A new form of Boussinesq equations for long waves in water of non-uniform depth

Abstract

Abstract The paper describes the non-linear transformation of long waves in shallow water of variable depth. Governing equations of the problem are derived under the assumption that the non-viscous fluid is incompressible and the fluid flow is a rotation free. A new form of Boussinesq-type equations is derived employing a power series expansion of the fluid velocity components with respect to the water depth. These non-linear partial differential equations correspond to the conservation of mass and momentum. In order to find the dispersion characteristic of the description, a linear approximation of these equations is derived. A second order approximation of the governing equations is applied to study a time dependent transformation of waves in a rectangular basin of water of variable depth. Such a case corresponds to a spatially periodic problem of sea waves approaching a near-shore zone. In order to overcome difficulties in integrating these equations, the finite difference method is applied to transform them into a set of non-linear ordinary differential equations with respect to the time variable. This final set of these equations is integrated numerically by employing the fourth order Runge - Kutta method.