An Approximate solution to the Boussinesq type equations describing periodic waves

The paper concerns the non-linear problem of description of shallow-water waves of finite amplitude. The description is based on the conservation-law formulation, which seems to be particularly convenient in constructing an approximate solution of the problem considered. The analysis is confined to the one-dimensional case of waves propagating in water of constant depth. In the model considered, vertical acceleration of the fluid is taken into account, and thus, the fundamental equations of the problem are similar to those given by Boussinesq (Abbott 1979). The equations differ from those frequently used in shallow-water hydrodynamics which are based on the assumption of hydrostatic pressure. An approximate solution of the problem is constructed by means of a perturbation scheme with the third order expansion of the equations with respect to a small parameter. It is demonstrated that the solution procedure may be successfully applied only within a certain range of the two ratios defining wave height to water depth and the depth to wave length, respectively.