|
|
| (52 intermediate revisions by 4 users not shown) |
| Line 1: |
Line 1: |
| − |
| |
| − |
| |
| − | \begin{opening}
| |
| − | \title{The Linear Wave Response of a Periodic Array of Floating \\
| |
| − | Elastic Plates}
| |
| − |
| |
| − | C. D. \surname{Wang}\email{cvewcd@nus.edu.sg}
| |
| − | \institute{Department of Civil Engineering, National University of Singapore,
| |
| − | Kent Ridge, Singapore 119260, Singapore.}
| |
| − | M. H. \surname{Meylan}\email{meylan@math.auckland.ac.nz}
| |
| − | \institute{Department of Mathematics, University of Auckland,
| |
| − | Private Bag 92019, Auckland, New Zealand.}
| |
| − | R. \surname{Porter}\email{Richard.Porter@bristol.ac.uk}
| |
| − | \institute{School of Mathematics, University of Bristol, Bristol,
| |
| − | BS8 1TW, UK.}
| |
| − | \institute{}
| |
| − |
| |
| − |
| |
| − |
| |
| − | \runningtitle{Response of a Periodic Array of Plates}
| |
| − | \runningauthor{Wang, Meylan \& Porter}
| |
| − |
| |
| − |
| |
| − | The problem of an infinite periodic array of identical floating elastic
| |
| − | plates subject to forcing from plane incident waves is considered.
| |
| − | This study is motivated by the problem of trying to model wave propagation
| |
| − | in the marginal ice zone, a region of ocean consisting of an arbitrary
| |
| − | packing of floating ice sheets. It is shown that the problem considered
| |
| − | can be formulated exactly in terms of the solution to an integral equation in
| |
| − | a manner similar to that used for the problem of wave scattering by a
| |
| − | single elastic floating plate, the key difference here being the use
| |
| − | of a modified periodic Green function. The convergence of this Green
| |
| − | function in its original form is poor, but can be accelerated by a
| |
| − | transformation. It is shown that the results from the method satisfy
| |
| − | energy conservation and that in the particular case of a fixed
| |
| − | rigid rectangular plate which spans the periodicity uniformly the
| |
| − | solution reduces to that for a two-dimensional rigid dock. We
| |
| − | also present solutions for a range of elastic plate geometries.
| |
| − |
| |
| − |
| |
| − | \keywords{Ice sheet, elastic plates, periodic Green function, water waves}
| |
| − |
| |
| − | \end{opening}
| |
| − |
| |
| | =Introduction= | | =Introduction= |
| | | | |
| − | The thin elastic plate of shallow draft floating on the surface of water
| + | We present here the solution to the [[:Category:Infinite Array|Infinite Array]] based |
| − | can be used to model a range of physical systems, for example ice floes
| + | on an infinite image system of [[Free-Surface Green Function|Free-Surface Green Functions]] |
| − | or so-called very large floating structures, which are of
| |
| − | particular interest in proposals to build offshore floating runways.
| |
| − | It is also of theoretical importance as one of the simplest models
| |
| − | of hydroelasticity. It is not surprising therefore that the floating
| |
| − | elastic plate has been the subject of a significant body of research. Much
| |
| − | of this research, especially that which was motivated by the construction
| |
| − | of very large floating structures, is summarised in the review papers
| |
| − | by Kashiwagi [[Kashiwagi00]] and Watanabe, Utsunomia, and Wang [[Watanabe04]]. The review
| |
| − | paper by Squire et al. [[Squire_Review]] summarises the research prior to 1995 which
| |
| − | was motivated by modelling sea ice floes, but does not include the more
| |
| − | recent work, Meylan and Squire [[jgrfloecirc]] and Meylan [[JGR02]], in which the solution
| |
| − | for wave scattering by a three-dimensional ice floe is presented.
| |
| | | | |
| − | Our aim here is to present a method of solution to the problem of wave
| + | =Problem Formulation= |
| − | scattering by an infinite periodic array of floating elastic plates of finite dimension.
| |
| − | This study is motivated by trying to further understanding of the
| |
| − | problem of wave scattering in the marginal ice zone. The marginal ice
| |
| − | zone consists of a vast field of broken ice floes each of which have a
| |
| − | thickness (typically <math>\approx 1</math>-<math>2</math>m) much smaller than the typical
| |
| − | horizontal lengthscale of the floe, or of the wavelength of waves
| |
| − | supported by the floe, and it is usual to model the ice floe as a thin
| |
| − | elastic plate. These ice floes are subject to intense wave forces whose
| |
| − | source originates from the open ocean. On account of the fact
| |
| − | that the floes are themselves able to support wave propagation,
| |
| − | wave energy is capable of travelling large distances within the marginal
| |
| − | ice zone where it can assist ice breakup.
| |
| − | | |
| − | Previous work aimed at trying to model wave interaction by isolated
| |
| − | three-dimensional regular and irregular ice floes has been performed in
| |
| − | Meylan and Squire [[jgrfloecirc]] and Meylan [[JGR02]], respectively. For more complicated
| |
| − | arrangements involving an arbitrary, but finite, number of irregular ice
| |
| − | floes see Peter and Meylan [[JFM_malte04]], although the number of floes that can be
| |
| − | dealt with is limited by numerical considerations. Although this latter
| |
| − | work includes the effect of interaction between neighbouring ice floes,
| |
| − | it does not allow one to capture the effects of a plane wave incident upon
| |
| − | a very large linear array of ice floes. One method of overcoming this, which
| |
| − | also allows significant analytic progress to be made is to consider
| |
| − | wave scattering by an infinite periodic array of identical ice floes,
| |
| − | being irregular in shape. Of course ice fields are not periodic, but
| |
| − | this arrangement does provide some information especially if some kind
| |
| − | of statistical averaging of the results over the floe geometries and
| |
| − | spacings is performed.
| |
| − | For example, we might expect that a random array of ice floes would
| |
| − | scatter equivalently to an average of periodic arrays of identical floes of
| |
| − | different spacings and geometries.
| |
| − | Furthermore, the solution readily lends itself
| |
| − | to an assumption that the rows of periodic ice floes are
| |
| − | wide-spaced. This method was used by Evans and Porter [[Evans95]]
| |
| − | and would allow accurate approximations to wave scattering by multiple
| |
| − | rows of non-identical ice floes. Central to such a procedure is
| |
| − | knowledge of the scattering process for each row in isolation.
| |
| − | | |
| − | The problem of determining the scattering of waves by periodic arrays
| |
| − | of obstacles subject to wave forcing has received considerable research
| |
| − | attention and spans a broad range of physical disciplines including
| |
| − | solid-state physics, acoustics, optics, etc. In many applications,
| |
| − | the interest centres on arrangements which are periodic in two
| |
| − | directions (for example, the study of crystallography).
| |
| − |
| |
| − | In the present context of water wave propagation and its interaction
| |
| − | with flexible surface structures, Chou [[Chou98]] has investigated
| |
| − | the effect of an infinite doubly-periodic array of elastic plates
| |
| − | on wave propagation. In the modelling of the plate equations,
| |
| − | Chou incorporates both bending stiffness and tension effects, so
| |
| − | that the discussion of the results not only includes the case of
| |
| − | pure bending of elastic plates in the absence of compression forces
| |
| − | (as considered here), but also, by setting the stiffness to zero,
| |
| − | pure tensional effects which would describe, for example, periodic arrays of
| |
| − | taught membranes. However, in contrast to the work described here,
| |
| − | the doubly-periodic configuration allows significant simplification
| |
| − | in the solution procedure by applying Floquet's theorem to reduce
| |
| − | the problem to one on a finite domain with periodic boundary
| |
| − | conditions. Moreover, problems involving infinite doubly-periodic
| |
| − | structures only offer information about the possibility of wave
| |
| − | propagation throughout the array (in the form of so-called pass-bands
| |
| − | or stop-bands) and cannot address the diffraction of plane waves from
| |
| − | infinity.
| |
| − | | |
| − | For arrays which are periodic in one direction only, the situation
| |
| − | is different and diffraction grating effects occur. Thus, for an
| |
| − | incident plane wave of a particular given wave frequency, a finite
| |
| − | number of distinct plane waves propagating away from the array at
| |
| − | certain discrete angles will occur. In the context of water waves and
| |
| − | fixed periodic arrays, Twersky [[Twersky52]] was able to solve the problem
| |
| − | of a periodic array of vertical circular cylinders. The uniformity of
| |
| − | the configuration in the depth coordinate implies that the resulting
| |
| − | equations also describe two-dimensional acoustic wave scattering, in
| |
| − | this case by circular cylinders. The problem of Twersky [[Twersky52]], who
| |
| − | used Schl\"{o}milch series to sum slowly-convergent series involving
| |
| − | Hankel functions, was re-considered by Linton and Evans [[Linton_evans93]] who
| |
| − | used a so-called multipole method. For periodic arrays of rectangular
| |
| − | cylinders extending uniformly through the depth, Fernyhough and Evans [[Fernyhough95]],
| |
| − | used domain decomposition and mode matching to derive an
| |
| − | integral equation formulation to the problem. In order to consider more
| |
| − | general cylinder profiles, boundary integral methods are inevitable and
| |
| − | require the use of a periodic Green function. In its most basic form,
| |
| − | the Green function consists of a series involving Hankel functions
| |
| − | which is slowly convergent and unsuitable for numerical computation.
