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| | Wave scattering by a [[Floating Elastic Plate]] on water of Variable Bottom Topography was treated in | | Wave scattering by a [[Floating Elastic Plate]] on water of Variable Bottom Topography was treated in |
| − | [[Wang and Meylan 2002]]. | + | [[Wang and Meylan 2002]] and is described in [[Floating Elastic Plate on Variable Bottom Topography]] |
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| − |
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| − |
| |
| − | \title{The Linear Wave Response of a Floating Thin Plate on Water of
| |
| − | Variable Depth}
| |
| − | \author{Cynthia D. Wang and Michael H. Meylan \\
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| − |
| |
| − | Institute of Information and Mathematical Sciences,\\
| |
| − | Massey University, New Zealand}
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| − |
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| − |
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| − |
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| − | We present a solution for the linear wave forcing of a floating
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| − | two-dimensional thin plate on water of variable depth. The solution method
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| − | is based on reducing the problem to a finite domain which contains both the
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| − | region of variable water depth and the floating thin plate. In this finite
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| − | region the outward normal derivative of the potential on the boundary is
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| − | expressed as a function of the potential. This is accomplished by using
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| − | integral operators for the radiating boundaries and the boundary under the
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| − | plate. Laplace's equation in the finite domain is solved using the boundary
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| − | element method and the integral equations are solved by numerical
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| − | integration. The same discretisation is used for the boundary element method
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| − | and to integrate the integral equations. The results show that there is a
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| − | significant region where the solution for a plate with a variable depth
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| − | differs from the simpler solutions for either variable depth but no plate or
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| − | a plate with constant depth. Furthermore, the presence of the plate
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| − | increases the frequency of influence of the variable depth.
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| − |
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| − |
| |
| − | =Introduction=
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| − |
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| − | The linear wave forcing of a floating thin plate can be used to model a wide
| |
| − | range of physical systems; for example very large floating structures \cite
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| − | {Kashiwagihydro98}, sea ice floes [[Squire_Review]] and breakwaters \cite
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| − | {Stoker}. For this reason it is the one of the best studied hydroelastic
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| − | problems and several solution methods have been developed. These methods
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| − | have focused on providing a fast solution and for this reason have
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| − | exclusively solved for water of constant depth. Furthermore, since all the
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| − | models tests have been conducted in water of constant depth, only the
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| − | constant depth solution is required to compare theory and experiment.
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| − | However, because of the large size of floating hydroelastic structures, it
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| − | is unlikely that the water depth will be constant under the entire
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| − | structure. For this reason the effect of a variation in the water depth
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| − | under a floating thin plate is investigated in this paper.
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| − |
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| − | As mentioned, the linear wave forcing of a floating thin plate has been
| |
| − | extensively studied and standard solution methods have now been developed.
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| − | These methods are based on expanding the plate motion in basis functions
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| − | (often thin plate or beam modes) and on solving the equations of motion for
| |
| − | the water using a Green function or by a further expansion in modes \cite
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| − | {Kashiwagihydro98}. A solution of the water equations by either a Green
| |
| − | function or an expansion in modes requires the water depth to be constant.
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| − | Therefore these standard methods are unsuitable to be extended to the case
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| − | of variable water depth.
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| − |
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| − | The linear wave scattering by variable depth (or bottom topography) in the
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| − | absence of a floating plate has been considered by many authors. Two
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| − | approaches have been developed. The first is analytical and the solution is
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| − | derived in an almost closed form ([[Porter95]], [[Staziker96]] and
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| − | [[Porter00]]). However this approach is unsuitable to be generalised to
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| − | the case when a thin plate is also floating on the water surface because of
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| − | the complicated free surface boundary condition which the floating plate
| |
| − | imposes. The second approach is numerical, an example of which is the method
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| − | developed by [[Liu82]], in which the boundary element method in a finite
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| − | region is coupled to a separation of variables solution in the semi-infinite
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| − | outer domains. This method is well suited to the inclusion of the plate as
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| − | will be shown. For both the analytic and numerical approach the region of
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| − | variable depth must be bounded.
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| − |
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| − | In this paper, a solution method for the linear wave forcing of a two
| |
| − | dimensional floating plate on water of variable depth will be derived from
| |
| − | [[Liu82]], [[Hazard]] and [[jgrfloe1d]]. The method is based on
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| − | dividing the water domain into two semi-infinite domains and a finite
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| − | domain. The finite domain contains both the plate and the region of variable
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| − | water depth. Laplace's equation in the finite domain is solved by the
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| − | boundary element method. Laplace's equation in the semi-infinite domains is
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| − | solved by separation of variables. The solution in the semi-infinite domains
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| − | gives an integral equation relating the normal derivative of the potential
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| − | and the potential on the boundary of the finite and semi-infinite domains.
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| − | The thin plate equation is expressed as an integral equation relating the
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| − | normal derivative of the potential and the potential under the surface of
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| − | the plate. The boundary element equations and the integral equations are
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| − | solved simultaneously using the same discretisation.
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| − |
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| − | =Problem Formulation=
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| − |
| |
| − | We consider a thin plate, floating on the water surface above a sea bed of
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| − | variable depth, which is subject to an incoming wave. The plate is
| |
| − | approximated as infinitely long in the <math>y</math> directions which reduces the
| |
| − | problem to two dimensions, <math>x</math> and <math>z</math>. The <math>x</math>-axis is horizontal and the <math>
| |
| − | z <math>-axis points vertically up with the free water surface at </math>z=0.</math> The
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| − | incoming wave is assumed to be travelling in the positive <math>x</math>-direction with
| |
| − | a single radian frequency <math>\omega </math>. The theory which will be developed
| |
| − | could be extended to oblique incident waves using the standard method \cite
| |
| − | {OhkusuISOPE}. However, to keep the treatment straightforward, this will not
| |
| − | be done. We assume that the wave amplitude is sufficiently small that the
| |
| − | problem can be approximated as linear. From the linearity and the single
| |
| − | frequency wave assumption it follows that all quantities can be written as
| |
| − | the real part of a complex quantity whose time dependence is <math>e^{-i\omega t}</math>
| |
| − | .
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| − |
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| − | The linear boundary value problem for the water is the following,
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| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
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| − | \nabla ^{2}\phi =0, \\
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| − | -\rho g\,w+i\omega \rho \phi =p,\qquad z=0, \\
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| − | \phi _{n}=0,\qquad z=d\left( x\right) ,
| |
| − | \end{matrix}
| |
| − | \right\} (1)
| |
| − | </math></center>
| |
| − | where <math>\rho </math> is the density of the water, <math>p</math> is the pressure on water
| |
| − | surface, <math>w</math> is the displacement of the water surface, <math>g</math> is the
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| − | gravitational acceleration, <math>d(x)</math> is the water depth and <math>\phi _{n}</math> is the
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| − | outward normal derivative of the potential. We assume that the water depth
| |
| − | is constant outside a finite region, <math>-l<x<l</math>, but allow the depth to be
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| − | different at either end. Therefore
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| − | <center><math>
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| − | d\left( x\right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | -H_{1},\;\;x<-l, \\
| |
| − | d\left( x\right) ,\;\;-l<x<l, \\
| |
| − | -H_{2},\;\;x>l,
| |
| − | \end{matrix}
| |
| − | \right.
| |
| − | </math></center>
| |
| − | where <math>H_{1}</math> and <math>H_{2}</math> are the water depths in the left and right hand
| |
| − | domains of constant depth.
