Introduction of a Simple Cnoidal Wave Formulation Based on Nonlinear Interaction of Wave-Wave Principles

Abstract

In this study, a simple and efficient approach based on nonlinear wave interaction fundamentals is theoretically proposed to generate surface profile of the cnoidal waves. The approach includes Newton-Raphson algorithm to calculate the Ursell parameter and using a simple formulation. The wave profile resulted by means of introduced approach is determined as a superposing of limited number of cosine harmonics without encountering difficulties of using elliptic or hyperbolic functions, or any complex and complicated differential equations. It is demonstrated that a cnoidal wave profile is a result of high order self nonlinear interaction of primary frequency. Some definite energy is transmitted to higher harmonics due to nonlinear interactions. The amount of transmitted energy is controlled by Ursell parameter. The desirable accuracy determines the number of included harmonics in the proposed formulation and relative error of approach can be predicted based on Fourier and least square analysis techniques. The outputs of the proposed method are verified with cnoidal resulted from elliptic functions and the high efficiency of new approximation is revealed for engineering applications. The calculation of wave parameters such as energy flux, volume flux and radiation stress for cnoidal wave can be approximated using the proposed method. Using this approach, a physical interpretation of the Bm parameter (introduced in the first order of cnoidal wave theory) is presented. The calculation of several parameters such as velocity vectors and dynamic pressure duo to cnoidal waves is very simple by means of proposed approach.

Keywords

• cnoidal waves
• nonlinear triad interactions
• harmonics superposition
• least square analysis