| |
| − | Hence Linton [[Linton98]] compared a number of different representations of
| |
| − | the periodic Green function, designed to increase the convergence
| |
| − | characteristics. Porter and Evans [[Porter_evans99]] used the work of Linton [[Linton98]]
| |
| − | to compute so-called Rayleigh-Bloch waves (or trapped waves) along a
| |
| − | periodic array of cylinders of arbitrary cross-section. A number of
| |
| − | papers in recent years have concentrated on similar ideas, to those of
| |
| − | Porter and Evans [[Porter_evans99]], a primary motivation being the connection between
| |
| − | large wave responses in large finite arrays of cylinders and the trapped
| |
| − | waves in infinite periodic arrays (see Maniar and Newman [[Maniar_newman97]]).
| |
| − | | |
| − | In contrast to the body of work cited in the previous paragraph, the
| |
| − | present work differs significantly in two respects. First, the problem
| |
| − | involving periodic elastic plates on the water surface implies that, though the periodicity extends only in one direction, the
| |
| − | problem uses fully three-dimensional plates. Secondly, the scattering obstacles are
| |
| − | neither fixed (as above) nor rigid, but are in fact themselves capable
| |
| − | of supporting wave motions. Thus the coupling between the plate motion
| |
| − | and the motion of the water adds an extra degree of complication to
| |
| − | the problem. We consider a special Green function which is build up out
| |
| − | of multiple images as was done in Kashiwagi [[kashiwagi91]] for the case of
| |
| − | wave scattering in a wave tank with rigid walls. Our method may be considered
| |
| − | as generalisation of Kashiwagi in which we allow the waves
| |
| − | to be incident at an angle and the body spacing to be arbitrary.
| |
| − | We also use a different method to sum the infinite series of
| |
| − | Green functions that was used by Kashiwagi.
| |
| − | | |
| − | The outline of the paper is as follows. In section 2 the
| |
| − | non-dimensional small-amplitude equations for time-harmonic motion
| |
| − | are formulated. In section 3, an integral equation is derived
| |
| − | connecting the motion of the plate to that of the water. In doing
| |
| − | so the periodic Green function is defined in terms of the
| |
| − | three-dimensional free-surface Green function. In section 4, a
| |
| − | variational principle for the motion of the plate is defined and
| |
| − | a Rayleigh-Ritz approximation is used to reduce it to a linear
| |
| − | system of equations. The same approximation is used in the integral
| |
| − | equation to provide a second set of linear equations coupling the
| |
| − | motion of the plate to the water. Section 5 concentrates on
| |
| − | improving the otherwise slow convergence of the periodic Green
| |
| − | function and section 6 describes how the far-field reflected and
| |
| − | transmitted waves are calculated from the integral formulation.
| |
| − | Finally, in section 7 a range of numerical results are presented
| |
| − | and discussed and the work is concluded in section 8.
| |
| − | | |
| − | =Problem Formulation: An Infinite Array of Elastic Plates= | |
| | | | |
| | We begin by formulating the problem. | | We begin by formulating the problem. |
| | Cartesian coordinates <math>(x,y,z)</math> are chosen with <math>z</math> vertically upwards | | Cartesian coordinates <math>(x,y,z)</math> are chosen with <math>z</math> vertically upwards |
| | such that <math>z=0</math> coincides with the mean free surface of the water. | | such that <math>z=0</math> coincides with the mean free surface of the water. |
| − | An infinite array of identical thin elastic plates | + | An infinite array of identical bodies |
| − | float on the surface of the water, periodically spaced along
| + | are periodically spaced along |
| | the <math>y</math>-axis with uniform separation <math>l</math>. The problem is to determine | | the <math>y</math>-axis with uniform separation <math>l</math>. The problem is to determine |
| − | the motion of the water and the plates when plane waves are obliquely-incident | + | the motion of the water and the bodies when plane waves are obliquely-incident |
| − | from <math>x=-\infty</math> upon the periodic array of plates. | + | from <math>x=-\infty</math> upon the periodic array of bodies. |
| | | | |
| − | The plates are assumed to be of zero draft and occupy <math>{\bf x} \in | + | The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies |
| − | \Delta_m<math>, </math>-\infty < m < \infty<math> on </math>z=0<math> where </math>{\bf x} = (x,y)</math> | + | that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y+ml) \in \Delta_m</math>, |
| − | represents the Cartesian vector lying in the mean free surface. The
| |
| − | plates are assumed to have arbitrary shape, and periodicity implies
| |
| − | that if <math>{\bf x} \in \Delta_0</math>, then <math>{\bf x}_m = (x,y+ml) \in \Delta_m</math>, | |
| | <math>-\infty < m < \infty</math>. | | <math>-\infty < m < \infty</math>. |
| − | This array of plates is shown in Figure (fig_array).
| |
| − |
| |
| | | | |
| − | ==Elastic plate equations==
| + | We assume that we have the [[Standard Linear Wave Scattering Problem]]. |
| − | | + | The incident wave |
| − | The equation of motion for the plate elevation <math>W(x,y,t)</math>, where <math>t</math>
| |
| − | is time, is given by the thin elastic plate (or Kirchhoff) equation,
| |
| − | <center><math>
| |
| − | D \nabla_h^{4} W + \rho_{i} h \frac{\partial^{2}W}{\partial t^{2}}
| |
| − | = p,
| |
| − | (floemot)
| |
| − | </math></center>
| |
| − | \cite[pp. 79--104]{Timoshenko_woinowsky-krieger59}
| |
| − | where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density, and <math>h</math>
| |
| − | the uniform thickness of the elastic plate. Here, <math>p</math> is the
| |
| − | excess pressure (i.e. excluding atmospheric pressure and the weight
| |
| − | of the plate) exerted by the fluid and <math>\nabla_h^2</math> is the
| |
| − | two-dimensional horizontal Laplacian in the plane <math>z=0</math> (we shall
| |
| − | reserve <math>\nabla^2</math> to mean the three-dimensional Laplacian). In addition,
| |
| − | on the free edges of the plates, <math>{\bf x} \in \partial \Delta_m</math>, boundary conditions
| |
| − | expressing the vanishing of bending moment and shearing stress apply,
| |
| − | which are written as
| |
| − | <center><math> (boundary1)
| |
| − | \left[ \nabla_h^2 - (1-\nu)
| |
| − | \left(\frac{\partial^2}{\partial s^2} + \kappa(s)
| |
| − | \frac{\partial}{\partial n} \right) \right] W = 0,
| |
| − | </math></center>
| |
| − | <center><math> (boundary2)
| |
| − | \left[ \frac{\partial}{\partial n} \nabla_h^2 +(1-\nu)
| |
| − | \frac{\partial}{\partial s}
| |
| − | \left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
| |
| − | -\kappa(s) \frac{\partial}{\partial s} \right) \right] W = 0,
| |
| − | </math></center>
| |
| − | where <math>\nu</math> is Poisson's ratio and
| |
| − | <center><math>
| |
| − | \nabla_h^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}
| |
| − | = \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2}
| |
| − | + \kappa(s) \frac{\partial}{\partial n}.
| |
| − | </math></center>
| |
| − | Here, <math>\kappa(s)</math> is the curvature of the boundary, <math>\partial \Delta_m</math>,
| |
| − | as a function of arclength <math>s</math> along <math>\partial \Delta_m</math>;
| |
| − | <math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
| |
| − | tangential and normal to the boundary <math>\partial \Delta_m</math>.
| |
| − | | |
| − | Equations ((boundary1)) and ((boundary2)) can be
| |
| − | established from Porter and Porter [[Porter_porter04]]
| |
| − | who considered the more complicated case of a plate of varying
| |
| − | thickness and derived the edge conditions from a variational principle
| |
| − | similar to the one used later in this paper. A direct derivation of the
| |
| − | edge conditions from the underlying constitutive equations for a thin
| |
| − | elastic plate can be found elsewhere, for example \cite[Chapter 8]{Veubeke79}.
| |
| − | We remark that in the case where the boundary of the elastic | |
| − | plates is piecewise linear, such that <math>\kappa(s)=0</math>, the equations
| |
| − | above reduce to the simpler form
| |
| − | <center><math>
| |
| − | \left( \frac{\partial^2}{\partial n^2} + \nu \frac{\partial^2}{\partial s^2}
| |
| − | \right) W = 0
| |
| − | </math></center>
| |
| − | and
| |
| − | <center><math>
| |
| − | \left( \frac{\partial^3}{\partial n^3} + (2-\nu)
| |
| − | \frac{\partial^3}{\partial n \partial s^2} \right) W = 0.