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| − |
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| − | A thin plate, of negligible draft, floats on the surface of the water and
| |
| − | occupies the region <math>-L\leq x\leq L\ </math>as is shown in Figure \ref{fig_region}
| |
| − | . Without loss of generality, we assume that <math>l</math> is sufficiently large that <math>
| |
| − | L<l.</math> For any point on the water surface not under the plate the pressure is
| |
| − | the constant atmospheric pressure whose time-dependent part is zero. Under
| |
| − | the plate, the pressure and the displacement are related by the
| |
| − | Bernoulli-Euler equation
| |
| − | <center><math>
| |
| − | D\frac{\partial ^{4}w}{\partial x^{4}}-\rho ^{\prime }a\omega
| |
| − | ^{2}\,w=p,\qquad -L\leq x\leq L= and = z=0, (2)
| |
| − | </math></center>
| |
| − | where <math>\rho ^{\prime }</math> is the density of the plate, <math>a</math> is the plate
| |
| − | thickness, and <math>D</math> is the bending rigidity of the plate. We assume that the
| |
| − | plate edges are free so the bending moment and shear must vanish at both
| |
| − | ends of the plate, i.e.
| |
| − | <center><math>
| |
| − | \frac{\partial ^{2}w}{\partial x^{2}}=\frac{\partial ^{3}w}{\partial x^{3}}
| |
| − | =0,\qquad =at= \ x=-L= and = x=L.
| |
| − | </math></center>
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| − | The kinematic boundary condition at the surface allows us to express the
| |
| − | displacement as a function of the outward normal derivative of the
| |
| − | potential,
| |
| − | <center><math>
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| − | w=\frac{i\phi _{n}}{\omega },\qquad z=0. (3)
| |
| − | </math></center>
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| − |
| |
| − | Substituting equations (2) and (3)\ into
| |
| − | equation (1), we obtain the following boundary value problem for
| |
| − | the potential only,
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | \nabla ^{2}\phi =0, \\
| |
| − | -\rho \left( g\phi _{n}-\omega ^{2}\,\phi \right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | 0,\;x\notin \left[ -L,L\right] ,\;z=0, \\
| |
| − | D\,\frac{\partial ^{4}\phi _{n}}{\partial x^{4}}-\rho ^{\prime }a\,\omega
| |
| − | ^{2}\phi _{n},\;x\in \left[ -L,L\right] ,\;z=0,
| |
| − | \end{matrix}
| |
| − | \right. \\
| |
| − | \phi _{n}=0,\qquad z=d\left( x\right) ,
| |
| − | \end{matrix}
| |
| − | \right\} (4)
| |
| − | </math></center>
| |
| − | together with the free plate edge conditions
| |
| − | <center><math>
| |
| − | \frac{\partial ^{2}\phi _{n}}{\partial x^{2}}=\frac{\partial ^{3}\phi _{n}}{
| |
| − | \partial x^{3}}=0,\qquad =at = x=\pm L=, \ \ = z=0.
| |
| − | (5)
| |
| − | </math></center>
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| − |
| |
| − | ==Radiation Boundary Conditions==
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| − |
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| − | Equation (4) is subject to radiation conditions as <math>
| |
| − | x\rightarrow \pm \infty .</math> We assume that there is a wave incident from the
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| − | left which gives rise to a reflected and transmitted wave. Therefore the
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| − | following boundary conditions for the potential apply as <math>x\rightarrow \pm
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| − | \infty </math>
| |
| − | <center><math>
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| − | \lim_{x\rightarrow -\infty }\phi \left( x,z\right) =\cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+H_{1}\right) \right) e^{ik_{t}^{\left(
| |
| − | 1\right) }x}+R\,\cosh \left( k_{t}^{\left( 1\right) }\left( z+H_{1}\right)
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| − | \right) e^{-k_{t}^{\left( 1\right) }x}, (6)
| |
| − | </math></center>
| |
| − | and
| |
| − | <center><math>
| |
| − | \lim_{x\rightarrow \infty }\phi \left( x,z\right) =T\cosh \left(
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| − | k_{t}^{\left( 2\right) }\left( z+H_{2}\right) \right) e^{k_{t}^{\left(
| |
| − | 2\right) }x}, (7)
| |
| − | </math></center>
| |
| − | where <math>R</math> and <math>T</math> are the reflection and transmission coefficients
| |
| − | respectively and <math>k_{t}^{\left( j\right) }\,\left( j=1,2\right) </math> are the
| |
| − | positive real solutions of the following equation
| |
| − | <center><math>
| |
| − | gk_{t}^{\left( j\right) }\tanh \left( k_{t}^{\left( j\right) }\,H_{j}\right)
| |
| − | =\omega ^{2}.
| |
| − | </math></center>
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| − |
| |
| − | ==(8)Non-dimensionalisation==
| |
| − |
| |
| − | Non-dimensional variables are now introduced. We non-dimensionalise the
| |
| − | space variables with respect to the water depth on the left hand side, <math>
| |
| − | H_{1},<math> and the time variables with respect to</math>\;\sqrt{g/H_{1}}</math>. The
| |
| − | non-dimensional variables, denoted by an overbar, are
| |
| − | <center><math>
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| − | \bar{x}=\frac{x}{H_{1}},\;\bar{z}=\frac{z}{H_{1}},\;\bar{t}=t\sqrt{\frac{g
| |
| − | }{H_{1}}},\;=and= \;\bar{\phi}=\frac{1}{H_{1}\sqrt{H_{1}g}}\,\phi .
| |
| − | </math></center>
| |
| − | Applying this non-dimensionalisation to equation (4)
| |
| − | we obtain
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | \nabla ^{2}\bar{\phi}=0, \\
| |
| − | \left( \bar{\phi}_{n}-\nu \bar{\phi}\right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | 0,\qquad \bar{x}\notin \left[ -L,L\right] ,\;\bar{z}=0, \\
| |
| − | -\beta \,\frac{\partial ^{4}\bar{\phi}_{n}}{\partial \bar{x}^{4}}+\gamma
| |
| − | \nu \,\bar{\phi}_{n},\qquad \bar{x}\in \left[ -L,L\right] ,\;\bar{z}=0,
| |
| − | \end{matrix}
| |
| − | \right. \\
| |
| − | \bar{\phi}_{n}=0,\qquad \bar{z}=\bar{d}\left( \bar{x}\right) ,
| |
| − | \end{matrix}
| |
| − | \right\} (9)
| |
| − | </math></center>
| |
| − | where
| |
| − | <center><math>
| |
| − | \beta =\frac{D}{\rho gH_{1}^{4}},\;\gamma =\frac{\rho ^{\prime }a}{\rho
| |
| − | H_{1}},\;=and= \;\nu =\frac{\omega ^{2}H_{1}}{g}. (10)
| |
| − | </math></center>
| |
| − | We will refer to <math>\beta </math> as the stiffness, <math>\gamma </math> as the mass and <math>\nu </math>
| |
| − | as the wavenumber. The non-dimensional water depth is
| |
| − |
| |
| − | <center><math>
| |
| − | \bar{d}\left( \bar{x}\right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | -1,\;\;\bar{x}<-\bar{l}, \\
| |
| − | \bar{d}\left( \bar{x}\right) ,\;\;-\bar{l}<\bar{x}<\bar{l}, \\
| |
| − | -H,\;\;\bar{x}>\bar{l},
| |
| − | \end{matrix}
| |
| − | \right.
| |
| − | </math></center>
| |
| − | where <math>H=H_{2}/H_{1}.</math> Equation (9) is also subject to the
| |
| − | non-dimensional free edge conditions of the plate (5) and the
| |
| − | radiation conditions (6) and (7). With the
| |
| − | understanding that all variables have been non-dimensionalised, from now
| |
| − | onwards we omit the overbar.
| |
| − |
| |
| − | =Reduction to a Finite Domain=
| |
| − |
| |
| − | We solve equation (9) by reducing the problem to a finite
| |
| − | domain which contains both the region of variable depth and the floating
| |
| − | thin plate. This finite domain <math>\Omega =\left\{ -l\leq x\leq l,\;d\left(
| |
| − | x\right) \leq z\leq 0\right\} </math> is shown in Figure \ref{fig_region}. We will
| |
| − | solve Laplace's equation in <math>\Omega </math> using the boundary element method. To
| |
| − | accomplish this we need to express the normal derivative of the potential on
| |
| − | the boundary of <math>\Omega \;(\partial \Omega )</math> as a function of the potential
| |
| − | on the boundary.
