WikiWaves - User contributions [en]

Dispersion Relation for a Floating Elastic Plate

2006-05-19T01:35:07Z

Twilliams: /* Non-dimensional form */

== 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>\,\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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 [[Tayler_1986a|Tayler 1986]] is to scale length with respect
to <math>L\,</math> and time with respect to <math>\sqrt{L/g}\,</math>. The
non-dimensional equations then become

<math> -\beta k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =
-\alpha \cosh(kH) \, </math>

where <math>H</math> is the nondimensional water depth, <math>\alpha = \omega^2,\, </math> <math>\beta = D/(\rho g L^4)\,</math>
and <math>\gamma = \rho_i h/(\rho L)\,</math>.

This reduces to the following form if the length parameter is set to the
[[Characteristic Length]] <math>L = (D/(\rho g))^{1/4}\,</math> (so that in
the notation above <math>\beta = 1\,</math>):

<math>-k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =-\alpha \cosh(kH).</math>

The most sophisticated form, which is due to [[Tim Williams]] and has certain theoretical and practical advantages, is the following,

<math>\cosh(k H)-(k^4+\varpi)k\sinh(k H)=0,</math>

where <math>\varpi=(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> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and
<math>\sigma\,</math> is the amount that the plate would be submerged relative to a region of open water. Note that if <math>\sigma\,</math> is greater than <math> 1/k_\infty, </math> or <math>2\pi\,</math> times the infinite depth open water wavelength, then the parameter <math>\varpi</math> becomes negative.

The dispersion relation relates the wavenumber <math> k/L\, </math> and thus wave speed <math> L\omega/k\, </math> to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-19T01:33:59Z

Twilliams: /* Non-dimensional form */

Dispersion Relation for a Floating Elastic Plate

2006-05-19T01:31:27Z

Twilliams: /* Non-dimensional form */

== 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>\,\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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 [[Tayler_1986a|Tayler 1986]] is to scale length with respect
to <math>L\,</math> and time with respect to <math>\sqrt{L/g}\,</math>. The
non-dimensional equations then become

<math> -\beta k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =
-\alpha \cosh(kH) \, </math>

where <math>H</math> is the nondimensional water depth, <math>\alpha = \omega^2,\, </math> <math>\beta = D/(\rho g L^4)\,</math>
and <math>\gamma = \rho_i h/(\rho L)\,</math>.

This reduces to the following form if the length parameter is set to the
[[Characteristic Length]] <math>L = ((\rho g)/D)^{1/4}\,</math> (so that in
the notation above <math>\beta = 1\,</math>):

<math>-k^5 \sinh(kH) - k \left(1 - \alpha \gamma \right) \sinh(kH) =-\alpha \cosh(kH).</math>

The most sophisticated form, which is due to [[Tim Williams]] and has certain theoretical and practical advantages, is the following,

<math>\cosh(k H)-(k^4+\varpi)k\sinh(k H)=0,</math>

where <math>\varpi=(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> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and
<math>\sigma\,</math> is the amount that the plate would be submerged relative to a region of open water. Note that if <math>\sigma\,</math> is greater than <math> 1/k_\infty, </math> or <math>2\pi</math> times the infinite depth open water wavelength, then the parameter <math>\varpi</math> becomes negative.

The dispersion relation relates the wavenumber <math> k/L\, </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-16T01:11:20Z

Twilliams: /* Non-dimensional form */

== 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>\,\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave traveling in open water of infinite depth, and <math> L\, </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> \gamma/L\, </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-16T01:06:35Z

Twilliams: /* Separation of Variables */

== 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>\,\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> \,g </math> is the acceleration due to gravity, <math> \,\rho_i </math> and <math> \,\rho </math>
are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> \omega\, </math> traveling in open water of infinite depth, and <math> L\, </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> \gamma/L\, </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T06:44:10Z

Twilliams: /* Non-dimensional form */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T06:27:15Z

Twilliams: /* Non-dimensional form */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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

<math>\cosh(k H)-\left(\beta k^4 + \gamma\right) k\sinh(k H)=0,</math>

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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T06:24:53Z

Twilliams: /* Solution of the dispersion equation */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The dispersion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T06:24:03Z

Twilliams: /* Separation of Variables */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) - k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T05:09:21Z

Twilliams: /* Solution of the dispersion equation */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) + k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T05:06:14Z

Twilliams: /* Non-dimensional form */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) + k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T05:04:55Z

Twilliams: /* Separation of Variables */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kH) + k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kH) =
-\rho \omega^2 \cosh(kH) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T05:01:35Z

Twilliams: /* Non-dimensional form */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+h) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kh) + k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kh) =
-\rho \omega^2 \cosh(kh) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(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> k_\infty </math> is the wave number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T01:39:40Z

Twilliams: /* Separation of Variables */

== 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>\omega</math> in terms of
the potential alone is

