Difference between revisions of "Dispersion Relation for a Floating Elastic Plate"
| Line 14: | Line 14: | ||
<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z} | <math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z} | ||
- \omega^2 \rho_i h \frac{\partial \phi}{\partial z} = | - \omega^2 \rho_i h \frac{\partial \phi}{\partial z} = | ||
| − | + | - \rho\left(g\frac{\partial \phi}{\partial z}+ \omega^2 \phi\right), \, z=0</math> | |
plus the equations within the fluid | plus the equations within the fluid | ||
| Line 35: | Line 35: | ||
<math> -D k^5 \sinh(kh) - k \omega^2 \rho_i h \sinh(kh) = | <math> -D k^5 \sinh(kh) - k \omega^2 \rho_i h \sinh(kh) = | ||
| − | + | -\rho \left(g k \sinh(kh) - \omega^2 \cosh(kh) \right) =0 \, </math> | |
== Solution of the dispersion equation == | == Solution of the dispersion equation == | ||
Revision as of 01:21, 15 May 2006
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 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} - \omega^2 \rho_i h \frac{\partial \phi}{\partial z} = - \rho\left(g\frac{\partial \phi}{\partial z}+ \omega^2 \phi\right), \, 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] 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 \omega^2 \rho_i h \sinh(kh) = -\rho \left(g k \sinh(kh) - \omega^2 \cosh(kh) \right) =0 \, }[/math]
Solution of the dispersion equation
The disperstion equation was first solved by Fox and Squire 1994 and the 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 is often given in non-dimensional form. The form used by Michael Meylan in many of his papers is
This reduces to the following form in the length parameter is set to
which is the method used in Chung and Fox.
The most sophisticated form, which is due to Tim Williams and has certain theoretical and practical advantages, is the following,
[math]\displaystyle{ f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0, }[/math]
where [math]\displaystyle{ H }[/math] is the nodimensional water depth, and
[math]\displaystyle{ \varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2). }[/math]
[math]\displaystyle{ k }[/math] is the waver number for a wave of radial frequency [math]\displaystyle{ omega }[/math] traveling in open water of infinite depth, [math]\displaystyle{ g }[/math] is the acceleration due to gravity, [math]\displaystyle{ \sigma }[/math] is the amount of the plate that is submerged, [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, and [math]\displaystyle{ L }[/math] is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber [math]\displaystyle{ gamma/L }[/math] and thus wave speed to the above parameters.