| |
| − |
| |
| − | ==Green's Function Solution for the Floating Thin Plate==
| |
| − |
| |
| − | We begin with the boundary condition under the plate which is the following
| |
| − | <center><math>
| |
| − | \beta \frac{\partial ^{4}\phi _{n}}{\partial x^{4}}-\left( \gamma \nu
| |
| − | -1\right) \phi _{n}=\nu \phi ,\;-L\leq x\leq L= and = z=0, (11)
| |
| − | </math></center>
| |
| − | together with the free edge boundary conditions (5).
| |
| − | Following [[jgrfloe1d]] we can transform equations (5) and
| |
| − | (11) to an integral equation using the Green function, <math>g,</math> which
| |
| − | satisfies
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | \beta \frac{\partial ^{4}}{\partial x^{4}}g\left( x,\xi \right) -\left(
| |
| − | \gamma \nu -1\right) g\left( x,\xi \right) =\nu \delta \left( x-\xi \right) ,
| |
| − | \\
| |
| − | \frac{\partial ^{2}}{\partial x^{2}}g\left( x,\xi \right) =\frac{\partial
| |
| − | ^{3}}{\partial x^{3}}g\left( x,\xi \right) =0,\qquad =at = x=-L,\;x=L.
| |
| − | \end{matrix}
| |
| − | \right\}
| |
| − | </math></center>
| |
| − | This gives us the following expression for <math>\phi _{n}</math> as a function of the
| |
| − | potential under the plate <math>\phi ,</math>
| |
| − | <center><math>
| |
| − | \phi _{n}\left( x\right) =\int_{-L}^{L}g\left( x,\xi \right) \;\phi \left(
| |
| − | \xi \right) \;d\xi . (12)
| |
| − | </math></center>
| |
| − | We will write this in operator notation as <math>\phi _{n}=\mathbf{g}\phi </math> where
| |
| − | <math>\mathbf{g}</math> denotes the integral operator with kernel <math>g\left( x,\xi
| |
| − | \right) .</math>
| |
| − |
| |
| − | ==Solution in the Semi-infinite Domains==
| |
| − |
| |
| − | We now solve Laplace's equation in the semi-infinite domains <math>\Omega
| |
| − | ^{-}=\left\{ x<-l,\;-1\leq z\leq 0\right\} <math> and </math>\Omega ^{+}=\left\{
| |
| − | x>l,\;-H\leq z\leq 0\right\} </math> which are shown in Figure (\ref{fig_region}).
| |
| − | Since the water depth is constant in these regions we can solve Laplace's
| |
| − | equation by separation of variables. The potential in the region <math>\Omega
| |
| − | ^{-} </math> satisfies the following equation
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | \nabla ^{2}\phi =0,\;\;\mathbf{x}\in \Omega ^{-}, \\
| |
| − | \phi _{n}-\nu \phi =0,\;\;z=0, \\
| |
| − | \phi _{n}=0,\;\;z=-1, \\
| |
| − | \phi =\tilde{\phi}\left( z\right) ,\;\;x=-\,l, \\
| |
| − | \lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) =\cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+1\right) \right) e^{ik_{t}^{\left( 1\right)
| |
| − | }x} \\
| |
| − | +R\,\cosh \left( k_{t}^{\left( 1\right) }\left( z+1\right) \right)
| |
| − | e^{-ik_{t}^{\left( 1\right) }x},
| |
| − | \end{matrix}
| |
| − | \right\} (13)
| |
| − | </math></center>
| |
| − | where <math>\mathbf{x}=\left( x,z\right) \ </math>and <math>\tilde{\phi}\left( z\right) </math> is
| |
| − | an arbitrary continuous function<math>.</math>\ Our aim is to find the outward normal
| |
| − | derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
| |
| − | z\right) </math>.
| |
| − |
| |
| − | We solve equation (13) by separation of variables \cite{Liu82,
| |
| − | Hazard} and obtain the following expression for the potential in the region <math>
| |
| − | \Omega ^{-},</math>
| |
| − | <center><math>\begin{matrix}
| |
| − | \phi \left( x,z\right) &=&\cosh \left( k_{t}^{\left( 1\right) }\left(
| |
| − | z+1\right) \right) \,e^{ik_{t}^{\left( 1\right) }x}+R\cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+1\right) \right) \,e^{-ik_{t}^{\left(
| |
| − | 1\right) }x} \notag \\
| |
| − | &&+\sum_{m=1}^{\infty }\left\langle \tilde{\phi}\left( z\right) ,\tau
| |
| − | _{m}^{\left( 1\right) }\left( z\right) \right\rangle \tau _{m}^{\left(
| |
| − | 1\right) }\left( z\right) e^{k_{m}^{\left( 1\right) }\left( x+l\right) }.
| |
| − | (14)
| |
| − | \end{matrix}</math></center>
| |
| − | The functions <math>\tau _{m}^{\left( 1\right) }\left( z\right) </math> (<math>m\geq 1)</math> are
| |
| − | the orthonormal modes given by
| |
| − | <center><math>
| |
| − | \tau _{m}^{\left( 1\right) }\left( z\right) =\left( \frac{1}{2}+\frac{\sin
| |
| − | \left( 2k_{m}^{\left( 1\right) }\,\right) }{4k_{m}^{\left( 1\right) }}
| |
| − | \right) ^{-\frac{1}{2}}\cos \left( k_{m}^{\left( 1\right) }\left(
| |
| − | z+1\right) \right) ,\;\;m\geq 1.
| |
| − | </math></center>
| |
| − | The evanescent eigenvalues <math>k_{m}^{\left( 1\right) }</math> are the positive real
| |
| − | solutions of the dispersion equation
| |
| − | <center><math>
| |
| − | -k_{m}^{\left( 1\right) }\,\tan \left( k_{m}^{\left( 1\right) }H_{j}\right)
| |
| − | =\nu ,\;\;m\geq 1, (15)
| |
| − | </math></center>
| |
| − | ordered by increasing size. The inner product in equation (13) is
| |
| − | the natural inner product for the region <math>-1\leq z\leq 0</math> given by
| |
| − | <center><math>
| |
| − | \left\langle \tilde{\phi}\left( z\right) ,\tau _{m}^{\left( j\right) }\left(
| |
| − | z\right) \right\rangle =\int_{-1}^{0}\tilde{\phi}\left( z\right) ,\tau
| |
| − | _{m}^{\left( j\right) }\left( z\right) dx. (16)
| |
| − | </math></center>
| |
| − |
| |
| − | The reflection coefficient is determined by taking an inner product of
| |
| − | equation (14) with respect to <math>\cosh \left( k_{t}^{\left( 1\right)
| |
| − | }\left( z+1\right) \right) .<math> This gives us the following expression for </math>R</math>
| |
| − | ,
| |
| − | <center><math>
| |
| − | R=\frac{\left\langle \tilde{\phi}\left( z\right) ,\cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+1\right) \right) \right\rangle }{\frac{1}{2
| |
| − | }+\sinh \left( 2k_{t}^{\left( 1\right) }\,\right) /4k_{0}^{\left( 1\right) }}
| |
| − | e^{-ik_{t}^{\left( 1\right) }l}-e^{-2ik_{t}^{\left( 1\right) }l}.