| |
| − | </math></center>
| |
| − | | |
| − | ==Equations of motion for the water==
| |
| − | | |
| − | We consider water of infinite depth. Under the usual assumptions of
| |
| − | small-amplitude waves above an incompressible and irrotational fluid,
| |
| − | there exists a velocity potential <math>\Phi(x,y,z,t)</math> satisfying Laplace's
| |
| − | equation together with the appropriate linearised boundary conditions,
| |
| − | namely
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}[c]{rcl}
| |
| − | \nabla^2 \Phi & = & 0,\qquad -\infty<z<0,
| |
| − | \\ \noalign{\vskip4pt}
| |
| − | \Phi, \; |\nabla \Phi| & \to & 0, \qquad z \to -\infty,
| |
| − | \\ \noalign{\vskip4pt}
| |
| − | {\displaystyle \frac{\partial \Phi}{\partial z}} & = &
| |
| − | {\displaystyle \frac{\partial W}{\partial t}},\qquad z=0,
| |
| − | \\ \noalign{\vskip4pt}
| |
| − | {\displaystyle -\rho \frac{\partial\Phi}{\partial t} - \rho gW}
| |
| − | & = & p,\qquad z=0,
| |
| − | \end{matrix}
| |
| − | \right\}
| |
| − | (watermot)
| |
| − | </math></center>
| |
| − | where the pressure, <math>p</math>, has already been introduced in
| |
| − | ((floemot)), <math>\rho</math> is the water density, and <math>g</math> is gravitational
| |
| − | acceleration.
| |
| − | <math>W</math> is the time-dependent displacement of the plate when <math>\mathbf{x}
| |
| − | \in \Delta_m<math>. When </math>\mathbf{x} \not\in \Delta_m</math>,
| |
| − | we take <math>p=0</math> in the above and now | |
| − | <math>W</math> represents the elevation of the water surface.
| |
| − | In addition to ((watermot)), radiation conditions at infinity need
| |
| − | to be applied and these will formally be introduced later.
| |
| − | | |
| − | ==Non-dimensionalising the variables==
| |
| − | | |
| − | We non-dimensionalise the spatial variables with respect to a length parameter
| |
| − | <math>L</math> (for example, <math>L</math> may be derived from the area of the plate or <math>L</math> may be
| |
| − | the characteristic length <math>\left( D/\rho g \right)^{1/4}</math>) and the time
| |
| − | variables with respect to <math>\sqrt{L/g.}</math>
| |
| − | The dimensionless variables are therefore given by
| |
| − | <center><math>
| |
| − | (\bar{x},\bar{y},\bar{z},\bar
| |
| − | {W}) = \frac{1}{L}(x,y,z,W),~
| |
| − | \bar{t} = t\sqrt{\frac{g}{L}},~
| |
| − | \bar{p} = \frac{p}{\rho g L}
| |
| − | \quad \mbox{and} \quad \bar{\Phi}
| |
| − | = \frac{\Phi}{L\sqrt{L\,g}}.
| |
| − | </math></center>
| |
| − | Using the dimensionless variables equation ((floemot)) combined
| |
| − | with the last equation of ((watermot)) becomes
| |
| − | <center><math>
| |
| − | \beta \bar{\nabla}_h^{4} \bar{W} + \gamma
| |
| − | \frac{\partial^{2}\bar{W}}{\partial\bar{t}^{2}} = \bar{p} =
| |
| − | -\left. \frac{\partial\bar{\Phi}}{\partial\bar{t}} \right|_{z=0}
| |
| − | -\bar{W},
| |
| − | (dimless_floepressmot_1)
| |
| − | </math></center>
| |
| − | where the dimensionless parameters associated with the motion of the
| |
| − | plate, <math>\beta</math> and <math>\gamma</math>, representing the `stiffness' and
| |
| − | `mass loading' of the plate respectively, are given by
| |
| − | <center><math>
| |
| − | \beta = \frac{D}{\rho g L^{4}} \quad \mbox{and} \quad
| |
| − | \gamma = \frac{\rho_{i}h}{\rho L}.
| |
| − | </math></center>
| |
| − | This notation is based on Tayler \cite[pp. 122--128]{Tayler86}.
| |
| − | | |
| − | Hereafter, we will work with dimensionless variables only but
| |
| − | omit the overbar from all variables for reasons of clarity. Note
| |
| − | that from now on we will use <math>l</math> to mean the non-dimensional
| |
| − | floe separation <math>\bar{l} = l/L</math>.
| |
| − | | |
| − | ==The single frequency equations==
| |
| − | | |
| − | We will consider the solution for a single frequency and we can therefore
| |
| − | represent the displacement and the potential as the real parts of complex
| |
| − | functions in which the time dependence is <math>\exp^{-i\omega t}</math> where
| |
| − | <math>\omega</math> is the dimensionless radian frequency, i.e.
| |
| − | <center><math>\begin{matrix}
| |
| − | W( x,y,t) & = &
| |
| − | \mbox{Re} \left[ \left(\frac{i}{\omega}\right) w(x,y) \exp^{-i\omega t} \right],\\
| |
| − | \Phi(x,y,z,t)
| |
| − | & = & \mbox{Re} \left[ \phi(x,y,z) \exp^{-i\omega t}\right],
| |
| − | \end{matrix}</math></center>
| |
| − | where we have introduced an additional scaling for <math>W</math> to simplify
| |
| − | the equations which follow.
| |
| − | | |
| − | Therefore equation ((dimless_floepressmot_1)) becomes
| |
| − | <center><math>
| |
| − | \beta \nabla_h^{4} w(x,y) + (1 - \omega^{2} \gamma) w(x,y) =
| |
| − | \omega^2 \phi(x,y,0).
| |
| − | (maineqn)
| |
| − | </math></center>
| |
| − | Combining the non-dimensionalisation with the time assumption
| |
| − | we also have from ((watermot))
| |
| − | <center><math>
| |
| − | \frac{\partial\phi}{\partial z} = w, \qquad z=0
| |
| − | (kineqn)
| |
| − | </math></center>
| |
| − | with
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}[c]{rcl}
| |
| − | \nabla^{2}\phi & = & 0,\qquad -\infty < z < 0,\\
| |
| − | \noalign{\vskip4pt}
| |
| − | \phi,\; |\nabla \phi| & \to & 0, \qquad z \to -\infty,\\
| |
| − | \noalign{\vskip4pt}
| |
| − | \end{matrix}
| |
| − | \right\}
| |
| − | (watermot2)
| |
| − | </math></center>
| |
| − | For <math>\mathbf{x} \not\in \Delta_m</math>, ((maineqn)) still holds,
| |
| − | but with <math>\beta = \gamma = 0</math> and combining with ((kineqn))
| |
| − | gives the usual free-surface condition
| |
| − | <center><math>
| |
| − | \frac{\partial\phi}{\partial z} - k \phi = 0, \qquad
| |
| − | \mbox{on <math>z=0</math>}, \qquad \mbox{where <math>k = \omega^2</math>}
| |
| − | </math></center>
| |
| − | and <math>k</math> is the dimensionless wavenumber (i.e. the dimensionless
| |
| − | wavelength is <math>\lambda=2\pi/k</math>).
