An Exact Analytical Solution to the Shallow Water Equations Near Beaches

M. Yourdkhani,Saber Zarrinkamar

doi:10.17951/aaa.2016.71.63

Abstract

<p>The nonlinear differential equation governing the dynamics of water waves can be well approximated by a linear counterpart in the case of shallow waters near beaches. The linear equation, which is of second order nature, cannot be exactly solved in many apparently simple cases. In our work, we consider the shape of system as a complete second-order polynomial which contains the constant (step-like), linear and quadratic shapes near the beach. We then apply some novel transformations and transform the problem into a form which can be solved in an exact analytical manner via the powerful Nikiforov-Uvarov technique. The eigenfunctions of the problem are obtained in terms of the Jacobi polynomials and the eigenvalue equation is reported for any arbitrary mode. </p>

Highlights

We frequently face wave equations in various fields of science and engineering including in water and ocean engineering
The shallow-water equations might be described by linear second-order partial differential equations
We use the powerful Nikiforov-Uvarov (NU for short) [11, 12, 13] technique, which has been widely used for various wave equations of physics, and thereby report the exact analytical eigenfunctions and the corresponding eigenvalue equation

Summary

INTRODUCTION

We frequently face wave equations in various fields of science and engineering including in water and ocean engineering. The shallow-water equations might be described by linear second-order partial differential equations. Shallow-water equations are based on conservation laws and have been so far analyzed by various numerical techniques [1]. In our work, bearing in mind the significance of shallow-water studies as well as the relative lack of analytical studies in the field, we consider a linear wave equation valid near beaches. By applying the common separation technique, we obtain the corresponding linear differential equation. We use the powerful Nikiforov-Uvarov (NU for short) [11, 12, 13] technique, which has been widely used for various wave equations of physics, and thereby report the exact analytical eigenfunctions and the corresponding eigenvalue equation

THE GEOMETRY OF THE SYSTEM AND THE GOVERNING EQUATION

THE NIKIFOROV-UVAROV METHOD

CONCLUSIONS