5.2 Surface Wave Energy

The energy equation for surface waves over an incompressible fluid like water subject to viscous dissipation is conveniently derived from Equation (1.5.12) in which ? = ? gz and ? = ? _o. When quantities of third order and higher are discarded, we have

Let the waves propagate in the x direction and integrate the equation over the interior of a thin, vertical slice of fluid, aligned with the wave crests, of thickness ? x and of unit lateral span (Figure 5.2.1). For viscous flow the velocity v = ( ? _x, 0, ? _z) must vanish at the bottom z = ? h, and the stress-free condition at the free surface must be applied in the form ( p ? _ij ? ? _ij) n _j ? pn _i ? ? _ij n _j = 0 at z = ?, where ? _z ? ? ?/ ? t. Retaining only second-order terms and considering the limit as ?x ? 0, we find

where

and

is the pressure reduced by the mean hydrostatic pressure ? ? _o gz.

Figure 5.2.1