Further Developments in a Nonlinear Theory of Water Waves for Finite and Infinite Depths

This paper is a companion to an earlier one (Green & Naghdi 1986, Phil. Trans. R. Soc. Lond. A 320, 37-70 (1986)) and deals with certain aspects of a nonlinear water-wave theory and its applications to waters of infinite and finite depths. A new procedure is used to establish a 1-1 correspondence between the lagrangian and eulerian formulations of the integral balance laws of a general thermomechanical theory of directed fluid sheets, as well as their associated jump conditions in the presence of any number of directors. (Such a correspondence between lagrangian and eulerian formulations was previously possible in the special case of a single constrained director.) These results are valid for both compressible and incompressible (not necessarily inviscid) fluids. Applications are then made to special cases of the general theory (including the jump conditions) for incompressible inviscid fluids of infinite depth (with two directors) and of finite depth (with three directors) and the nature of the results are illustrated with particular reference to a wedge-like boat.