| |
| − | The velocity potential also satisfies the Sommerfeld radiation | |
| − | condition as <math>|\mathbf{x}| \rightarrow\infty</math>,
| |
| − | <center><math>
| |
| − | \lim_{|\mathbf{x}| \rightarrow \infty}
| |
| − | \sqrt{|\mathbf{x}| }\left( \frac{\partial}{\partial |\mathbf{x}| }
| |
| − | - i k \right) \left( \phi-\phi^{\rm in} \right) = 0,
| |
| − | (sommerfeld)
| |
| − | </math></center>
| |
| − | (see Wehausen and Laitone [[Weh_Lait]]). In equation ((sommerfeld)),
| |
| − | <math>\phi^{\rm in}</math> is the incident wave
| |
| | potential given by | | potential given by |
| | <center><math> | | <center><math> |
| | \phi^{{\rm in}} = \frac{A}{k} | | \phi^{{\rm in}} = \frac{A}{k} |
| − | \exp^{ik (x\cos\theta+y\sin\theta)}\,\exp^{kz}, | + | e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz}, |
| − | (phi_inc)
| |
| | </math></center> | | </math></center> |
| | where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of | | where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of |
| − | propagation of the wave. | + | propagation of the wave (with <math>\theta = 0</math> corresponding to normal incidence. |
| | | | |
| | =Transformation to an Integral Equation= | | =Transformation to an Integral Equation= |
| | | | |
| − | We now Floquet's theorem (Scott [[Scott98]]) (also called {\it the assumption of | + | We now Floquet's theorem ([[Scott 1998]]) (also called ''the assumption of periodicity'' |
| − | periodicity} in the water wave context) which states the
| + | in the water wave context) which states the |
| | displacement from adjacent plates differ only by a phase factor. | | displacement from adjacent plates differ only by a phase factor. |
| − | If the potential under the {\em central} plate <math>\Delta_{0}</math> is given by <math>\phi( \mathbf{x}_{0},0)</math>, | + | If the potential under the ''central'' plate <math>\Delta_{0}</math> is given by <math>\phi( \mathbf{x}_{0},0)</math>, |
| | <math>\mathbf{x}_{0}\in\Delta_{0}</math>, then by Floquet's theorem the potential | | <math>\mathbf{x}_{0}\in\Delta_{0}</math>, then by Floquet's theorem the potential |
| | satisfies | | satisfies |
| | <center><math> | | <center><math> |
| − | \phi(\mathbf{x}_{m},0) = \phi(\mathbf{x}_{0},0) \exp^{im\sigma l}, | + | \phi(\mathbf{x}_{m},0) = \phi(\mathbf{x}_{0},0) e^{im\sigma l}, |
| − | (floq1)
| |
| | </math></center> | | </math></center> |
| | and the displacement of the plate <math>\Delta_{m}</math> satisfies | | and the displacement of the plate <math>\Delta_{m}</math> satisfies |
| | <center><math> | | <center><math> |
| − | w(\mathbf{x}_{m}) = w(\mathbf{x}_{0}) \exp^{im\sigma l}, | + | w(\mathbf{x}_{m}) = w(\mathbf{x}_{0}) e^{im\sigma l}, |
| − | (floq2)
| |
| | </math></center> | | </math></center> |
| | where <math>\mathbf{x}_{m} \in \Delta_{m}</math>, <math>-\infty < m < \infty</math> and | | where <math>\mathbf{x}_{m} \in \Delta_{m}</math>, <math>-\infty < m < \infty</math> and |
| − | the phase difference is <math>\sigma = k\sin\theta</math> (see, for example, | + | the phase difference is <math>\sigma = k\sin\theta</math> (see, for example, [[Linton 1998]]). |
| − | Linton [[Linton98]]).
| |
| | | | |
| | A standard approach to the solution of the equations of motion for | | A standard approach to the solution of the equations of motion for |
| − | the water ((maineqn)), ((kineqn)), ((watermot2)) | + | the water is the [[Green Function Solution Method]] in which |
| − | is to transform these equations into a boundary integral
| + | we transform the equations into a boundary integral |
| − | equation using the free-surface Green function for infinite depth (see | + | equation using the [[Free-Surface Green Function]]. In doing so we obtain |
| − | [[Weh_Lait,Kim65]]). In doing so we obtain
| |
| | <center><math> | | <center><math> |
| − | \phi(\mathbf{x},0) = \phi^{\rm in} (\mathbf{x},0) | + | \phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0) |
| − | +\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}} G(\mathbf{x},0 | + | +\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}} |
| − | ;\mbox{\boldmath<math>\xi</math>}) \left[ k\phi(\mbox{\boldmath<math>\xi</math>},0)
| + | \left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi |
| − | - w (\mbox{\boldmath<math>\xi</math>}) \right] d\mbox{\boldmath<math>\xi</math>}
| |
| − | (phi_w_m)
| |
| − | </math></center>
| |
| − | where <math>\mbox{\boldmath</math>\xi<math>} = (\xi,\eta)</math> and
| |
| − | <math>G(\mathbf{x},z;\mbox{\boldmath</math>\xi<math>})</math> is
| |
| − | the free-surface Green function satisfying
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}[c]{rcl} | |
| − | \nabla^2 G & = & 0,\qquad -\infty < z < 0,
| |
| − | \\ \noalign{\vskip4pt}
| |
| − | \displaystyle{\frac{\partial G}{\partial z}} - k G
| |
| − | & = & -\delta(\mathbf{x}-\mbox{\boldmath<math>\xi</math>}),\qquad z=0,
| |
| − | \\ \noalign{\vskip4pt}
| |
| − | G, \; |\nabla G| & \to & 0,\qquad z \to -\infty.
| |
| − | \end{matrix}
| |
| − | \right\} | |
| | </math></center> | | </math></center> |
| − | which, on <math>z=0</math>, is given by
| + | <math>G(\mathbf{x},\xi)</math> is |
| − | <center><math>\begin{matrix}
| + | the [[Free-Surface Green Function]] This |
| − | G(\mathbf{x},0;\mbox{\boldmath<math>\xi</math>}) & = &
| + | can be written alternatively as |
| − | -\frac{1}{4\pi} \Bigg( \frac {2}{\left|
| + | <center><math> |
| − | \mathbf{x}-\mbox{\boldmath<math>\xi</math>}\right|}
| + | \phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x}) |
| − | -\pi k \Big[ \mathbf{H}_{0}\left( k
| + | +\int_{\Delta_{0}} |
| − | \left| \mathbf{x}-\mbox{\boldmath<math>\xi</math>}\right| \right)
| + | \sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l} \phi(\xi) |
| − | \\
| + | - G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi |
| − | & & \quad \quad \quad
| |
| − | +Y_{0}\left( k\left| \mathbf{x}-\mbox{\boldmath<math>\xi</math>} \right|
| |
| − | \right) -2i\pi J_{0}\left( k
| |
| − | \left| \mathbf{x}-\mbox{\boldmath<math>\xi</math>} \right| \right) \Big] \Bigg),
| |
| − | (green)
| |
| − | \end{matrix}</math></center>
| |
| − | In the above <math>\mathbf{H}_{0}</math> is the Struve function of order zero,
| |
| − | <math>J_{0}</math> is the Bessel function of the first kind of order zero and <math>Y_{0}</math>
| |
| − | is the Bessel function of the second kind of order zero
| |
| − | (Abramowitz and Stegun \cite[Chapters 9 and 12]{abr_ste}).
| |
| − | | |
| − | Using ((floq1)) and ((floq2)) in ((phi_w_m))
| |
| − | we obtain
| |
| − | <center><math>\begin{matrix} | |
| − | (bem_eq_1)
| |
| − | \phi (\mathbf{x},0) & = & \phi^{\rm in}(\mathbf{x},0) | |
| − | \\ | |
| − | & & \!
| |
| − | +\!\! \sum_{m=-\infty}^{\infty} \int_{\Delta_{0}} \!\!
| |
| − | G (\mathbf{x},0
| |
| − | ;\mbox{\boldmath<math>\xi</math>}+(0,ml)) \left[ k\phi(\mbox{\boldmath<math>\xi</math>},0)
| |
| − | - w(\mbox{\boldmath<math>\xi</math>}) \right] \exp^{im\sigma l}
| |
| − | \,d\mbox{\boldmath<math>\xi</math>}
| |
| − | \end{matrix}</math></center>
| |
| − | which can be written alternatively as
| |
| − | <center><math> (bem_eq_2)
| |
| − | \phi(\mathbf{x},0) = \phi^{\rm in}(\mathbf{x},0)
| |
| − | +\int_{\Delta_{0}} G_{\mathbf{P}} (\mathbf{x}
| |
| − | ;\mbox{\boldmath<math>\xi</math>}) \left[ k\phi(\mbox{\boldmath<math>\xi</math>},0)
| |
| − | - w(\mbox{\boldmath<math>\xi</math>}) \right] \,d\mbox{\boldmath<math>\xi</math>},
| |
| | </math></center> | | </math></center> |
| | where the kernel <math>G_{\mathbf{P}}</math> (referred to as the | | where the kernel <math>G_{\mathbf{P}}</math> (referred to as the |
| − | {\it periodic Green function}) is given by
| + | ''periodic Green function'') is given by |
| − | <center><math>
| |
| − | G_{\mathbf{P}} (\mathbf{x};\mbox{\boldmath<math>\xi</math>})
| |
| − | = \sum_{m=-\infty}^{\infty} G \left(\mathbf{x},0;\mbox{\boldmath<math>\xi</math>}+(0,ml)
| |
| − | \right) \exp^{im\sigma l}.
| |
| − | (G_periodic)
| |
| − | </math></center>
| |
| − | | |
| − | =Solution of the Integral Equation=
| |
| − | | |
| − | The integral equation ((bem_eq_2)) is identical to the integral
| |
| − | equation for a single plate (see Meylan [[JGR02]]) except for the
| |
| − | modification to the Green function. Furthermore, the periodic Green
| |
| − | function <math>G_{\mathbf{P}}</math> has the same singularity at <math>\mathbf{x}=
| |
| − | \mbox{\boldmath<math>\xi</math>}</math> as the standard free-surface Green function
| |
| − | <math>G</math>. We solve equation ((bem_eq_2)) using a higher-order method,
| |
| − | which is explained in detail in Wang and Meylan [[Wang04]]. We will
| |
| − | briefly outline the solution method here.