| |
| − | (17)
| |
| − | </math></center>
| |
| − | The normal derivative of the potential on the boundary of <math>\Omega ^{-}</math> and <math>
| |
| − | \Omega <math> </math>\left( x=-l\right) </math> is calculated using equation (14) and
| |
| − | we obtain,
| |
| − | <center><math>
| |
| − | \left. \phi _{n}\right| _{x=-l}=\mathbf{Q}_{1}\tilde{\phi}\left( z\right)
| |
| − | -2ik_{t}^{\left( 1\right) }\cosh \left( k_{t}^{\left( 1\right) }\left(
| |
| − | z+1\right) \right) \,e^{-ik_{t}^{\left( 1\right) }l},
| |
| − | </math></center>
| |
| − | where the outward normal is with respect to the <math>\Omega </math> domain. The
| |
| − | integral operator <math>\mathbf{Q}_{1}</math> is given by
| |
| − | <center><math>\begin{matrix}
| |
| − | \mathbf{Q}_{1}\tilde{\phi}\left( z\right) &=&\sum_{m=1}^{\infty
| |
| − | }k_{m}^{\left( 1\right) }\left\langle \tilde{\phi}\left( z\right) ,\tau
| |
| − | _{m}^{\left( 1\right) }\left( z\right) \right\rangle \,\tau _{m}^{\left(
| |
| − | 1\right) }\left( z\right) (18) \\
| |
| − | &&+ik_{t}\frac{\left\langle \tilde{\phi}\left( z\right) ,\cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+1\right) \right) \right\rangle \cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+1\right) \right) }{\frac{1}{2}+\sinh
| |
| − | \left( 2k_{t}^{\left( 1\right) }\,\right) /4k_{t}^{\left( 1\right) }}.
| |
| − | \notag
| |
| − | \end{matrix}</math></center>
| |
| − | We can combine the two terms of equation (18) and express <math>\mathbf{Q}
| |
| − | _{1}</math> as
| |
| − | <center><math>
| |
| − | \mathbf{Q}_{1}\tilde{\phi}\left( z\right) =\sum_{m=0}^{\infty }k_{m}^{\left(
| |
| − | 1\right) }\left\langle \tilde{\phi}\left( z\right) ,\tau _{m}^{\left(
| |
| − | 1\right) }\left( z\right) \right\rangle \,\tau _{m}^{\left( 1\right) }\left(
| |
| − | z\right) (19)
| |
| − | </math></center>
| |
| − | where <math>k_{0}^{\left( 1\right) }=ik_{t}^{\left( 1\right) }</math> and
| |
| − | <center><math>
| |
| − | \tau _{0}^{\left( 1\right) }\left( z\right) =\left( \frac{1}{2}+\frac{\sin
| |
| − | \left( 2k_{0}^{\left( 1\right) }\right) }{4k_{0}^{\left( 1\right) }}\right)
| |
| − | ^{-\frac{1}{2}}\cos \left( k_{0}^{\left( 1\right) }\left( z+1\right)
| |
| − | \right) .
| |
| − | </math></center>
| |
| − | As well as providing a more compact notation, equation (19)
| |
| − | will be useful in the numerical calculation of <math>\mathbf{Q}_{1}.</math>
| |
| − |
| |
| − | Similarly, we now consider the potential in the region <math>\Omega ^{+}</math> which
| |
| − | satisfies
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | \nabla ^{2}\phi =0,\;\;\mathbf{x}\in \Omega ^{+}, \\
| |
| − | \phi _{n}-\nu \phi =0,\;\;z=0, \\
| |
| − | \phi _{n}=0=,= \;\;z=-H, \\
| |
| − | \phi =\tilde{\phi}\left( z\right) ,\;\;x=\,l, \\
| |
| − | \lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) =T\cosh \left(
| |
| − | k_{t}^{\left( 2\right) }\left( z+H_{2}\right) \right) e^{ik_{t}^{\left(
| |
| − | 2\right) }x}.
| |
| − | \end{matrix}
| |
| − | \right\} (20)
| |
| − | </math></center>
| |
| − | Solving equation (20) by separation of variables as before we obtain
| |
| − | <center><math>
| |
| − | \left. \phi _{n}\right| _{x=l}=\mathbf{Q}_{2}\tilde{\phi}\left( z\right) ,
| |
| − | </math></center>
| |
| − | where the outward normal is with respect to the <math>\Omega </math> domain. The
| |
| − | integral operator <math>\mathbf{Q}_{2}</math> is given by
| |
| − | <center><math>
| |
| − | \mathbf{Q}_{2}\tilde{\phi}\left( z\right) =\sum_{m=0}^{\infty }k_{m}^{\left(
| |
| − | 2\right) }\left\langle \tilde{\phi}\left( z\right) ,\tau _{m}^{\left(
| |
| − | 2\right) }\left( z\right) \right\rangle \tau _{m}^{\left( 2\right) }\left(
| |
| − | z\right) . (21)
| |
| − | </math></center>
| |
| − | The orthonormal modes <math>\tau _{m}^{\left( 2\right) }</math> are given by
| |
| − | <center><math>
| |
| − | \tau _{m}^{\left( 2\right) }\left( z\right) =\left( \frac{H}{2}+\frac{\sin
| |
| − | \left( 2k_{m}^{\left( 2\right) }\,H\right) }{4k_{m}^{\left( 2\right) }}
| |
| − | \right) ^{-\frac{1}{2}}\cos \left( k_{m}^{\left( 2\right) }\left(
| |
| − | z+H\right) \right) ,
| |
| − | </math></center>
| |
| − | The eigenvalues <math>k_{m}^{\left( 2\right) }</math> are the positive real solutions <math>
| |
| − | \left( m\geq 1\right) <math> and positive imaginary solutions </math>\left( m=0\right) </math>
| |
| − | of the dispersion equation
| |
| − | <center><math>
| |
| − | -k_{m}^{\left( 2\right) }\,\tan \left( k_{m}^{\left( 2\right) }H\right) =\nu
| |
| − | .
| |
| − | </math></center>
| |
| − | The inner product is the same as that given by equation (16)
| |
| − | except that the integration is from <math>z=-H</math> to <math>z=0.</math> The transmission
| |
| − | coefficient, <math>T,</math> is given by
| |
| − | <center><math>
| |
| − | T=\frac{\left\langle \tilde{\phi}\left( z\right) ,\cosh \left(
| |
| − | k_{t}^{\left( 2\right) }\left( z+1\right) \right) \right\rangle }{\frac{1}{2
| |
| − | }+\sinh \left( 2k_{t}^{\left( 2\right) }H\right) /4k_{t}^{\left( 2\right) }}
| |
| − | e^{-ik_{t}^{\left( 2\right) }l}. (22)
| |
| − | </math></center>
| |
| − |
| |
| − | ==The equation for the finite domain==
| |
| − |
| |
| − | We now consider the finite domain <math>\Omega .</math> In this domain, Laplace's
| |
| − | equation is subject to the boundary conditions given by equations (\ref
| |
| − | {integral_plate}), (18) and (21) as well as the free surface and
| |
| − | sea floor boundary conditions. This gives the following equation for the
| |
| − | potential in <math>\Omega ,</math>
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | \nabla ^{2}\phi =0,\;\;\;\mathbf{x}\in \Omega , \\
| |
| − | \phi _{n}=\mathbf{Q}_{1}\phi -2ik_{t}^{\left( 1\right) }\cosh \left(
| |
| − | k_{t}^{\left( 1\right) }\left( z+1\right) \right) \,e^{-ik_{t}^{\left(
| |
| − | 1\right) }l},\;\;\;\mathbf{x}\in \partial \Omega _{1}, \\
| |
| − | \phi _{n}=\mathbf{Q}_{2}\phi ,\;\;\;\mathbf{x}\in \partial \Omega _{2}, \\
| |
| − | \phi _{n}=\nu \phi ,\;\;\;\mathbf{x}\in \partial \Omega _{3}\cup \partial
| |
| − | \Omega _{5}, \\
| |
| − | \phi _{n}=\mathbf{g}\phi ,\;\;\;\mathbf{x}\in \partial \Omega _{4}, \\
| |
| − | \phi _{n}=0,\;\;\;\mathbf{x}\in \partial \Omega _{6}.
