Nonhydrostatic waves in channels of varying depth

Abstract

Abstract The periods of standing waves in closed canals can be influenced by nonhydrostatic effects if the shallow-water criterion is not satisfied In the analysis this can be handled by an expansion with respect to the square of the aspect ratio whereby the vertical accelerations can be taken into account. From the zeroth- and first-order sets of equations of motion and boundary conditions a linear partia1 differential equation governing nonviscous infmitesimal surface displacements is established. This equation (a reformulation of the linearised Boussinesq equations extended to allow for the effects of large changes of depth) together with boundary conditions constitutes, after separation of variables, an eigenvalue problem determining the periods of the standing waves. Interest has been directed towards cases with linear and quadratic bottom profiles where the depth displays a singularity, i.e. the surface intersects the bottom. The semianalytical procedure for determining the eigenvalues involves a Frobenius expansion of the general solution, whereby the coefficients of the series become functionally dependent on the eigenvalues. By applying the boundary conditions to the series solution the eigenvalues can be evaluated by numerical iteration. In order to show the effects of different degrees of nonhydrostaticity, computational as well as experimental results on the gravest mode of the abovementioned cases are presented. For varying aspect ratios of the system, the experimental data is in good agreement with the predictions.