| |
| − | | |
| − | We expand the plate potential and displacement as
| |
| − | <center><math>
| |
| − | \left.
| |
| − | \begin{matrix}{c}
| |
| − | {\displaystyle
| |
| − | w(\mathbf{x})\approx \sum_{i=1}^{3q}w_{i}\chi _{i}(\mathbf{x}) =
| |
| − | \vec{\chi}^{T}(\mathbf{x})\vec{w}
| |
| − | } \\
| |
| − | {\displaystyle
| |
| − | \phi (\mathbf{x},0)\approx \sum_{i=1}^{3q}\phi_{i}\chi_{i}(\mathbf{x})
| |
| − | =\vec{\chi}^{T}(\mathbf{x})\vec{\phi}
| |
| − | }
| |
| − | \end{matrix}
| |
| − | \right\}
| |
| − | (expansion)
| |
| − | </math></center>
| |
| − | where <math>\vec{w}</math> and <math>\vec{\phi}</math>
| |
| − | <center><math>\begin{matrix}
| |
| − | \vec{w}=\left[\begin{matrix}{c}
| |
| − | w\<center><math>1pt]
| |
| − | \mbox{\Large <math>\frac{\partial{w}}{\partial{x}}</math>} \<center><math>6pt]
| |
| − | \mbox{\Large <math>\frac{\partial{w}}{\partial{y}}</math>} \end{matrix}\right],
| |
| − | \quad \vec{\phi}=\left[\begin{matrix}{c}
| |
| − | \phi \<center><math>1pt]
| |
| − | \mbox{\Large <math>\frac{\partial{\phi}}{\partial{x}}</math>} \<center><math>6pt]
| |
| − | \mbox{\Large <math>\frac{\partial{\phi}}{\partial{y}}</math>} \end{matrix}\right],
| |
| − | \end{matrix}</math></center>
| |
| − | are vectors representing the values of the displacement and potential, respectively, and their horizontal derivatives at the <math>q</math> nodes and <math>\vec{\chi}</math> is a vector of the associated basis functions. We can express <math>\vec{\chi}^{T}(\mathbf{x})</math> in terms of the FEM basis functions for non-conforming square element denoted by <math>\mathbf{N}_{d}\left( \mathbf{x} \right)</math>,
| |
| − | ([[Wang04]] and \cite[pp. 229--243)]{Petyt}) as
| |
| − | <center><math>
| |
| − | \vec{\chi}^{T}(\mathbf{x}) =
| |
| − | \left( \sum_{d=1}^{N}\mathbf{N}_{d}\left( \mathbf{x} \right)
| |
| − | \left[ o\right]_{d}\right)
| |
| − | (chi_expansion)
| |
| − | </math></center>
| |
| − | where <math>N</math> is the number of panels and the matrix <math>\left[ o\right]_{d}</math> is
| |
| − | the \emph{assembler matrix
| |
| − | \index{Finite Element Method (FEM)!assembler matrix (see also assembler matrix)}
| |
| − | \index{assembler matrix}}. Both <math>\mathbf{N}_{d}(\mathbf{x})</math> (which is a <math>1 \times 12</math> matrix) and <math>\left[ o\right]_{d}</math> (which is a <math>12\times 3q</math> matrix) are discussed in detail by Wang and Meylan [[Wang04]]. Following Wang and Meylan [[Wang04]] we call the square element {\it panel}.
| |
| − | | |
| − | The equation governing the motion of the plate ((maineqn)) and
| |
| − | the time-harmonic non-dimensionalised versions of the boundary conditions
| |
| − | at the edge of the plate ((boundary1)), ((boundary2)) are
| |
| − | equivalent to the variational principle
| |
| − | [[Porter_porter04,AppliedOR01]] and \cite[pp. 185--187]{Hildebrand65},
| |
| − | <math>\delta L = 0</math>
| |
| − | where
| |
| − | <center><math>\begin{matrix}
| |
| − | L(w) & = &
| |
| − | \frac{1}{2} \int_{\Delta_{0}}
| |
| − | \beta \left[ \left(\nabla_h^2 w\right)^2 - 2(1-\nu)
| |
| − | \left( \frac{\partial^2 w}{\partial x^2} \frac{\partial^2 w}{\partial y^2}
| |
| − | - \left( \frac{\partial^2 w}{\partial x \partial y} \right)^2 \right)
| |
| − | \right]
| |
| − | \\
| |
| − | & & \qquad + \left( 1- k \gamma \right) w^{2}
| |
| − | - 2 k w \phi|_{z=0} \,d\mathbf{x}.
| |
| − | \end{matrix}</math></center>
| |
| − | The three terms in the integrand above represent, respectively, the
| |
| − | strain energy of the plate, the acceleration, and the dynamic pressure
| |
| − | on the plate. Apart from the plate equation itself other natural
| |
| − | conditions of <math>\delta L = 0</math> are the free edge conditions described
| |
| − | by ((boundary1)), and ((boundary2)). Thus, using the variational
| |
| − | principle means that the edge conditions are satisfied indirectly as
| |
| − | part of the approximation.
| |
| − | | |
| − | If we now substitute the approximation
| |
| − | for <math>w</math> and <math>\phi</math> in ((expansion)) into the above and minimise
| |
| − | (i.e. apply <math>\delta L = 0</math>) we obtain
| |
| − | <center><math>
| |
| − | \beta \,\mathbb{K}\,\mathbf{\vec{w}}+(1- k \gamma)
| |
| − | \,\mathbb{M}\,\mathbf{\vec{w}}
| |
| − | = k \,\mathbb{M}\,\mathbf{\vec{\phi}},
| |
| − | (comp_eqn)
| |
| − | </math></center>
| |
| − | where the stiffness matrix <math>\mathbb{K}</math>, is given by
| |
| − | <center><math>\begin{matrix}
| |
| − | \mathbb{K} & = &
| |
| − | \int_{\Delta _{0}}\Bigg[ \frac{\partial ^{2}\vec{\chi}}{\partial x^{2}}
| |
| − | \frac{\partial ^{2}\vec{\chi}^{T}}{\partial x^{2}}
| |
| − | + \frac{\partial ^{2} \vec{\chi}}{\partial y^{2}}
| |
| − | \frac{\partial ^{2}\vec{\chi}^{T}}{\partial y^{2}}
| |
| − | \\
| |
| − | & &
| |
| − | + 2 \left( 1-\nu \right) \frac{\partial ^{2}\vec{\chi}}{\partial x\partial y}
| |
| − | \frac{\partial ^{2}\vec{\chi}^{T}}{\partial x\partial y}
| |
| − | +\nu \frac{ \partial ^{2}\vec{\chi}}{\partial x^{2}}
| |
| − | \frac{\partial ^{2}\vec{\chi}^{T}}{\partial y^{2}}
| |
| − | +\nu \frac{\partial ^{2}\vec{\chi}}{\partial y^{2}}
| |
| − | \frac{\partial^{2}\vec{\chi}^{T}}{\partial x^{2}}\Bigg] d\mathbf{x}
| |
| − | \end{matrix}</math></center>
| |
| − | and the mass matrix <math>\mathbb{M}</math> is given by
| |
| − | <center><math>
| |
| − | \mathbb{M}=\int_{\Delta _{0}}\vec{\chi}(\mathbf{x})\vec{\chi}^{T}(\mathbf{x}
| |
| − | )\,d\mathbf{x}.
| |
| − | </math></center>
| |
| − | The integral equation ((bem_eq_2)) is transformed by substituting the
| |
| − | approximations for <math>w</math> and <math>\phi</math> given by ((expansion)) to obtain
| |
| | <center><math> | | <center><math> |
| − | \vec{\chi}^{T}(\mathbf{x})\vec{\phi} = | + | G^{\mathbf{P}} (\mathbf{x};\xi) |
| − | \vec{\chi}^{T}(\mathbf{x})\vec{\phi}^{\rm in}
| + | = \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}. |
| − | + \int_{\Delta _{0}}G_{\mathbf{P}}(\mathbf{x};\mbox{\boldmath<math>\xi</math>}) | |
| − | \left[ k\vec{\chi}^{T}(\mbox{\boldmath<math>\xi</math>})\vec{\phi}
| |
| − | - \vec{\chi}^{T}(\mbox{\boldmath<math>\xi</math>})\vec{w}\right]
| |
| − | \,d \mbox{\boldmath<math>\xi</math>}
| |
| | </math></center> | | </math></center> |
| − | (where where <math>\vec{\phi}^{\rm in}</math> is the representation of
| |
| − | <math>{\phi }^{\rm in}</math> in the basis functions <math>\chi</math>) and then multiplying
| |
| − | this equation by <math>\vec{\chi}(\mathbf{x})</math> and integrating over <math>\Delta_{0}</math>.