| |
| − | \end{matrix}
| |
| − | \right\} (23)
| |
| − | </math></center>
| |
| − | This boundary value problem is shown in Figure \ref{fig_sigma}. The boundary
| |
| − | of <math>\Omega </math> (<math>\partial \Omega )</math> has been divided into the six boundary
| |
| − | regions <math>\partial \Omega _{i}</math> shown in the figure. They are respectively,
| |
| − | the boundary of <math>\Omega ^{-}</math> and <math>\Omega </math> (<math>\partial \Omega _{1}),</math> the
| |
| − | boundary of <math>\Omega ^{+}</math> and <math>\Omega </math> (<math>\partial \Omega _{2}),</math> the free
| |
| − | water surface to the left (<math>\partial \Omega _{3}</math>) and right of the plate (<math>
| |
| − | \partial \Omega _{5}<math>), the plate (</math>\partial \Omega _{4}</math>), and the sea
| |
| − | floor (<math>\partial \Omega _{6}</math>). Equation (23) is the
| |
| − | boundary value problem which we will solve numerically.
| |
| − |
| |
| − | =Numerical Solution Method=
| |
| − |
| |
| − | We have reduced the problem to Laplace's equation in a finite domain subject
| |
| − | to certain boundary conditions (23). These boundary
| |
| − | conditions give the outward normal derivative of the potential as a function
| |
| − | of the potential but this is not always a point-wise condition; on some
| |
| − | boundaries it is given by an integral equation. We must solve both Laplace's
| |
| − | equation and the integral equations numerically. We will solve Laplace's
| |
| − | equation by the boundary element method and the integral equations by
| |
| − | numerical integration. However, the same discretisation of the boundary will
| |
| − | be used for both numerical solutions.
| |
| − |
| |
| − | We begin by applying the boundary element method to equation (\ref
| |
| − | {finitedomain}). This gives us the following equation relating the potential
| |
| − | and its outward normal derivative on the boundary <math>\partial \Omega </math>
| |
| − | <center><math>
| |
| − | \frac{1}{2}\phi \left( \mathbf{x}\right) =\int_{\partial \Omega }\left(
| |
| − | G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
| |
| − | ^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
| |
| − | _{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime },\;\;
| |
| − | \mathbf{x}\in \partial \Omega . (24)
| |
| − | </math></center>
| |
| − | In equation (24) <math>G\left( \mathbf{x},\mathbf{x}^{\prime
| |
| − | }\right) </math> is the free space Green function given by
| |
| − | <center><math>
| |
| − | G\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\frac{1}{2\pi }\ln \,\left|
| |
| − | \mathbf{x}-\mathbf{x}^{\prime }\right| , (25)
| |
| − | </math></center>
| |
| − | and <math>G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) </math> is the outward
| |
| − | normal derivative of <math>G</math> (with respect to the <math>\mathbf{x}^{\prime }</math>
| |
| − | coordinate).
| |
| − |
| |
| − | We solve equation (24) by a modified constant panel method
| |
| − | which reduces it to the following matrix equation
| |
| − | <center><math>
| |
| − | \frac{1}{2}\vec{\phi}=\mathbf{G}_{n}\vec{\phi}-\mathbf{G}\vec{\phi}_{n}.
| |
| − | (26)
| |
| − | </math></center>
| |
| − | In equation (26) <math>\vec{\phi}\mathcal{\ }</math>and <math>\vec{\phi}
| |
| − | _{n}</math> are vectors which approximate the potential and its normal derivative
| |
| − | around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
| |
| − | are matrices corresponding to the Green function and the outward normal
| |
| − | derivative of the Green function respectively. The method used to calculate
| |
| − | the elements of the matrices <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math> will be
| |
| − | discussed in section \ref{Green}.
| |
| − |
| |
| − | The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the
| |
| − | potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
| |
| − | \partial \Omega </math> in equation (23). This can be expressed as
| |
| − | <center><math>
| |
| − | \vec{\phi}_{n}=\mathbf{A}\,\vec{\phi}-\vec{f}, (27)
| |
| − | </math></center>
| |
| − | where <math>\mathbf{A}</math> is the block diagonal matrix given by
| |
| − | <center><math>
| |
| − | \mathbf{A}\mathbb{=}\left[
| |
| − | \begin{matrix}{cccccc}
| |
| − | \mathbf{Q}_{1} & & & & & \\
| |
| − | & \mathbf{Q}_{2} & & & & \\
| |
| − | & & \nu \,\mathbf{I} & & & \\
| |
| − | & & & \mathbf{g} & & \\
| |
| − | & & & & \nu \,\mathbf{I} & \\
| |
| − | & & & & & 0
| |
| − | \end{matrix}
| |
| − | \right] , (28)
| |
| − | </math></center>
| |
| − | <math>\mathbf{Q}_{1}</math>, <math>\mathbf{Q}_{2}</math>, and <math>\mathbf{g}</math> are matrix
| |
| − | approximations of the integral operators of the same name and <math>\vec{f}</math> is
| |
| − | the vector
| |
| − | <center><math>
| |
| − | \vec{f}=\left[
| |
| − | \begin{matrix}{c}
| |
| − | 2ik_{t}^{\left( 1\right) }\cosh \left( k_{t}^{\left( 1\right) }\left(
| |
| − | z+1\right) \right) \,e^{ik_{t}^{\left( 1\right) }l} \\
| |
| − | 0 \\
| |
| − | \vdots \\
| |
| − | 0
| |
| − | \end{matrix}
| |
| − | \right] . (29)
| |
| − | </math></center>
| |
| − | \ The methods used to construct the matrices <math>\mathbf{Q}_{1},\mathbf{Q}_{2}</math>
| |
| − | , and <math>\mathbf{g}</math> will be described in sections 31 and \ref
| |
| − | {numericalg} respectively.
| |
| − |
| |
| − | Substituting equation (27) into equation (\ref
| |
| − | {panelEqn_boundary}) we obtain the following matrix equation for the
| |
| − | potential
| |
| − | <center><math>
| |
| − | \left( \frac{1}{2}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
| |
| − | \vec{f}
| |
| − | </math></center>
| |
| − | which can be solved straightforwardly. The reflection and transmission
| |
| − | coefficients are calculated from <math>\vec{\phi}</math> using equations (\ref
| |
| − | {reflection}) and (22) respectively.
| |
| − |
| |
| − | ==Numerical Calculation of <math>\mathbf{G==</math> and <math>\mathbf{G}_{n}</math>\label
| |
| − | {Green}}
| |
| − |
| |
| − | The boundary element equation (24) is solved numerically by
| |
| − | a modified constant panel method. In this method, the boundary is divided
| |
| − | into panels over which the potential, <math>\phi ,</math> or its outward normal
| |
| − | derivative, <math>\phi _{n},</math> are assumed to be constant. The free-space Green's
| |
| − | function, <math>G,</math> and its normal derivative, <math>G_{n}</math> are more rapidly varying
| |
| − | and have a singularity at <math>\mathbf{x}=\mathbf{x}^{\prime }</math>. For this
| |
| − | reason, over each panel, while <math>\phi </math> and <math>\phi _{n}</math> are assumed constant,
| |
| − | <math>G</math> and <math>G_{n}</math> are integrated exactly. For example, we use the following
| |
| − | approximation to calculate the integral of <math>G</math> and <math>\phi </math> over a single
| |
| − | panel
| |
| − | <center><math>
| |
| − | \int_{\mathbf{x}_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}
| |
| − | ^{\prime }\right) \phi \left( \mathbf{x}^{\prime }\right) d\mathbf{x}
| |
| − | ^{\prime }\approx \phi \left( \mathbf{x}_{i}\right) \int_{\mathbf{x}
| |
| − | _{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}^{\prime }\right)
| |
| − | d\mathbf{x}^{\prime }, (30)
| |
| − | </math></center>
| |
| − | where <math>\mathbf{x}_{i}</math> is the midpoint of the panel and <math>h</math> is the panel
| |
| − | length. The integral on the right hand side of equation (30),
| |
| − | because of the simple structure of <math>G</math>, can be calculated exactly.
