Dispersion Relation for a Floating Elastic Plate

This is an generalisation of the Dispersion Relation for a Free Surface, to the case where the surface condition is given by a Floating Elastic Plate

Separation of Variables

The dispersion equation arises when separating variables subject to the boundary conditions for a Floating Elastic Plate of infinite extent. The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. The equations are described in detail in the Floating Elastic Plate page and we begin with the equations The equations of motion for the Frequency Domain Problem with radial frequency [math]\displaystyle{ \,\omega }[/math] in terms of the potential alone is

[math]\displaystyle{ D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z} + \left(\rho g - \omega^2 \rho_i d \right) \frac{\partial \phi}{\partial z} = - \rho \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{ \,d }[/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{ -D k^5 \sinh(kh) - k \left(\rho g - \omega^2 \rho_i d \right) \sinh(kh) = -\rho \omega^2 \cosh(kh) \,\,\,(1) }[/math]

This is the dispersion equation for a floating elastic plate.

We can also separate variables as

[math]\displaystyle{ \phi(x,z) = e^{kx} \cos k(z+h) \, }[/math]

in which case the dispersion relations becomes

[math]\displaystyle{ D k^5 \sin(kh) + k \left(\rho g - \omega^2 \rho_i h \right) \sin(kh) = -\rho \omega^2 \cos(kh) }[/math]

Solution of the dispersion equation

The dispersion equation was first solved by Fox and Squire 1994 and they determined that the solution consists of one real, two complex, and infinite number of imaginary roots plus their negatives. Interestingly the eigenfunctions form an over complete set for [math]\displaystyle{ L_2[-H,0]\, }[/math]. Also, there are some circumstances (non-physical) in which the complex roots become purely imaginary. The solution of this dispersion equation is far from trivial and the optimal solution method has been developed by Tim Williams and is described below.

Non-dimensional form

The dispersion equation (1) is often given in non-dimensional form. The form used by Tayler 1986 is to scale length with respect to [math]\displaystyle{ L\, }[/math] and time with respect to [math]\displaystyle{ \sqrt{L/g}\, }[/math]. The non-dimensional equations then become

[math]\displaystyle{ -\beta k^5 \sinh(kh) - k \left(1 - \alpha \gamma \right) \sinh(kh) = -\alpha \cosh(kh) \, }[/math]

where [math]\displaystyle{ h }[/math] is the nondimensional water depth, [math]\displaystyle{ \alpha = L\omega^2/g,\, }[/math] [math]\displaystyle{ \beta = D/(\rho g L^4)\, }[/math] and [math]\displaystyle{ \gamma = \rho_i h/(\rho L)\, }[/math]. This equation has four parameters [math]\displaystyle{ \alpha,\beta,\gamma,H\, }[/math]. We can trivially remove one of these by setting the [math]\displaystyle{ L=h\, }[/math]. Alternatively we can set the length parameter to the Characteristic Length [math]\displaystyle{ L = (D/(\rho g))^{1/4}\, }[/math] (so that in the notation above [math]\displaystyle{ \beta = 1\, }[/math] and we also have three parameters). It should be noted that in many cases, especially for ice floes, we have infinitely deep water and the parameter [math]\displaystyle{ \gamma }[/math] is small so that the Characteristic Length can be see as a single parameter which define the ice properties. It is then very useful to compare the value of the Characteristic Length for real ice and ice in the model basin (which much be deliberately made weaker to keep the Characteristic Length similar).

It turns out that by choosing [math]\displaystyle{ L^5=D/(\rho\omega^2)\, }[/math] we obtain a dispersion equation with only two parameters and an alternative (and more straightforward) derivation of this is as follow. Tim Williams noticed that the dispersion equation can be written as

[math]\displaystyle{ -\frac{D}{\rho \omega^2} k^5 \sinh(kh) - k \frac{\left(\rho g - \omega^2 \rho_i h \right)} {\rho \omega^2} \sinh(kh) = - \cosh(kh) }[/math]

and if we simply non-dimensionalise with respect to a length [math]\displaystyle{ L^5=D/(\rho\omega^2)\, }[/math] we obtain the following dispersion equation which depends on only two parameters

[math]\displaystyle{ -k^5 \sinh(kh) - k \frac{\rho g - \omega^2 \rho_i h } {\rho \omega^2 L} \sinh(kh) = - \cosh(kh). }[/math]

This can be written as

[math]\displaystyle{ \cosh(k h)-(k^4+\varpi)k\sinh(k h)=0, }[/math]

where

[math]\displaystyle{ \varpi=\frac{\rho g - \omega^2 \rho_i h }{\rho \omega^2 L} =(1-k_\infty\sigma)/(k_\infty L),\quad k_\infty=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2). }[/math]

[math]\displaystyle{ k_\infty }[/math] is the wave number for a wave traveling in open water of infinite depth, and [math]\displaystyle{ \sigma\, }[/math] is the amount that the plate would be submerged relative to a region of open water. Note that if [math]\displaystyle{ \sigma\, }[/math] is greater than [math]\displaystyle{ 1/k_\infty, }[/math] or [math]\displaystyle{ 2\pi\, }[/math] times the infinite depth open water wavelength, then the parameter [math]\displaystyle{ \varpi }[/math] becomes negative. This is a highly non-physical situation since for the plate to float [math]\displaystyle{ \rho_i\lt \rho }[/math] and we are therefore requiring the thickness to be similar in size to the wavelength which violates our assumption of Negligible Submergence. However, this situation does have important theoretical applications, for example connecting the Free-Surface Green function for a Floating Elastic Plate with the Green function for an Elastic Plate in a vaccum.

The dispersion relation relates the wavenumber [math]\displaystyle{ k/L\, }[/math] and thus wave speed [math]\displaystyle{ L\omega/k\, }[/math] to the above

Matlab Code

A program to calculate solutions to the dispersion relation for a free surface dispersion_elastic_surface.m