<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
+ \left(\rho g
- \omega^2 \rho_i h \right) \frac{\partial \phi}{\partial z} =
- \rho \omega^2 \phi, \, z=0</math>

plus the equations within the fluid

<math>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+h) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<math> -D k^5 \sinh(kh) + k \left(\rho g - \omega^2 \rho_i h \right) \sinh(kh) =
-\rho \omega^2 \cosh(kh) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> \sigma </math> is the amount of the plate that is submerged, <math> \rho_i </math> and <math> \rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Floating Elastic Plate

2006-05-15T01:34:18Z

Twilliams: /* Equations of Motion */

= Introduction =

The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
more than 100 years ago and a discussion of the problem appears in [[Stoker_1957a|Stoker 1957]]. The problem can
be divided into the two and three dimensional formulations which are closely related.

= Two Dimensional Problem =

= Equations of Motion =

The equation for a elastic plate which is governed by Kirkoffs equation is given
by the following

<math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>

where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the plate,
<math>h</math> is the thickness of the plate (assumed constant), <math> p</math> is the pressure
and <math>\eta</math> is the plate displacement.

The pressure is given by the linearised Bernouilli equation at the wetted surface (assuming zero
pressure at the surface), i.e.

<math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>

where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
is the velocity potential. The velocity potential is governed by Laplace's equation through out
the fluid domain subject to the free surface condition and the condition of no flow through the
bottom surface. If we denote the region of the fluid surface covered in the plate (or possible
multiple plates) by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
[[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
[[Finite Depth]] are the following. At the surface
we have the dynamic condition

<math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
i\omega \rho \phi, \, z=0, \, x\in P</math>

<math>0=
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>

and the kinematic condition

<math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>

the equation within the fluid is [[Laplace's Equation]]

<math>\nabla^2\phi =0 </math>

and we have the no-flow condition through the bottom boundary

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

(so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
free surface or plate covered surface are at <math>z=0</math>).
<math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
the thickness and flexural rigidity of the plate.

Finally we need to include
some boundary conditions at the edge of the plate. The most common boundary conditions
in pratical applications are that the edges are free, this means that we have the additional
conditions that

<math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>

at the edges of the plate.

= Solution Method =

There are many different methods to solve the corresponding equations ranging from highly analytic such
as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
applicable and have advantages in different situations.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T01:24:43Z

Twilliams: /* Separation of Variables */

== 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>\omega</math> in terms of
the potential alone is

<math>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>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+h) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<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) \, </math>

== Solution of the dispersion equation ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> \sigma </math> is the amount of the plate that is submerged, <math> \rho_i </math> and <math> \rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T01:23:18Z

Twilliams: /* Separation of Variables */

== 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>\omega</math> in terms of
the potential alone is

<math>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>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+h) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<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 ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> \sigma </math> is the amount of the plate that is submerged, <math> \rho_i </math> and <math> \rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-15T01:21:37Z

Twilliams: /* Separation of Variables */

== 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>\omega</math> in terms of
the potential alone is

<math>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>\nabla^2\phi =0 </math>

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

where <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> 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>\phi(x,z) = e^{ikx} \cosh k(z+h) \,</math>

If we then apply the condition at <math>z=0</math> we see that the constant <math>k</math>
(which corresponds to the wavenumber) is given by

<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 ==

The disperstion equation was first solved by [[Fox_Squire_1994a|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>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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> \sigma </math> is the amount of the plate that is submerged, <math> \rho_i </math> and <math> \rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-12T05:33:09Z

Twilliams: /* Dispersion Relation for a Floating Elastic Plate */

The (nondimensional) dispersion relation for a floating thin elastic plate can be written

<math>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-12T05:30:56Z

Twilliams: /* Dispersion Relation for a Floating Elastic Plate */

==Dispersion Relation for a Floating Elastic Plate==

The (nondimensional) dispersion relation for a floating thin elastic plate can be written

<math>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-12T05:29:21Z

Twilliams: /* Dispersion Relation for a Floating Elastic Plate */

Dispersion Relation for a Floating Elastic Plate

2006-05-12T05:28:42Z

Twilliams: /* Dispersion Relation for a Floating Elastic Plate */

Dispersion Relation for a Floating Elastic Plate

2006-05-12T03:53:23Z

Twilliams: /* Dispersion Relation for a Floating Elastic Plate */

==Dispersion Relation for a Floating Elastic Plate==

The (nondimensional) dispersion relation for a floating thin elastic plate can be written

<math>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where <math>H</math> is the nodimensional water depth, and

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma </math> and thus speed (<math> =omega/gamma </math>) to the above parameters.

Dispersion Relation for a Floating Elastic Plate

2006-05-12T03:52:05Z

Twilliams:

==Dispersion Relation for a Floating Elastic Plate==

The (nondimensional) dispersion relation for a floating thin elastic plate can be written

<math>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>

where

<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>

<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma </math> and thus speed (<math> =omega/gamma </math>) to the above parameters.