| |
| − | This gives us
| |
| − | <center><math>
| |
| − | \mathbb{M}\,\mathbf{\vec{\phi}} = \mathbb{M} \vec{\phi}^{\rm in}
| |
| − | + k \,\mathbb{G}_{\mathbf{P}}\,\mathbf{\vec{\phi}}
| |
| − | - \,\mathbb{G}_{\mathbf{P}}\,\mathbf{\vec{w}},
| |
| − | (phi_plate)
| |
| − | </math></center>
| |
| − | where <math>\mathbb{G}_{\mathbf{P}}</math>, is given by
| |
| − | <center><math>
| |
| − | \mathbb{G}_{\mathbf{P}} =
| |
| − | \int_{\Delta _{0}}\int_{\Delta _{0}}\vec{\chi}(\mathbf{x})
| |
| − | G_{\mathbf{P}}(\mathbf{x},\mbox{\boldmath<math>\xi</math>})
| |
| − | \vec{\chi}^{T}(\mathbf{x})\,d\mathbf{x} d\mbox{\boldmath<math>\xi</math>}.
| |
| − | </math></center>
| |
| − | We can use equation ((chi_expansion)) to express the matrices <math>\mathbb{K}
| |
| − | <math>, </math>\mathbb{M}<math>, and </math>\mathbb{G}_{\mathbf{P}}</math> as
| |
| − | <center><math>\begin{matrix}
| |
| − | \mathbb{K} &=&\sum_{d=1}^{N}\left[ o\right] _{d}^{T}[\int_{\Delta
| |
| − | _{d}}\left( \frac{\partial ^{2}\mathbf{N}_{d}^{T}}{\partial x^{2}}\,\frac{
| |
| − | \partial ^{2}\mathbf{N}_{d}}{\partial x^{2}}\right) +\nu \,\left( \frac{
| |
| − | \partial ^{2}\mathbf{N}_{d}^{T}}{\partial x^{2}}\,\frac{\partial ^{2}\mathbf{
| |
| − | N}_{d}}{\partial y^{2}}+\frac{\partial ^{2}\mathbf{N}_{d}^{T}}{\partial y^{2}
| |
| − | }\,\frac{\partial ^{2}\mathbf{N}_{d}}{\partial x^{2}}\right) \\
| |
| − | &&+2\left( 1-\nu \right) \,\left( \frac{\partial ^{2}\mathbf{N}_{d}^{T}}{
| |
| − | \partial x\partial y}\,\frac{\partial ^{2}\mathbf{N}_{d}}{\partial x\partial
| |
| − | y}\right) +\left( \frac{\partial ^{2}\mathbf{N}_{d}^{T}}{\partial y^{2}}\,
| |
| − | \frac{\partial ^{2}\mathbf{N}_{d}}{\partial y^{2}}\right) d\mathbf{x]}\left[
| |
| − | o\right] _{d},
| |
| − | \end{matrix}</math></center>
| |
| − | <center><math>
| |
| − | \mathbb{M}=\sum_{d=1}^{N}\left[ o\right] _{d}^{T}\left[ \int_{\Delta _{d}}
| |
| − | \mathbf{N}_{d}^{T}\left( \mathbf{x}\right) \mathbf{\,N}_{d}\left( \mathbf{x}
| |
| − | \right) \,d\mathbf{x}\right] \left[ o\right] _{d},
| |
| − | </math></center>
| |
| − | and
| |
| − | <center><math>
| |
| − | \mathbb{G}_{\mathbf{P}}=\sum_{e=1}^{p}\sum_{d=1}^{p}\left[ o\right] _{e}^{T}
| |
| − | \left[ \int_{\Delta _{e}}\int_{\Delta _{d}}\mathbf{N}_{e}^{T}\left( \mathbf{x
| |
| − | }\right) G_{\mathbf{P}}(\mathbf{x},\mathbf{\xi })\mathbf{N}_{d}\left(
| |
| − | \mathbf{\xi }\right) \,d\mathbf{x}d\mathbf{\xi }\right] \left[ o\right] _{d}.
| |
| − | </math></center>
| |
| − | These are the equations which are used to find to calculate
| |
| − | the matrices <math>\mathbb{K}</math>, <math>\mathbb{M}</math>, and <math>\mathbb{G}</math>. The solution
| |
| − | for <math>\vec{w}</math> and <math>\vec{\phi}</math> is then found by solving
| |
| − | equation ((comp_eqn)) simultaneously with ((phi_plate)).
| |
| | | | |
| | =Accelerating the Convergence of the Periodic Green Function= | | =Accelerating the Convergence of the Periodic Green Function= |
| | | | |
| | The spatial representation of the periodic Green function given by | | The spatial representation of the periodic Green function given by |
| − | equation ((G_periodic)) is slowly convergent | + | equation is slowly convergent |
| | and in the far field the terms decay in | | and in the far field the terms decay in |
| | magnitude like <math>O(n^{-1/2})</math>. In this section we | | magnitude like <math>O(n^{-1/2})</math>. In this section we |
| | show how to accelerate the convergence. We begin with the asymptotic | | show how to accelerate the convergence. We begin with the asymptotic |
| − | approximation of the three-dimensional Green function ((green)) | + | approximation of the Three-dimensional [[Free-Surface Green Function]] |
| | far from the source point, | | far from the source point, |
| | <center><math> | | <center><math> |
| − | G(\mathbf{x},0;\mbox{\boldmath<math>\xi</math>}) \sim -\frac{ik}{2} | + | G(\mathbf{x},\xi) \sim -\frac{ik}{2} |
| − | \,H_{0}( k |\mathbf{x}-\mbox{\boldmath<math>\xi</math>}|), | + | \,H_{0}( k |\mathbf{x}-\xi|), |
| − | \qquad \mbox{as <math>|\mathbf{x}-\mbox{\boldmath</math>\xi<math>}| \to \infty</math>}, | + | |\mathbf{x}-\xi| \to \infty |
| − | (asyG)
| |
| | </math></center> | | </math></center> |
| − | [[Weh_Lait]] where <math>H_0 \equiv H_{0}^{(1)}</math> is the Hankel function | + | [[Wehausen and Laitone 1960]] where <math>H_0 \equiv H_{0}^{(1)}</math> is the Hankel function |
| − | of the first kind of order zero [[abr_ste]]. In Linton [[Linton98]] | + | of the first kind of order zero [[Abramowitz and Stegun 1964]]. In Linton [[Linton 1998]] |
| | various methods were described in which the convergence of the periodic | | various methods were described in which the convergence of the periodic |
| | Green functions was improved. One such method, which suits the particular | | Green functions was improved. One such method, which suits the particular |
| | problem being considered here, involves writing the periodic | | problem being considered here, involves writing the periodic |
| | Green function as | | Green function as |
| − | <center><math>\begin{matrix} | + | <center><math> |
| − | G_{\mathbf{P}} (\mathbf{x};\mbox{\boldmath<math>\xi</math>}) | + | G_{\mathbf{P}} (\mathbf{x};\xi) |
| − | &\! = &\!\!\!\! \displaystyle{\sum_{m=-\infty}^{\infty}} \! | + | = \sum_{m=-\infty}^{\infty} |
| − | \left[ G\left(\mathbf{x};\mbox{\boldmath<math>\xi</math>}+(0,ml)\right)
| + | \left[ |
| − | + \frac{ik}{2} H_{0} \Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big)
| + | G\left(\mathbf{x};\xi)+(0,ml)\right) |
| − | \right] \exp^{im\sigma l}
| + | + \frac{ik}{2} H_{0} |
| − | \\
| + | \Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) e^{im\sigma l} |
| − | &\! &\! -\displaystyle{\sum_{m=-\infty}^{\infty}} | + | \right] |
| | + | -\sum_{m=-\infty}^{\infty} |
| | \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big) | | \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big) |
| − | \exp^{im\sigma l} | + | e^{im\sigma l} |
| − | (near_accelerated)
| + | </math></center> |
| − | \end{matrix}</math></center>
| |
| | where <math>c</math> is a numerical smoothing parameter, introduced to avoid the | | where <math>c</math> is a numerical smoothing parameter, introduced to avoid the |
| − | singularity at <math>\mathbf{x} = \mbox{\boldmath</math>\xi<math>}</math> in the Hankel | + | singularity at <math>\mathbf{x} = \xi</math> in the Hankel |
| | function and | | function and |
| | <center><math> | | <center><math> |
| − | X = x-\xi,\quad \mbox{and} \quad | + | X = x-\xi,\quad \mathrm{and} \quad |
| | Y_{m} = (y-\eta)-ml. | | Y_{m} = (y-\eta)-ml. |
| | </math></center> | | </math></center> |
| − | Furthermore we use the fact that second slowly convergent sum in ((near_accelerated)) can be transformed to | + | Furthermore we use the fact that second slowly convergent sum can be transformed to |
| − | <center><math> (help_accelerated) | + | <center><math> |
| − | -\frac{i}{l} \sum_{m=-\infty}^{\infty} | + | -\sum_{m=-\infty}^{\infty} |
| − | \frac{\exp^{ik \mu_{m} |X+c| }\,\exp^{i \sigma_{m} Y_0}}{\mu_{m}} | + | \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big) |
