A HIGHER ORDER THEORY FOR SYMMETRICAL GRAVITY WAVES

Peter L. Monkmeyer

Abstract


A higher order theory is presented for symmetrical, non-linear gravity waves As a consequence of the generality employed, the theory includes the full range of possible wave lengths, water depths and wave heights that may be encountered, and brings them into one unified formulation Thus, the theory encompasses both linear and non-linear waves, including Airy waves, Stokes waves, cnoidal waves and the solitary wave Based on the work of Nekrasov, a complex potential in the form of an infinite series is developed to describe the flow field The potential satisfies the bottom (horizontal) condition as well as the kinematic surface condition exactly Furthermore, the dynamic surface condition is satisfied by numerical calculation of the series coefficients which appear in the complex potential The calculation of these coefficients is accomplished by solving a set of non-linear algebraic equations, with the aid of a Newton-Raphson iteration procedure and matrix inversion Coefficients of the complex potential have been obtained for a fifth order analysis and preliminary results are presented in tabular form A brief discussion of the characteristics of the waves, including wave speed, wave shape and the height of the highest possible wave follows.

Keywords


symmetrical waves; gravity waves; higher order theory

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