The energies of the evanescent modes as well as the propagating mode are derived when predicting the transformation of water waves over a varying depth using the linear wave theory. The energies of evanescent modes have three types of components, i.e., compounds of propagating and evanescent modes, those of identically equal evanescent modes, and those of identically different evanescent modes. For the case of waves propagating over a downward or upward step, the eigenfuncion expansion method is used to estimate and compare the wave energies of evanescent modes with those of the propagating mode. Both the kinetic and potential energies of the evanescent modes decrease exponentially with the horizontal distance. For most cases of the step, it is all right to ignore the energies of evanescent modes except an upward step with very small values of the depth ratio of up- to down-wave regions at a relative water depth of up-wave regions with 0.11π 1 1 k h =.


As wind waves generated in deep water approach the nearshore zone, they experience many important physical phenomena caused by bathymetric variations, nonlinear interactions among different wave components and interferences with man-made coastal structures. Among these, the bathymetric variations play a significant role in the change of the wave climate. Thus, coastal engineers should have a proper tool for estimating the wave climate as accurately as possible to design a coastal structure in nearshore areas. We consider the Laplace equation that is the mass transport equation of incompressible fluid and irrotational flow. In solving the equation, we need the kinematic boundary condition at the bottom and both the kinematic and dynamic boundary conditions on the free surface. The solutions with the assumption of linear free surface consist of a propagating mode and an infinite number of evanescent modes.

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