| |
| − |
| |
| − | ==Numerical Calculation of <math>\mathbf{Q==_{1}</math> and <math>\mathbf{Q}_{2}
| |
| − | (31)</math>}
| |
| − |
| |
| − | We will discuss the numerical approximation of the operator <math>\mathbf{Q}_{1}</math>
| |
| − | . The operator <math>\mathbf{Q}_{2}</math> is approximated in a similar fashion. We
| |
| − | begin with equation (19) truncated to a finite number (<math>N</math>) of
| |
| − | evanescent modes,
| |
| − | <center><math>
| |
| − | \mathbf{Q}_{1}\phi =\sum_{m=0}^{N}k_{m}^{\left( 1\right) }\left\langle \phi
| |
| − | \left( z\right) ,\tau _{m}^{\left( 1\right) }\left( z\right) \right\rangle
| |
| − | \tau _{m}^{\left( 1\right) }\left( z\right) .
| |
| − | </math></center>
| |
| − | We calculate the inner product
| |
| − | <center><math>
| |
| − | \left\langle \phi \left( z\right) ,\tau _{m}^{\left( 1\right) }\left(
| |
| − | z\right) \right\rangle =\int\nolimits_{-1}^{0}\phi \left( z\right) \,\tau
| |
| − | _{m}^{\left( 1\right) }\left( z\right) \,dz
| |
| − | </math></center>
| |
| − | with the same panels as we used to approximate the integrals of the Green
| |
| − | function and its normal derivative in subsection \ref{Green}. Similarly, we
| |
| − | assume that <math>\phi </math> is constant over each panel and integrate <math>\tau
| |
| − | _{m}^{\left( 1\right) }\left( z\right) </math> exactly. This gives us the
| |
| − | following matrix factorisation of <math>\mathbf{Q}_{1},</math>
| |
| − | <center><math>
| |
| − | \mathbf{Q}_{1}\,\vec{\phi}=\mathbf{S}\,\mathbf{R}\,\vec{\phi}.
| |
| − | </math></center>
| |
| − | The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
| |
| − | <center><math>\begin{matrix}
| |
| − | s_{im} &=&\tau _{m}^{\left( 1\right) }\left( z_{i}\right) , \\
| |
| − | r_{mj} &=&k_{m}^{\left( 1\right) }\int_{z_{j}-h/2}^{z_{j}+h/2}\tau
| |
| − | _{m}^{\left( 1\right) }\left( s\right) ds
| |
| − | \end{matrix}</math></center>
| |
| − | where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
| |
| − | and <math>h</math> is the panel length. The integral operator <math>\mathbf{Q}_{2}</math> is
| |
| − | approximated by a matrix in exactly the same manner.
| |
| − |
| |
| − | ==Numerical Calculation of <math>\mathbf{g(32)==</math>}
| |
| − |
| |
| − | The method used to approximate <math>\mathbf{g}</math> by a matrix is similar to the
| |
| − | methods used for <math>\mathbf{Q}_{1}</math> and <math>\mathbf{Q}_{2}</math> and follows \cite
| |
| − | {jgrfloe1d} with some modification. The Green function for the plate can be
| |
| − | expressed as
| |
| − | <center><math>
| |
| − | g\left( x,\xi \right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | A_{1}e^{i\alpha x}+B_{1}e^{-i\alpha x}+C_{1}e^{\alpha x}+D_{1}e^{-\alpha
| |
| − | x},\qquad x<\xi , \\
| |
| − | A_{2}e^{i\alpha x}+B_{2}e^{-i\alpha x}+C_{2}e^{\alpha x}+D_{2}e^{-\alpha
| |
| − | x},\qquad x>\xi ,
| |
| − | \end{matrix}
| |
| − | \right.
| |
| − | </math></center>
| |
| − | where the coefficients are determined by solving a linear system \cite
| |
| − | {jgrfloe1d}. Again the same panels are used to approximate the integral
| |
| − | operator by a matrix as were used for the boundary integral equation (\ref
| |
| − | {integral_eqn}). Over each panel we assume that the potential is constant
| |
| − | and integrate the Green function <math>g</math> exactly.
| |
| − |
| |
| − | =Results=
| |
| − |
| |
| − | We will now present some results, concentrating on comparing the reflection
| |
| − | coefficient for constant and variable depth profiles. This will allow us to
| |
| − | determine when the variable depth profile has a significant effect. To
| |
| − | reduce the number of figures we restrict ourselves to four values of the
| |
| − | stiffness <math>\beta </math> and two variable depth profiles.
| |
| − |
| |
| − | ==Profiles for the variable depth==
| |
| − |
| |
| − | We will consider two different profiles for the variable depth. The first
| |
| − | will be the profile which was used by [[Staziker96]]. This corresponds to
| |
| − | a rise from a uniform depth to a maximum height of half the uniform depth at
| |
| − | <math>x=0</math>. The formula for this profile is the following
| |
| − | <center><math>
| |
| − | d\left( x\right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | -1,\;\;x<-l, \\
| |
| − | -\left( \frac{1}{2}\left( \frac{x+l}{l}\right) ^{2}-\frac{x+l}{l}
| |
| − | +1\right) ,\;\;-l<x<l, \\
| |
| − | -1,\;\;x>l.
| |
| − | \end{matrix}
| |
| − | \right. (33)
| |
| − | </math></center>
| |
| − | Following [[Staziker96]] we will refer to this variable depth profile as
| |
| − | the ''hump. ''
| |
| − |
| |
| − | In the second profile the depth rises linearly. The depth in the right hand
| |
| − | region is half the depth in the left hand region. The formula for this
| |
| − | profile is
| |
| − | <center><math>
| |
| − | d\left( x\right) =\left\{
| |
| − | \begin{matrix}{c}
| |
| − | -1,\;\;x<-l, \\
| |
| − | \frac{x+l}{4l}-1,\;\;-l<x<l, \\
| |
| − | -\frac{1}{2},\;\;x>l.
| |
| − | \end{matrix}
| |
| − | \right. (34)
| |
| − | </math></center>
| |
| − | We will refer to this variable depth profile as the ''simple slope'' .
