5.3 Viscous Damping of Surface Waves

Let us now consider the dissipation term on the right-hand side of energy equation (5.2.2). The dissipation per unit surface area is just equal to

where the integration is over the two-dimensional region beneath one wave, as in Figure 5.2.2. The motion has thus far been assumed to be entirely irrotational, and in this case the rate of strain e _ij decreases rapidly with distance from the free surface, so that the main contribution to the dissipation integral is the interior damping that occurs within a layer of fluid of width ~ ? near the surface. However, viscous effects cannot strictly be neglected, and for a wave of frequency ? the motion is actually rotational within a boundary layer of width on the bottom boundary at z = ? h ( 4.4, Example 1), which is typically very small compared with both the wavelength ? and the depth h. The velocity gradient and e _ij can become very large within the boundary layer provided that ? ? h (when the wave experiences significant bottom drag), and it is then necessary to consider separately the contribution of the boundary layer, to overall damping.

There is a similar narrow boundary layer at the free surface, where irrotational and rotational components of the flow are coupled by the linearised stress-free boundary conditions

However, there is no rapid variation in velocity across this layer, and the contribution...