Abstract
Two time-dependent equations for wave propagation on rapidly varying topography are developed using different theoretical approaches and are shown to be identical. The developed equations include higher-order bottom effect terms proportional to the square of bottom slope and to the bottom curvature. Without these higher-order terms, the equations developed are reduced to the time-dependent mild-slope equations of Smith and Sprinks and Radder and Dingemans, respectively. For a monochromatic wave, the equation reduces to the extended refraction-diffraction equation of Massel or the modified mild-slope equation of Chamberlain and Porter, which in turn, without the higher-order terms, reduces to the Berkhoff's mild-slope equation. For a monochromatic wave, the theory is verified against other theoretical and experimental results related to the waves propagating over a plane slope with different inclination and over a patch of periodic ripples. For random waves, numerical tests are made for the transmission of unidirectional random waves normally incident on a finite ripple patch.
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