| |
| − |
| |
| − | ==Convergence study==
| |
| − |
| |
| − | We now present a convergence study. Since we have two parameters, the panel
| |
| − | size used to discretise the boundary and the number of evanescent modes (<math>N</math>
| |
| − | ) used to approximate the integral equations <math>\mathbf{Q}_{1}</math> and <math>\mathbf{Q}
| |
| − | _{2},</math> we must present two convergence studies considering each parameter
| |
| − | separately. We begin by considering the panel size used to discretise the
| |
| − | boundary. We expect that the panel size should be proportional to the
| |
| − | wavelength and therefore inversely proportional to the wavenumber <math>
| |
| − | k_{t}^{\left( 1\right) }</math> (assuming that the water depth at either end is of
| |
| − | a similar size). We therefore use the following formula for the panel size
| |
| − | <center><math>
| |
| − | =panel size = =\frac{1}{\kappa \,k_{t}^{\left( 1\right) }},
| |
| − | </math></center>
| |
| − | where <math>\kappa </math> is a constant of proportionality which will be determined
| |
| − | from the convergence study. Table 35 shows the absolute value
| |
| − | of the reflection coefficient for a plate of length <math>L=2.5,</math> stiffness <math>
| |
| − | \beta =1,<math> and mass </math>\gamma =0<math> for </math>\nu =1,</math> 2, and 3 for a constant depth
| |
| − | and for the hump with <math>l=2.5.</math> The number of evanescent modes, <math>N,</math> was
| |
| − | fixed to be 5. Three values of <math>\kappa </math> were considered, <math>\kappa =10,</math> 20,
| |
| − | and 40. The results in Table 35 show that good convergence is
| |
| − | achieved when <math>\kappa =20.</math>
| |
| − |
| |
| − | Table 37 shows a similar convergence study for the number of
| |
| − | evanescent modes. We have considered <math>0,</math> 5 and <math>10</math> evanescent modes (<math>N</math>)
| |
| − | and fixed <math>\kappa </math> to be <math>\kappa =20</math>. All other parameters as the same as
| |
| − | those in Table 35. The results in Table 37 show
| |
| − | that good convergence is achieved when the number of modes is 5. For all
| |
| − | subsequent calculations the panel size will be determined by setting <math>\kappa
| |
| − | =20<math> and </math>N=5.</math>
| |
| − |
| |
| − | ==Comparison with existing results==
| |
| − |
| |
| − | Before presenting our results for a plate on water of variable depth we will
| |
| − | make comparisons with two results from the literature. This is to establish
| |
| − | that our method gives the correct solution for the simpler cases of either
| |
| − | variable depth but no plate, or a plate floating on constant depth. We begin
| |
| − | by comparing our results with [[Staziker96]] who solved for wave
| |
| − | scattering by variable depth only. One problem which they solved was to
| |
| − | determine the absolute value of the reflection coefficient for a hump depth
| |
| − | profile with fixed frequency <math>\nu =1</math> and variable hump length <math>l.</math> The
| |
| − | solution to this problem by our method is shown in Figure (\ref{staziker1.ps}
| |
| − | ). This figure is identical to Figure 2 in [[Staziker96]] (p. 290) which
| |
| − | establishes that our method gives the correct solution for a variable depth
| |
| − | in the absence of the plate.
| |
| − |
| |
| − | The second comparison is with [[jgrfloe1d]] in which the problem of a
| |
| − | thin plate on water of constant depth was solved (to model an ice floe). One
| |
| − | problem which they solved was the absolute value of the reflection
| |
| − | coefficient as a function of plate length for fixed <math>\nu .</math> The dimensional
| |
| − | parameters which they used were, density <math>\rho ^{\prime }=922.5\,</math>kgm<math>^{-3},</math>
| |
| − | thickness <math>h=1</math>m, and bending rigidity <math>D=</math>5.4945<math>\times 10^{8}</math>kgm<math>^{2}</math>s<math>
| |
| − | ^{-2}.<math> The water density was 1025kgm</math>^{-3}</math> and the incoming wave was
| |
| − | chosen to have wavelength <math>100</math>m. The solution to this problem by our method
| |
| − | is shown in Figure (\ref{fig_mike_refl}). This figure is identical to Figure
| |
| − | 3 in [[jgrfloe1d]] (p. 895) which establishes that our method gives the
| |
| − | correct result for a plate on water of constant depth.
| |
| − |
| |
| − | ==Reflection==
| |
| − |
| |
| − | We will consider the absolute value of the reflection coefficient as a
| |
| − | function of wavenumber <math>\nu </math> for various values of the parameters. Figure (
| |
| − | \ref{plot4hump}) shows the absolute value of the reflection coefficient as a
| |
| − | function of <math>\nu </math> with <math>\beta =0.01,</math> <math>0.1,</math> 1 and 10 and <math>\gamma =0</math> for
| |
| − | the hump depth profile. Both the length of the hump and the length of the
| |
| − | plate were fixed to be <math>l=L=2.5</math>. The solution for the plate and hump (solid
| |
| − | line), plate with constant depth (dashed line) and hump only (dotted line)
| |
| − | are shown. The two simpler solutions are drawn so that the full solution may
| |
| − | be compared to these simpler cases. Figure (\ref{plot4hump}) shows the
| |
| − | existence of two asymptotic regimes. When <math>\nu </math> is small (low frequency or
| |
| − | large wavelength)\ the reflection is dominated by the hump and the plate is
| |
| − | transparent. For large <math>\nu </math> the reflection is dominated by the plate and
| |
| − | the hump is transparent. As the value of the stiffness <math>\beta </math> is increased
| |
| − | the hump dominated region becomes smaller and the plate dominated region
| |
| − | becomes larger. This not unexpected because increasing the value of <math>\beta </math>
| |
| − | increases the influence of the plate. It is apparent that, especially for
| |
| − | smaller values of stiffness, there is a large region where the hump and
| |
| − | plate solution is significantly different from the plate only solution, even
| |
| − | though the hump only reflection is practically zero. This is because the
| |
| − | wavelength under the plate is larger than the open water wavelength so the
| |
| − | depth variation is felt more strongly when the plate is present.
| |
| − |
| |
| − | Figure (\ref{plot4plane}) is equivalent to Figure (\ref{plot4hump}) except
| |
| − | that the depth profile is the simple slope and the results are also very
| |
| − | similar<math>.</math> Figure (\ref{plot4hump_gamma}) is equivalent to Figure (\ref
| |
| − | {plot4hump}) except that the value of stiffness is fixed (<math>\beta =0.1)</math> and
| |
| − | the value of <math>\gamma </math> is varied. This figure shows that for realistic
| |
| − | (small) values of <math>\gamma </math> this parameter is not significant. This explains
| |
| − | why <math>\gamma </math> is often neglected (e.g. [[OhkusuISOPE]]) and why we have
| |
| − | chosen <math>\gamma =0</math> for Figures (\ref{plot4hump}) and (\ref{plot4plane}).
| |
| − |
| |
| − | Figure (\ref{plot4ahump}) is equivalent to figure (\ref{plot4hump}) except
| |
| − | that the hump has been moved by <math>L</math> to the left so that the minimum depth is
| |
| − | directly underneath the left (incoming) end of the plate. Comparing figure (
| |
| − | \ref{plot4ahump}) with figure (\ref{plot4ahump}) we see that moving the hump
| |
| − | has made a significant change to the reflection coefficient for low
| |
| − | frequencies, especially as the stiffness <math>\beta </math> is increased.
| |
| − |
| |
| − | ==Displacements==
| |
| − |
| |
| − | Finally we investigate the displacement of the plate for some of the regimes
| |
| − | we have considered. We present the displacement for the variable and
| |
| − | constant depth profiles so that we may compare the effect of the variable
| |
| − | depth. We divide the displacement by <math>i\omega /k_{t}^{\left( 1\right) }\sinh
| |
| − | \left( k_{t}^{\left( 1\right) }\right) </math> so that the incoming wave now has
| |
| − | unit amplitude in displacement at the water surface. Figure (\ref
| |
| − | {deflbeta4hump_nu05}) shows the displacement of the plate for <math>\nu =0.5</math> and
| |
| − | <math>\beta =0.01,</math> <math>0.1,</math> 1 and 10 and <math>\gamma =0.</math> The plate length is <math>L=2.5</math>.
| |
| − | The solid line is the real part of the displacement and the dotted line is
| |
| − | the imaginary part of the displacement. The thicker line is the solution
| |
| − | with the hump depth profile (<math>l=2.5)</math> and the thinner line is the solution
| |
| − | with a constant depth. It is apparent from this figure, that the bending of
| |
| − | the plate is increased by the presence of the hump, but that this effect is
| |
| − | not very strong. Figures (\ref{deflbeta4hump_nu1}) and (\ref
| |
| − | {deflbeta4hump_nu2}) are identical to Figure (\ref{deflbeta4hump_nu05})
| |
| − | except that <math>v=1</math> and <math>\nu =2</math> respectively. These figures also show only a
| |
| − | slight increase in the bending of the plate due to the hump. It appears
| |
| − | that, while the variable depth does have a significant effect on the
| |
| − | reflection coefficient, the effect on the plate displacement is not
| |
| − | necessarily as strong.
