, inviscid fluid, the rest of the paper (arranged as parts B, C, and D) deals with applications to various water wave problems under gravity. These include steady-state solutions for surface disturbance of a stream by pressure when the fluid depth may be any one of the above-mentioned three categories and a detailed study of nonlinear stern waves over waters of infinite depth.