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| {{elastic plate modes code}} | | {{elastic plate modes code}} |
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− | == Alternative Solution Method using Green Functions for a Uniform Plate ==
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− | We can also solve the equation by a closely related method which was given in
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− | [[Meylan and Squire 1994]].
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− | We can transform the equations to
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− | <center><math>
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− | \phi(x) = \phi^{\rm I}(x) + \int_{-L}^{L}G(x,\xi)
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− | \left(
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− | \alpha\phi(\xi) - \partial_z\phi(\xi)
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− | \right)\mathrm{d} \xi
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− | </math></center>
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− |
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− | Expanding as before
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− | <center>
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− | <math>
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− | \partial_z \phi = i\omega \sum \xi_n X_n
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− | </math>
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− | </center>
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− | we obtain
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− | <center><math>
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− | -i\omega \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)\xi_n X_n
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− | </math>
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− | </center>
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− | This leads to the following equation
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− | <center>
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− | <math>
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− | \partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1} \phi(\xi)\mathrm{d}\xi
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− | </math>
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− | </center>
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− | or
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− | <center>
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− | <math>
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− | \partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} g(x,\xi) \phi(\xi)\mathrm{d}\xi
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− | </math>
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− | </center>
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− | where
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− | <center>
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− | <math>
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− | g(x,\xi) = \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}
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− | </math>
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− | </center>
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− | which is the Green function for the plate.
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− |
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− | [[Category:Floating Elastic Plate]]
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Introduction
The problem of a two-dimensional floating body which has negligible submergence
is solved using a green function.
The problem of a dock is solved in Green Function Method for a Finite Dock
and for a floating elastic plate is solved
in Green Function Methods for Floating Elastic Plates
Equations for a Finite Plate in Frequency Domain
We consider the problem of small-amplitude waves which are incident on finite floating body occupying water surface for [math]\displaystyle{ -L\lt x\lt L }[/math].
The submergence of the body is considered negligible.
We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction.
[math]\displaystyle{
\Delta \phi = 0, \;\;\; -h \lt z \leq 0,
}[/math]
[math]\displaystyle{
\partial_z \phi = 0, \;\;\; z = - h,
}[/math]
[math]\displaystyle{
\partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L
}[/math]
where [math]\displaystyle{ \alpha = \omega^2 }[/math]. The equation under the body consists of
the kinematic condition
[math]\displaystyle{
\mathrm{i}\omega w = \partial_z \phi,\,\,\, z=0,\,\,-L\leq x\leq L
}[/math]
plus the kinematic condition. The body motion is expanded using the modes for
heave and pitch.
Using the expression [math]\displaystyle{ \partial_n \phi =\partial_t w }[/math], we can form
[math]\displaystyle{
\frac{\partial \phi}{\partial z} = i\omega \sum_{n=0,1} \xi_n X_n(x)
}[/math]
where [math]\displaystyle{ \xi_n \, }[/math] are coefficients to be evaluated.
The functions [math]\displaystyle{ X_n(x) }[/math] are given by
[math]\displaystyle{
X_0 = \frac{1}{\sqrt{2L}}
}[/math]
and
[math]\displaystyle{
X_1 = \sqrt{\frac{3}{2L^3}} x
}[/math]
Note that this numbering is non-standard for a floating body and comes
from Eigenfunctions for a Uniform Free Beam.
Equation in Terms of the Modes
The equations are
[math]\displaystyle{
\Delta \phi = 0, \;\;\; -h \lt z \leq 0,
}[/math]
[math]\displaystyle{
\partial_z \phi = 0, \;\;\; z = - h,
}[/math]
[math]\displaystyle{
\partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L
}[/math]
[math]\displaystyle{
\mathrm{i}\omega\sum_{n=0,1}\zeta_{n}X_{n} =\partial_{z}\phi,\,\,x\in
(-L,L),\,\, z=0,
}[/math]
[math]\displaystyle{
\left(- \alpha\gamma + 1\right) \zeta_{n}=-i\omega
\int_{-L}^{L}\phi X_{n}\mathrm{d}x, \,\,x\in
(-L,L),\,\, z=0,
}[/math]
We solve for the potential (and displacement) as the sum of
the diffracted and radiation potentials in the standard way,
as for a rigid body.