| | + | e^{im\sigma l} |
| | + | -\frac{i}{l} \sum_{m=-\infty}^{\infty} |
| | + | \frac{e^{ik \mu_{m} |X+c| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}} |
| | </math></center> | | </math></center> |
| − | [[Linton98,Jorgenson90,Singh90]] where | + | [[Linton 1998]] where |
| | <math>\sigma_{m} = \sigma + 2 m \pi/l</math> and | | <math>\sigma_{m} = \sigma + 2 m \pi/l</math> and |
| | <center><math> | | <center><math> |
| Line 721: |
Line 127: |
| | where the positive real or positive imaginary part of | | where the positive real or positive imaginary part of |
| | the square root is taken. | | the square root is taken. |
| − | Combining equations ((near_accelerated)) and ((help_accelerated)) | + | Combining these equations |
| | we obtain the accelerated version of the periodic Green function | | we obtain the accelerated version of the periodic Green function |
| − | <center><math>\begin{matrix} | + | <center><math> |
| − | G_{\mathbf{P}} (\mathbf{x};\mbox{\boldmath<math>\xi</math>}) | + | G_{\mathbf{P}} (\mathbf{x};\xi) |
| − | &\! = &\!\!\!\! \displaystyle{\sum_{m=-\infty}^{\infty}} \! | + | = \sum_{m=-\infty}^{\infty} |
| − | \left[ G\left(\mathbf{x};\mbox{\boldmath<math>\xi</math>}+(0,ml)\right) | + | \left[ G\left(\mathbf{x};\xi+(0,ml)\right) |
| | + \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) | | + \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) |
| − | \right] \exp^{im\sigma l} | + | \right] e^{im\sigma l} |
| − | \\
| + | -\frac{i}{l}\sum_{m=-\infty}^{\infty} |
| − | &\! &\! -\frac{i}{l}\displaystyle{\sum_{m=-\infty}^{\infty}
| + | \frac{e^{ik \mu_{m}|X+cl| }e^{i\sigma_{m}Y_0}}{\mu_{m}}. |
| − | \frac{\exp^{ik \mu_{m}|X+cl| }\exp^{i\sigma_{m}Y_0}}{\mu_{m}}}. | + | </math></center> |
| − | (Gp_fast)
| + | The convergence of the two sums depends on the value |
| − | \end{matrix}</math></center>
| |
| − | The convergence of the two sums in ((Gp_fast)) depends on the value | |
| | of <math>c</math>. For small <math>c</math> the first sum converges rapidly while the second converges | | of <math>c</math>. For small <math>c</math> the first sum converges rapidly while the second converges |
| | slowly. For large <math>c</math> the second sum converges rapidly while the first converges | | slowly. For large <math>c</math> the second sum converges rapidly while the first converges |
| Line 740: |
Line 144: |
| | The smoothing parameter <math>c</math> must be carefully chosen to balance these two | | The smoothing parameter <math>c</math> must be carefully chosen to balance these two |
| | effects. Of course, the convergence also depends strongly on how close together the | | effects. Of course, the convergence also depends strongly on how close together the |
| − | points <math>\mathbf{x}</math> and <math>\mbox{\boldmath</math>\xi<math>}</math> are. | + | points <math>\mathbf{x}</math> and <math>\xi</math> are. |
| | | | |
| | Note that some special combinations of wavelength <math>\lambda</math> and angle | | Note that some special combinations of wavelength <math>\lambda</math> and angle |
| | of incidence <math>\theta</math> cause the periodic Green function to diverge | | of incidence <math>\theta</math> cause the periodic Green function to diverge |
| − | ( [[Scott98]],[[Jorgenson90]]). This singularity is closely | + | ( [[Scott 1998]]). This singularity is closely |
| | related to the diffracted waves and will be explained shortly. | | related to the diffracted waves and will be explained shortly. |
| | | | |
| Line 750: |
Line 154: |
| | | | |
| | We begin with the accelerated periodic Green function, equation | | We begin with the accelerated periodic Green function, equation |
| − | ((Gp_fast)) setting <math>c=0</math> and considering the case when <math>X</math> is large
| + | setting <math>c=0</math> and considering the case when <math>X</math> is large |
| | (positive or negative). We also note that for <math>m</math> sufficiently small | | (positive or negative). We also note that for <math>m</math> sufficiently small |
| | or large <math>i\mu_m</math> will be negative and the corresponding terms will | | or large <math>i\mu_m</math> will be negative and the corresponding terms will |
| | decay. Therefore | | decay. Therefore |
| − | <center><math> (G_p_accelerate) | + | <center><math> |
| − | G_{\mathbf{P}} (\mathbf{x};\mbox{\boldmath<math>\xi</math>}) | + | G_{\mathbf{P}} (\mathbf{x};\xi) |
| − | \sim - \frac{i}{l} \sum_{m=-M}^{N} \frac{\exp^{ik\mu_{m}|X|}\, \exp^{i\sigma_{m}Y_0}} | + | \sim - \frac{i}{l} \sum_{m=-M}^{N} \frac{e^{ik\mu_{m}|X|}\, e^{i\sigma_{m}Y_0}} |
| − | {\mu_{m}}, \qquad \mbox{as <math>X \to \pm \infty</math>} | + | {\mu_{m}}, X \to \pm \infty |
| − | (new_G_p_trunc)
| |
| | </math></center> | | </math></center> |
| | where the integers <math>M</math> and <math>N</math> satisfy the following | | where the integers <math>M</math> and <math>N</math> satisfy the following |
| Line 765: |
Line 168: |
| | \left. | | \left. |
| | \begin{matrix} | | \begin{matrix} |
| − | [c]{c}
| |
| | \sigma_{-M-1}<-k<\sigma_{-M},\\ | | \sigma_{-M-1}<-k<\sigma_{-M},\\ |
| | \sigma_{N}<k<\sigma_{N+1}. | | \sigma_{N}<k<\sigma_{N+1}. |
| | \end{matrix} | | \end{matrix} |
| − | \right\} (cut_off) | + | \right\} |
| | </math></center> | | </math></center> |
| − | Equations ((M_N)) can be written as
| + | These equations can be written as |
| | <center><math> | | <center><math> |
| | \frac{l}{2\pi}\left(\sigma+k-2\pi \right) < M < \frac{l}{2\pi}\left( | | \frac{l}{2\pi}\left(\sigma+k-2\pi \right) < M < \frac{l}{2\pi}\left( |
| | \sigma+k \right), | | \sigma+k \right), |
| − | (diffracted_M)
| |
| | </math></center> | | </math></center> |
| | and | | and |
| Line 781: |
Line 182: |
| | \frac{l}{2\pi}\left( k - \sigma \right) > N > \frac{l}{2\pi} | | \frac{l}{2\pi}\left( k - \sigma \right) > N > \frac{l}{2\pi} |
| | \left( k-\sigma - 2\pi \right) | | \left( k-\sigma - 2\pi \right) |
| − | (diffracted_N)
| |
| | </math></center> | | </math></center> |
| − | [[Linton98]]. | + | [[Linton 1998]]. |
| | It is obvious that <math>G_{\mathbf{P}}</math> will diverge if <math>\sigma_m = \pm k</math>; | | It is obvious that <math>G_{\mathbf{P}}</math> will diverge if <math>\sigma_m = \pm k</math>; |
| | these values correspond to cut-off frequencies which are an expected | | these values correspond to cut-off frequencies which are an expected |
| Line 792: |
Line 192: |
| | The diffracted waves are the plane waves which are observed as <math>x \to \pm | | The diffracted waves are the plane waves which are observed as <math>x \to \pm |
| | \infty</math>. Their amplitude and form are obtained by substituting the limit | | \infty</math>. Their amplitude and form are obtained by substituting the limit |
| − | of the periodic Green function ((new_G_p_trunc)) as <math>x\to\pm\infty</math> | + | of the periodic Green function as <math>x\to\pm\infty</math> |
| − | into the boundary integral equation for the potential ((bem_eq_2)). | + | into the boundary integral equation for the potential. |
| | This gives us | | This gives us |
| | <center><math> | | <center><math> |
| | \lim_{x\to\pm\infty} | | \lim_{x\to\pm\infty} |
