Dispersion Relation for a Free Surface

Separation of Variables

The dispersion equation arises when separating variables subject to the boundary conditions for a free surface. The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. We begin with the equations for the Frequency Domain Problem with radial frequency [math]\displaystyle{ \,\omega }[/math] in terms of the potential alone which are

[math]\displaystyle{ g \frac{\partial \phi}{\partial z} = - \omega^2 \phi, \, z=0 }[/math]

plus the equations within the fluid

[math]\displaystyle{ \nabla^2\phi =0 }[/math]

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-H }[/math]

where [math]\displaystyle{ \,g }[/math] is the acceleration due to gravity, [math]\displaystyle{ \,\rho_i }[/math] and [math]\displaystyle{ \,\rho }[/math] are the densities of the plate and the water respectively, [math]\displaystyle{ \,h }[/math] and [math]\displaystyle{ \,D }[/math] are the thickness and flexural rigidity of the plate.

We then look for a separation of variables solution to Laplace's Equation and obtain the following expression for the velocity potential

[math]\displaystyle{ \phi(x,z) = e^{ikx} \cosh k(z+H) \, }[/math]

If we then apply the condition at [math]\displaystyle{ z=0 }[/math] we see that the constant [math]\displaystyle{ \,k }[/math] (which corresponds to the wavenumber) is given by

[math]\displaystyle{ - k \sinh(kH) = - \omega^2 \cosh(kH) \,\,\,(1) }[/math]

This is the dispersion equation for a free surface.