[math]\displaystyle{
\phi=\phi^{\mathrm{D}}+\phi^{\mathrm{R}} ,\,
}[/math]
We begin with the diffraction potential [math]\displaystyle{ \phi^{\mathrm{D}} }[/math] which
satisfies the following equations
[math]\displaystyle{
\Delta\phi^{\mathrm{D}} =0,\,\,-h\lt z\lt 0,
}[/math]
[math]\displaystyle{
\partial_{z}\phi^{\mathrm{D}} =0,\,\,z=-h,
}[/math]
[math]\displaystyle{
\partial_{z}\phi^{\mathrm{D}} =\alpha \phi^{\mathrm{D}},\,\,x\notin(-L,L),\,\,
z=0,
}[/math]
[math]\displaystyle{
\partial_{z}\phi^{\mathrm{D}} =0,\,\,x\in(-L,L),\,\,z=0.
}[/math]
[math]\displaystyle{ \phi^{\mathrm{D}} }[/math] satisfies the Sommerfeld Radiation Condition
[math]\displaystyle{
\frac{\partial}{\partial x} \left(\phi^{\mathrm{D}}-\phi^{\rm
I} \right) \pm k_0\left( \phi^{\mathrm{D}}-\phi^{\rm I}\right)
= 0
,\,\,\mathrm{as}
\,\,x\rightarrow\infty.
}[/math]
[math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math]
is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,
[math]\displaystyle{
\phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \,
}[/math]
where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is
the positive imaginary solution of the Dispersion Relation for a Free Surface
(note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math])
and
[math]\displaystyle{
\phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h}
}[/math]
We now consider the radiation potentials [math]\displaystyle{ \phi^{\mathrm{R}} }[/math]. We can express the radiation potential as:
[math]\displaystyle{
\phi^{\mathrm{R}}=\sum_{n=0,1}\zeta_n \phi_n^{\mathrm{R}}
}[/math]
which satisfy the following equations
[math]\displaystyle{
\Delta\phi_n^{\mathrm{R}} =0,\,\,-h\lt z\lt 0,
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} =0,\,\,z=-h,
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} =\alpha\phi_n^{\mathrm{R}},\,\,x\notin(-L,L),\, \,
z=0
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} = i\omega X_{n},\,\,x\in(-L,L),\,\,z=0.
}[/math]
The radiation condition for the radiation potential is
[math]\displaystyle{
\frac{\partial\phi_n^{\mathrm{R}}}{\partial x}\pm ik\phi_n^{\mathrm{R}}=0,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty.
}[/math]
[math]\displaystyle{
\Delta\phi_n^{\mathrm{R}} =0,\,\,-h\lt z\lt 0,
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} =0,\,\,z=-h,
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} =\alpha\phi_n^{\mathrm{R}},\,\,x\notin(-L,L),\, \,
z=0
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} = i\omega X_{n},\,\,x\in(-L,L),\,\,z=0.
}[/math]
The radiation condition for the radiation potential is
[math]\displaystyle{
\frac{\partial\phi_n^{\mathrm{R}}}{\partial x}\pm ik\phi_n^{\mathrm{R}}=0,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty.
}[/math]
Therefore we find the potential as
[math]\displaystyle{
\left( - \alpha\gamma + 1\right) \zeta_{n}=-i\omega
\int_{-L}^{L}\phi^{\mathrm{D}} X_{n}\mathrm{d}x +
\sum_{m=0,1}\left(\omega^2 a_{mn}(\omega) - i\omega b_{mn}(\omega)\right)
\zeta_{m},
}[/math]
where the functions [math]\displaystyle{ a_{mn}(\omega) }[/math] and [math]\displaystyle{ b_{mn}(\omega) }[/math] are given by
[math]\displaystyle{
\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,
}[/math]
and they are referred to as the added mass and damping coefficients (see Added-Mass, Damping Coefficients And Exciting Forces)
respectively.
