Green Function Method for a Floating Body on the Surface

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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 of the Plate

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{ i\omega\sum_{n=0}^{\infty}\zeta_{n}X_{n} =\partial_{z}\phi,\,\,x\in (-L,L),\,\, z=0, }[/math] [math]\displaystyle{ \sum_{n=0}^{\infty}\zeta_{n}\left( 1+\beta\lambda_{n}^{4}\right) X_{n}-\alpha\gamma\sum_{n=0}^{\infty}\zeta_{n}X_{n} = -i\omega \phi,\,\,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]

Under these assumptions, the equations become

[math]\displaystyle{ \Delta\phi =0,\,\,-h\lt z\lt 0, }[/math] [math]\displaystyle{ \partial_{z}\phi =0,\,\,z=-h, }[/math] [math]\displaystyle{ \alpha\phi =\partial_{z}\phi,\,\,x\notin(-L,L),\ \ z=0, }[/math] [math]\displaystyle{ i\omega\sum_{n=0}^{\infty}\zeta_{n}X_{n} =\partial_{z}\phi,\,\,x\in (-L,L),\,\, z=0, }[/math] [math]\displaystyle{ \sum_{n=0}^{\infty}\zeta_{n}\left( 1+\beta\lambda_{n}^{4}\right) X_{n}-\alpha\gamma\sum_{n=0}^{\infty}\zeta_{n}X_{n} = -i\omega \phi,\,\,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]

As the plate is floating on the surface, we can denote it as follows:

[math]\displaystyle{ \phi^{\rm I}|_{z=0} = e^{-i kx} \, }[/math]

We now consider the scattered potentials [math]\displaystyle{ \phi^{\mathrm{S}} }[/math]. The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:

[math]\displaystyle{ \phi^{\mathrm{D}}=\phi^{\mathrm{I}}+\phi^{\mathrm{S}} \, }[/math]

from this, we can construct the following conditions:

[math]\displaystyle{ \Delta\phi^{\mathrm{S}} =0,\,\,-h\lt z\lt 0, }[/math] [math]\displaystyle{ \partial_{z}\phi^{\mathrm{S}} =0,\,\,z=-h, }[/math] [math]\displaystyle{ \partial_{z}\phi^{\mathrm{S}} =\alpha\phi^{\mathrm{S}},\,\,x\notin(-L,L),\, \, z=0 }[/math] [math]\displaystyle{ \partial_{z}\phi^{\mathrm{S}} = -\partial_{z}\phi^{\mathrm{I}},\,\,x\in(-L,L),\,\,z=0. }[/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}^{\infty}\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]

Therefore we find the potential as

[math]\displaystyle{ \phi=\phi^{\mathrm{D}} +\sum_{n=0}^{\infty}\zeta_{n}\phi_n^{\mathrm{R}}, }[/math]

so that

[math]\displaystyle{ \sum_{n=0}^{\infty}\left( 1+\beta\lambda_{n}^{4} - \alpha\gamma\right) \zeta_{n}X_{n}=-i\omega \left( \phi^{\mathrm{D}}+\sum_{n=0}^{\infty}\zeta_{n}\phi_n^{\mathrm{R}} \right). }[/math]

If we multiply by [math]\displaystyle{ X_m }[/math] and take an inner product over the plate we obtain

[math]\displaystyle{ \left( 1+\beta\lambda_{n}^{4} - \alpha\gamma\right) \zeta_{n}=-i\omega \int_{-L}^{L}\phi^{\mathrm{D}} X_{n}\mathrm{d}x + \sum_{m=0}^{\infty}\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 for the equivalent definition for a rigid body). respectively. This equation is solved by truncating the number of modes.

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

• matrix_G_surface.m
• two_d_finite_depth_Green_surface.m
• dispersion_free_surface.m
• beam_ev.m
• beam_em.m

Alternative Solution Method using Green Functions for a Uniform Plate

We can also solve the equation by a closely related method which was given in Meylan and Squire 1994. We can transform the equations to

[math]\displaystyle{ \phi(x) = \phi^{\rm I}(x) + \int_{-L}^{L}G(x,\xi) \left( \alpha\phi(\xi) - \partial_z\phi(\xi) \right)\mathrm{d} \xi }[/math]

Expanding as before

[math]\displaystyle{ \partial_z \phi = i\omega \sum \xi_n X_n }[/math]

we obtain

[math]\displaystyle{ -i\omega \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)\xi_n X_n }[/math]

This leads to the following equation

[math]\displaystyle{ \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 }[/math]

or

[math]\displaystyle{ \partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} g(x,\xi) \phi(\xi)\mathrm{d}\xi }[/math]

where

[math]\displaystyle{ g(x,\xi) = \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1} }[/math]

which is the Green function for the plate.