BERKHOFF APPROXIMATION IN A PROBLEM ON SURFACE GRAVITY-WAVE PROPAGATION IN A BASIN WITH BOTTOM IRREGULARITIES

The problem of the propagation of small-amplitude surface gravity wave s in a basin of constant mean depth with one-and two-dimensional botto m roughness is solved in the framework of the Berkhoff model by a mean -field method. In both cases the solutions obtained are compared with the solutions of sets of exact linearized equations of the hydrodynami cs of an incompressible fluid. The comparison of the exact and approxi mate mean-held attenuation coefficients has shown that the Berkhoff ap proximation is appropriate for the solution of this problem in the cas e of shallow water for an arbitrary correlation length of bottom irreg ularities and in the case of arbitrary depth and large-scale irregular ities. An explanation is given for the limits of applicability of the Berkhoff approximation which are connected with the weak variability o f the vertical structure of the wave field in shallow water and in a b asin with large-scale depth fluctuations. The mean-field attenuation c oefficients reach their maximum values in the region k(0)h(0) greater than or equal to 1 (where k(0) is the wavenumber of the surface gravit y wave in a basin of constant depth h(0)). The location of these maxim s is practically independent of the correlation length of the bottom i rregularities. For the case of one-dimensional irregularities the effe ct of bottom roughness on the surface gravity wave velocity is investi gated. It is shown that the surface wave in a basin with an uneven bot tom may propagate more slowly, as well as faster than the wave in a ba sin with an even bottom, depending on the relations between the wavele ngth, depth and correlation length of the bottom imperfections.