### A THEORY FOR WAVES OF FINITE HEIGHT

#### Abstract

A theory for waves of finite height, presented in this paper, is an exact theory, to any order for which it is extended. The theory is represented by a summation liarmdnic series, each term of which is in an unexpanded form. The terms of the series when expanded result in an approximation of the exact theory, and this approximation is identical to Stokes' wave theory extended to the same order. The theory represents irrotational - divergenceless flow. The procedure is to select the form of equations for the coordinates of the particles in anticipation of later operations to be performed in the evaluation of the coefficients of the series. The horizontal and vertical components of these coordinates are given respectively by the following; (equations given).

From the above equations it is possible to deduce the expressions for velocity potential and stream function. The horizontal and vertical components of particle velocity are obtained by differentiating £ and ^with respect to time. Along the free surface z -1?!a 0 and z = Vs and all expressions reduce to simple forms, which in turn saves considerable work in the evaluation of the coefficients. The coefficients are evaluated by use of Bernoulli's equation. The final form of the solution is given by two sets of equations. One set of equations (same as above) is used to compute the particle position and the second set (the first time derivatives of the above) is used to compute the components of particle velocity at the particle position. That is, the particles and velocities are referenced to the lines of the stream function and the velocity potential. Expanding the two sets of equations, by approximation methods, results in one set of equation for computing particle velocity and no equations are required for the particle position.The unexpanded form requiring two sets of equations, being an exact solution, is more accurate theoretically, than the Stokes or the expanded form to the same order. Coefficients have been formulated for all terms of the order one to five for both the unexpanded and the expanded form of the theory, and are presented in tabular form as functions of d/L, as consecutive equations.

From the above equations it is possible to deduce the expressions for velocity potential and stream function. The horizontal and vertical components of particle velocity are obtained by differentiating £ and ^with respect to time. Along the free surface z -1?!a 0 and z = Vs and all expressions reduce to simple forms, which in turn saves considerable work in the evaluation of the coefficients. The coefficients are evaluated by use of Bernoulli's equation. The final form of the solution is given by two sets of equations. One set of equations (same as above) is used to compute the particle position and the second set (the first time derivatives of the above) is used to compute the components of particle velocity at the particle position. That is, the particles and velocities are referenced to the lines of the stream function and the velocity potential. Expanding the two sets of equations, by approximation methods, results in one set of equation for computing particle velocity and no equations are required for the particle position.The unexpanded form requiring two sets of equations, being an exact solution, is more accurate theoretically, than the Stokes or the expanded form to the same order. Coefficients have been formulated for all terms of the order one to five for both the unexpanded and the expanded form of the theory, and are presented in tabular form as functions of d/L, as consecutive equations.

#### Keywords

finite height; harmonic series; Stokes' theory;

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