Gravity effects in the initial value problems of water wave theory

Abstract

The theory described in this paper is directed towards obtaining a general expression for the development of the free surface of a fluid, subsequent to a given initial state and prescribed boundary conditions, as a power series in g, the gravitational acceleration. In an earlier paper [4], a result, applicable to the particular case of the entry of a thin wedge into an incompressible fluid, was obtained and gave the shape of the free surface as such a power series. This series was valid for values of the ratio ut/x < 1, where u, was the (constant) velocity of entyr of the wedge; x the horizontal distance from the vertex of the wedge; and t the times elapsed after entry. This particular problem was first investigated by Mackie [6] who derived an asymptotic solution.

Presidential Address 1968: Initial value problems in two-dimensional water wave theory.

“…Equation (17) agrees with a previous result [9] for the shape of the free surface, as a power series in g, due to a wavemaker U(y,t). By virtue of equations ( 12), ( 13) and ( 14), we may also determine from (24) and (25) similar expressions for the free surface profile due to pressure distributions of the form (x,t) =8(0)8(x).…”

“…Solutions to these problems have been given, Jae ohs JEON independently, by Miles [12] and Mackie [10]. For both the pressure and the wavemaker problems, solutions may be obtained by a double transform method using an even Fourier transform in x and a Laplace transform in ¢ [9].…”