Mild-slope approximation for long waves generated by short waves

The mild-slope approximation has become a popular basis for calculating inf initesimal surface waves on slowly varying depth. It is less restrictive he nce more advantageous than the ray and parabolic approximations for describ ing diffraction and refraction by bathymetry and/or by complex coastlines. Since its computation involves only two horizontal coordinates, the mild-sl ope equation is also numerically less demanding than the solution of fully three-dimensional equations for a horizontal area with sides much greater t han the typical wavelength. By consideration of nonlinear effects of the se cond order, the mild-slope approximation for long waves over slowly varying depth is derived here, in order to provide a convenient basis for predicti ng long-period resonance in a large harbor by short-period wind waves.