Boussinesq-type Equations for Long Waves in Water of Variable Depth

The paper deals with the problem of the transformation of long gravitational waves propagating in water of variable depth. The main attention of the paper is focused on the derivation of equations describing this phenomenon. These equations are derived under the assumption that the non-viscous fluid is incompressible and rotation free, and that the fluid velocity components may be expressed in the form of the power series expansions with respect to the water depth. This procedure makes it possible to transform the original two-dimensional problem into a one-dimensional one, in which all unknown variables depend on time and a horizontal coordinate. The partial differential equations derived correspond to the conservation of mass and momentum. The solution of these equations is constructed by the finite difference method and an approximate discrete integration in the time domain. In order to estimate the accuracy of this formulation, theoretical results obtained for a specific problem were compared with experimental measurements carried out in a laboratory flume. The comparison shows that the proposed theoretical formulation is an accurate description of long waves propagating in water of variable depth.