The linearised Navier-Stokes equation permits the velocity to be split into a potential part, , and a rotational part, [78, 79, 77, 80, 81]. The solution for is simply the inviscid solution which has already been given in equation (6.17). To find the rotational velocity we note [81, 80] that provided
is small in both fluids; the motion is essentially irrotational except near viscous boundary layers which are of thickness and which occur at solid boundaries and at the interface. Thus we look for solutions for which satisfy the Navier-Stokes equation for the rotational part of the velocity,
and the appropriate boundary conditions. We only consider solutions which have a significant value in the boundary layer and a negligible value outside the boundary layer. Since we expect [79, 80] that will be an order of magnitude smaller then we will solve for , the component in the direction of wave propagation. The solution for can then be found by integrating [80]. Looking first at the area of the interface we consider solutions of the form [78, 80]
Now consider the boundary conditions at the interface
where and .
Next consider the fluid adjacent to the solid boundaries. We look for solutions, for the rotational velocity close to the boundary layer, of the form [78, 80]
Substituting these into the no-slip boundary condition
gives the constants
The final solution for can be found by summing all the horizontal velocity terms. In doing this we neglect in terms ( ) except where it appears in an exponential multiplied by t, which can be large. This gives
and
where we have set and . Note that the total velocity can be taken as the sum of the three velocity terms, equations (6.17), (6.48) and (6.52) since the rotational velocities must be negligible far from their respective boundary layers. The first term in both equation (6.55) and equation (6.56) is the inviscid solution and the second and third terms are the rotational corrections at the interface and the solid boundaries respectively. The vertical velocity can be found from the continuity equation and is
and
It can easily be seen that integrating the first term in equations (6.55) and (6.56) gives the inviscid solution . The integral of the second and third terms are smaller than the original terms by a factor and so give only a very small correction term.
Figures 6-2 and 6-3 show the different components of the horizontal and vertical velocities for a wave with amplitude , , f = 1.4, and . The solid line in figure 6-2 shows the irrotational velocity which has a discontinuity at z = 0 and is symmetric under a rotation of about (z=0, u=0). The magnitude of the irrotational velocity is maximum at z=0. The dashed line is the sum of the irrotational velocity and the rotational component at the interface. This velocity distribution changes continuously from its smallest value slightly below z=0 to its maximum value slightly above z=0. In most of the boundary layer the rotational velocity has the opposite sign to the irrotational velocity and a smaller magnitude. Thus it has the effect of reducing the velocity from its inviscid value. At the outer region of the boundary layer, where the rotational component is small but not negligible, the sign of the rotational component is the same as the irrotational component and so the combined velocity is larger than the inviscid velocity. Further from the interface the rotational component becomes negligible and the solid and dashed lines appear to merge. The velocity represented by the dashed line is not symmetric about (z=0, u=0). This is due to the ratio being f and not unity. The dotted line represents the final solution for the velocity incorporating the rotational components at the interface and the solid boundaries. It is coincidental with the dashed line except close to the solid boundaries at the top and bottom of the wave. Here, as with the rotational velocity at the interface, the sign of U' is opposite to the sign of u close to the boundary, where its magnitude is largest. In this region the rotational component acts to reduce the magnitude of the total velocity, reducing it to zero on the boundary where the two components have the same magnitude. Further from the boundary there is a region where the signs are the same and the rotational component adds to the magnitude of the horizontal velocity. The magnitude of the rotational component is small here and the effect is not as obvious in figure 6-2 as it is near the interface. This is because is increasing away from the solid boundary but it is decreasing away from the interface. There is much less difference between the inviscid vertical velocity and the viscous vertical velocity shown in figure 6-3. The main difference is near the interface where the magnitude of the velocity is maximum. The magnitude of the viscous solution is slightly smaller and slightly more rounded at z = 0.