Transformation of Long Surface and Tsunami-Like Waves in the Ocean with a Variable Bathymetry

We consider a transformation of long linear waves in an ocean with a variable depth. We calculate the transformation coefficients (the coefficients of transmission and reflection) as the functions of frequency and the total depth drop for three typical models of bottom profile variation: (i) piecewise-linear, (ii) piecewise-quadratic, and (iii) hyperbolic tangent profiles. For all these cases, exact solutions are obtained, analysed and graphically illustrated. This allows us to derive the transformation coefficients in the analytic form for the subsequent comparison with the results of approximate, numerical, or experimental study. We show that the results obtained are in agreement with both the energy flux conservation and Lamb’s formulae in the limiting case of zero frequency. We also study wave transformation on the underwater barriers and trenches of different shapes and compare the results obtained.