Solution for the Radiation and Diffracted Potential
We use the Free-Surface Green Function for two-dimensional waves, with singularity at
the water surface since we are only
interested in its value at [math]\displaystyle{ z=0 }[/math]
(details about this method can be found in Integral Equation for the Finite Depth Green Function at Surface).
Using this we can transform the system of equations to
[math]\displaystyle{
\phi^{\mathrm{D}}(x) = \phi^{\mathrm{I}}(x) + \int_{-L}^{L}G(x,\xi) \alpha\phi^{\mathrm{D}}(\xi) \mathrm{d} \xi
}[/math]
and
[math]\displaystyle{
\phi_n^{\mathrm{R}}(x) = \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi_n^{\mathrm{R}}(\xi) - i\omega X_n(\xi)
\right)\mathrm{d} \xi
}[/math]
Reflection and Transmission Coefficients
The Reflection and Transmission Coefficients represent the ratio of the amplitude of the reflected or transmitted wave to the amplitude of the incident wave. Conservation of energy means that [math]\displaystyle{ |R|^2+|T|^2=1\, }[/math].
A diagram depicting the area
[math]\displaystyle{ \Omega\, }[/math] which is bounded by the rectangle
[math]\displaystyle{ \partial \Omega \, }[/math]. The rectangle
[math]\displaystyle{ \partial \Omega \, }[/math] is bounded by
[math]\displaystyle{ -h \leq z \leq 0 \, }[/math] and
[math]\displaystyle{ -\infty \leq x \leq \infty \, }[/math] or
[math]\displaystyle{ -N \leq x \leq N\, }[/math]
We can calculate the Reflection and Transmission coefficients by
applying Green's theorem to [math]\displaystyle{ \phi\, }[/math] and [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math]
[math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math]
is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,
[math]\displaystyle{
\phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \,
}[/math]
where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is
the positive imaginary solution of the Dispersion Relation for a Free Surface
(note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math])
and
[math]\displaystyle{
\phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h}
}[/math]
We assume that [math]\displaystyle{ A=1 }[/math]. This gives us
[math]\displaystyle{
\iint_{\Omega}(\phi\Delta\phi^{\mathrm{I}} - \phi^{\mathrm{I}}\Delta\phi)\mathrm{d}x\mathrm{d}z
= \int_{\partial\Omega}(\phi \partial_n \phi^{\rm I} - \phi^{\rm I}\partial_n\phi)\mathrm{d}s = 0,
}[/math]
This means that (using the far field behaviour of the potential [math]\displaystyle{ \phi }[/math])
[math]\displaystyle{
\int_{\partial\Omega_{B}}
(\phi \partial_n \phi^{\rm I} - \phi^{\rm I}\partial_n\phi)\mathrm{d}s
+ 2k_0 R \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z = 0,
}[/math]
For the present case the body is present only on the surface and we therefore have
[math]\displaystyle{
\int_{-L}^{L} e^{-k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x
+ 2k_0 R \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z = 0
}[/math]
Therefore
[math]\displaystyle{
R = -\frac{\int_{-L}^{L} e^{-k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x }
{2 k_0 \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z}.
}[/math]
and using a wave incident from the right we obtain
[math]\displaystyle{
T = 1 - \frac{\int_{-L}^{L} e^{k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x }
{2 k_0 \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z}.
}[/math]
Note that an expression for the integral in the denominator can be found in
Eigenfunction Matching for a Semi-Infinite Dock
Matlab Code
A program to calculate the solution in elastic modes can be found here
elastic_plate_modes.m
Additional code
This program requires