| |
| − |
| |
| − | =Summary=
| |
| − |
| |
| − | We have presented a solution for the linear wave forcing of a floating thin
| |
| − | plate on water of variable depth. The solution method was based on reducing
| |
| − | the problem to a finite domain which contained both the region of variable
| |
| − | water depth and the floating thin plate. In this finite region the outward
| |
| − | normal derivative of the potential around the boundary was expressed as a
| |
| − | function of the potential. This was accomplished by using integral operators
| |
| − | for the boundary under the plate and the radiating boundaries. The integral
| |
| − | operator for the plate was calculated using a Green function as described in
| |
| − | [[jgrfloe1d]]. The integral operators for the radiation boundary
| |
| − | conditions were calculated by solving Laplace's equation in the
| |
| − | semi-infinite outer domains using separation of variables as described in
| |
| − | [[Liu82, Hazard]]. Laplace's equation in the finite domain was solved
| |
| − | using the boundary element method.
| |
| − |
| |
| − | The results showed that, for certain parameter regimes, there was
| |
| − | significant difference between the absolute value of the reflection
| |
| − | coefficient for the variable depth and constant depth profiles. Furthermore,
| |
| − | the region of influence of the variable depth was increased by the presence
| |
| − | of the plate due to the increased wavelength under the plate. Finally, there
| |
| − | was a slight increase in the bending of the plate due to the presence of the
| |
| − | variable depth profile.
| |
| − |
| |
| − | \bibliographystyle{IEEE}
| |
| − | \bibliography{mike,others}
| |
| − | \pagebreak
| |
| − |
| |
| − | =Tables=
| |
| − |
| |
| − |
| |
| − |
| |
| − | \begin{table}[h] \centering
| |
| − |
| |
| − | (35)
| |
| − | \begin{tabular}{lll}
| |
| − | <math>\nu =1</math> & & \\ \hline
| |
| − | \multicolumn{1}{|c}{<math>\kappa </math>} & \multicolumn{1}{|c}{plate only} &
| |
| − | \multicolumn{1}{|c|}{plate and hump} \\ \hline
| |
| − | \multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.2983746166} &
| |
| − | \multicolumn{1}{|c|}{0.2491046427} \\ \hline
| |
| − | \multicolumn{1}{|c}{20} & \multicolumn{1}{|c}{0.2957175612} &
| |
| − | \multicolumn{1}{|c|}{0.2470511349} \\ \hline
| |
| − | \multicolumn{1}{|c}{40} & \multicolumn{1}{|c}{0.2948480461} &
| |
| − | \multicolumn{1}{|c|}{0.2465182527} \\ \hline
| |
| − | <math>\nu =2</math> & & \\ \hline
| |
| − | \multicolumn{1}{|c}{<math>\kappa </math>} & \multicolumn{1}{|c}{plate only} &
| |
| − | \multicolumn{1}{|c|}{plate and hump} \\ \hline
| |
| − | \multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.3457203891} &
| |
| − | \multicolumn{1}{|c|}{0.1934227806} \\ \hline
| |
| − | \multicolumn{1}{|c}{20} & \multicolumn{1}{|c}{0.3461627544} &
| |
| − | \multicolumn{1}{|c|}{0.1947144005} \\ \hline
| |
| − | \multicolumn{1}{|c}{40} & \multicolumn{1}{|c}{0.3463681703} &
| |
| − | \multicolumn{1}{|c|}{0.1952205755} \\ \hline
| |
| − | <math>\nu =3</math> & & \\ \hline
| |
| − | \multicolumn{1}{|c}{<math>\kappa </math>} & \multicolumn{1}{|c}{plate only} &
| |
| − | \multicolumn{1}{|c|}{plate and hump} \\ \hline
| |
| − | \multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.0277944414} &
| |
| − | \multicolumn{1}{|c|}{0.2605833203} \\ \hline
| |
| − | \multicolumn{1}{|c}{20} & \multicolumn{1}{|c}{0.0249319083} &
| |
| − | \multicolumn{1}{|c|}{0.2568361963} \\ \hline
| |
| − | \multicolumn{1}{|c}{40} & \multicolumn{1}{|c}{0.0236686063} &
| |
| − | \multicolumn{1}{|c|}{0.2550926942} \\ \hline
| |
| − | \end{tabular}
| |
| − | \caption{<math>\left|R\right|</math> for <math>\kappa</math> = 10, 20, and 40 and
| |
| − | <math>\nu </math> = 1,2, and 3 for the plate only and the plate and hump.
| |
| − | <math>\beta=1</math>, <math>\gamma =0</math> and 5 evanscent modes are
| |
| − | used. (36)}
| |
| − |
| |
| − |
| |
| − | \end{table}
| |
| − |
| |
| − | \pagebreak
| |
| − |
| |
| − |
| |
| − |
| |
| − | \begin{table}[t] \centering
| |
| − |
| |
| − | (37)
| |
| − | \begin{tabular}{ccc}
| |
| − | <math>\nu =1</math> & & \\ \hline
| |
| − | \multicolumn{1}{|c}{<math>N</math>} & \multicolumn{1}{|c}{plate only} &
| |
| − | \multicolumn{1}{|c|}{plate and hump} \\ \hline
| |
| − | \multicolumn{1}{|c}{0} & \multicolumn{1}{|c}{0.2934754434} &
| |
| − | \multicolumn{1}{|c|}{0.2448842047} \\ \hline
| |
| − | \multicolumn{1}{|c}{5} & \multicolumn{1}{|c}{0.2957175612} &
| |
| − | \multicolumn{1}{|c|}{0.2470511349} \\ \hline
| |
| − | \multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.2957697025} &
| |
| − | \multicolumn{1}{|c|}{0.2470814786} \\ \hline
| |
| − | <math>\nu =2</math> & & \\ \hline
| |
| − | \multicolumn{1}{|c}{<math>N</math>} & \multicolumn{1}{|c}{plate only} &
| |
| − | \multicolumn{1}{|c|}{plate and hump} \\ \hline
| |
| − | \multicolumn{1}{|c}{0} & \multicolumn{1}{|c}{0.3392954189} &
| |
| − | \multicolumn{1}{|c|}{0.2021691049} \\ \hline
| |
| − | \multicolumn{1}{|c}{5} & \multicolumn{1}{|c}{0.3461627544} &
| |
| − | \multicolumn{1}{|c|}{0.1947144005} \\ \hline
| |
| − | \multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.3459741914} &
| |
| − | \multicolumn{1}{|c|}{0.1944525347} \\ \hline
| |
| − | <math>\nu =3</math> & & \\ \hline
| |
| − | \multicolumn{1}{|c}{<math>N</math>} & \multicolumn{1}{|c}{plate only} &
| |
| − | \multicolumn{1}{|c|}{plate and hump} \\ \hline
| |
| − | \multicolumn{1}{|c}{0} & \multicolumn{1}{|c}{0.0262065602} &
| |
| − | \multicolumn{1}{|c|}{0.2296877448} \\ \hline
| |
| − | \multicolumn{1}{|c}{5} & \multicolumn{1}{|c}{0.0249319083} &
| |
| − | \multicolumn{1}{|c|}{0.2568361963} \\ \hline
| |
| − | \multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.0259483771} &
| |
| − | \multicolumn{1}{|c|}{0.2578144528} \\ \hline
| |
| − | \end{tabular}
| |
| − | \caption{<math>\left|R\right|</math> for 0, 5, and 10 evanescent modes (<math>N</math>)
| |
| − | and <math>\nu </math> = 1,2, and 3 for the plate only and the plate and hump.
| |
| − | <math>\beta=1</math>, <math>\gamma =0</math> and
| |
| − | <math>\kappa = 20</math>.(38)}
| |
| − |
| |
| − |
| |
| − | \end{table}
| |
| | | | |
| | | | |