| | \phi^{s} ( \mathbf{x},0 ) = - \frac{i}{l} \sum_{m=-M}^{N} \int_{\Delta_0} | | \phi^{s} ( \mathbf{x},0 ) = - \frac{i}{l} \sum_{m=-M}^{N} \int_{\Delta_0} |
| − | \frac{\exp^{ik\mu_{m} |X| } \exp^{i\sigma_{m}Y_0}}{\mu_{m}} | + | \frac{e^{ik\mu_{m} |X| } e^{i\sigma_{m}Y_0}}{\mu_{m}} |
| − | \left[ k\phi(\mbox{\boldmath<math>\xi</math>},0) | + | \left[ k\phi(\xi,0) |
| − | - w(\mbox{\boldmath<math>\xi</math>}) \right] | + | - w(\xi) \right] |
| − | d\mbox{\boldmath<math>\xi</math>}, | + | d\xi, |
| − | (phi_s)
| |
| | </math></center> | | </math></center> |
| | where <math>\phi^{s} = \phi-\phi^{\rm in}</math> is the scattered wave which | | where <math>\phi^{s} = \phi-\phi^{\rm in}</math> is the scattered wave which |
| Line 808: |
Line 207: |
| | <center><math> | | <center><math> |
| | \lim_{x\to-\infty}\phi^{s} | | \lim_{x\to-\infty}\phi^{s} |
| − | (\mathbf{x},0) = A_{m}^{-}\,\exp^{ik\mu_{m}x}\exp^{i\sigma_{m}y}, | + | (\mathbf{x},0) = A_{m}^{-}\,e^{ik\mu_{m}x}e^{i\sigma_{m}y}, |
| − | (phi_m_min)
| |
| | </math></center> | | </math></center> |
| − | where the amplitudes <math>A_{m}^{-}</math> are identified from ((phi_s)) | + | where the amplitudes <math>A_{m}^{-}</math> are |
| − | as
| |
| | <center><math> | | <center><math> |
| | A_{m}^{-} = -\frac{i}{\mu_{m}l} \int_{\Delta_0} | | A_{m}^{-} = -\frac{i}{\mu_{m}l} \int_{\Delta_0} |
| − | \exp^{ik\mu_{m}\xi } \exp^{-i\sigma_{m}\eta} | + | e^{ik\mu_{m}\xi } e^{-i\sigma_{m}\eta} |
| − | \left[ k\phi\left( \mbox{\boldmath<math>\xi</math>}\right) | + | \left[ k\phi\left( \xi\right) |
| − | - w (\mbox{\boldmath<math>\xi</math>}) \right] | + | - w (\xi) \right] |
| − | d\mbox{\boldmath<math>\xi</math>}. | + | d\xi. |
| − | (Ad_m)
| |
| | </math></center> | | </math></center> |
| | Likewise as <math>x \to \infty</math> the scattered wave is given by | | Likewise as <math>x \to \infty</math> the scattered wave is given by |
| | <center><math> | | <center><math> |
| | \lim_{x\to\infty}\phi^{s} (\mathbf{x},0) = | | \lim_{x\to\infty}\phi^{s} (\mathbf{x},0) = |
| − | A_{m}^{+} \exp^{-ik\mu_{m}x} \exp^{i\sigma_{m}y}, | + | A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y}, |
| − | (phi_m_plus)
| |
| | </math></center> | | </math></center> |
| | where <math>A_{m}^{+}</math> are | | where <math>A_{m}^{+}</math> are |
| | <center><math> | | <center><math> |
| | A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0} | | A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0} |
| − | \exp^{-ik\mu_{m}\xi }\exp^{-i\sigma_{m}\eta} | + | e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta} |
| − | \left[ k\phi (\mbox{\boldmath<math>\xi</math>},0) | + | \left[ k\phi (\xi,0) |
| − | - w(\mbox{\boldmath<math>\xi</math>}) \right] | + | - w(\xi) \right] |
| − | d\mbox{\boldmath<math>\xi</math>}. | + | d\xi. |
| − | (Ad_p)
| |
| | </math></center> | | </math></center> |
| | The diffracted waves propagate at various angles with respect to the | | The diffracted waves propagate at various angles with respect to the |
| Line 864: |
Line 258: |
| | <center><math> | | <center><math> |
| | R = A_{0}^{-} | | R = A_{0}^{-} |
| − | = -\frac{i}{\mu_{0}l}\int_{\Delta_0}\exp^{ik (\xi\cos\theta | + | = -\frac{i}{\mu_{0}l}\int_{\Delta_0}e^{ik (\xi\cos\theta |
| − | -\eta\sin\theta)}\left[ k\phi(\mbox{\boldmath<math>\xi</math>},0) | + | -\eta\sin\theta)}\left[ k\phi(\xi,0) |
| − | - w(\mbox{\boldmath<math>\xi</math>})\right] | + | - w(\xi)\right] |
| − | d\mbox{\boldmath<math>\xi</math>}. | + | d\xi. |
| − | (R)
| |
| | </math></center> | | </math></center> |
| | The coefficient, <math>T</math>, for the fundamental transmitted wave for | | The coefficient, <math>T</math>, for the fundamental transmitted wave for |
| Line 874: |
Line 267: |
| | <center><math> | | <center><math> |
| | T = 1 + A_{0}^{+} | | T = 1 + A_{0}^{+} |
| − | = 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} \exp^{-ik(\xi\cos\theta | + | = 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta |
| − | +\eta\sin\theta)}\left[ k\phi(\mbox{\boldmath<math>\xi</math>},0) | + | +\eta\sin\theta)}\left[ k\phi(\xi,0) |
| − | - w(\mbox{\boldmath<math>\xi</math>}) \right] | + | - w(\xi) \right] |
| − | d\mbox{\boldmath<math>\xi</math>}. | + | d\xi. |
| − | (T)
| |
| | </math></center> | | </math></center> |
| | | | |
| Line 889: |
Line 281: |
| | <center><math> | | <center><math> |
| | \cos\theta = \left( |R|^2+|T|^2 \right) \cos\theta | | \cos\theta = \left( |R|^2+|T|^2 \right) \cos\theta |
| − | + \sum_{\stackrel{\scriptstyle{m=-M} }{m ~ \neq 0}}^{N} | + | + \sum_{m=-M,\,m \neq 0}^{N} |
| | \left( |A_{m}^{-}|^2 \cos\psi_{m}^{-} +|A_{m}^{+}|^2 | | \left( |A_{m}^{-}|^2 \cos\psi_{m}^{-} +|A_{m}^{+}|^2 |
| | \cos\psi_{m}^{+} \right). | | \cos\psi_{m}^{+} \right). |
| − | (NRGbal)
| |
| | </math></center> | | </math></center> |
| − | The energy balance equation ((NRGbal)) can be used as an accuracy | + | The energy balance equation can be used as an accuracy |
| | check on the numerical results. | | check on the numerical results. |
| − |
| |
| − | =Results=
| |
| − |
| |
| − | We tested the convergence of our accelerated version of the Green function
| |
| − | and we use <math>c = 0.05</math> and <math>44</math> terms in the first sum (the spatial term) and <math>46</math> terms in the second sum (the spectral term) of ((Gp_fast))).
| |
| − | These values were determined by a convergence study, the details of which
| |
| − | can be found in Wang [[wangphd04]].
| |
| − | These values will be used in all our subsequent
| |
| − | calculations. We consider four geometries for the plates which are shown
| |
| − | in Figure (floes4jfs).
| |
| − |
| |
| − |
| |
| − | ==Scattering from a Dock==
| |
| − |
| |
| − | Aside from the energy balance equation or wide spacing, it is difficult
| |
| − | to compare our results to establish their validity. However,
| |
| − | there is one case in which we can make comparisons. If we
| |
| − | consider the case when we have the dock boundary condition
| |
| − | under the plate (so that <math>w=0</math>) and the plates are square and joined
| |
| − | then the problem reduces to a two dimensional dock problem
| |
| − | which is discussed extensively in \cite[Chapter 2]{Linton_mciver01}.
| |
| − | To impose the condition of
| |
| − | a dock we simply solve equation ((phi_plate)) setting
| |
| − | <math>\vec{w}</math> to zero (we do not require equation ((comp_eqn))),
| |
| − | choosing a square plate (geometry 1) and setting the plate separation to <math>l=4</math>.
| |
| − |
| |
| − | Figure (fig_RnT4stiff) shows the reflection and
| |
| − | transmission coefficients for a square plate (geometry 1) with
| |
| − | the plate separation <math>l=4</math> and the dock boundary condition
| |
| − | (crosses) and the solution to the two-dimensional dock problem using
| |
| − | the method of [[Linton_mciver01]] (solid and dashed lines).
| |
| − | As expected the results agree.
| |
| − |
| |
| − | Table (table_stiff) shows the values of the coefficient
| |
| − | <math>A_{m}^{\pm}</math> for a dock of geometry 1 with <math>{\lambda=4}</math>,
| |
| − | <math>{l=6}</math> and <math>{\theta=\pi/6}</math>. These results are given
| |
| − | to assist in numerical comparisons.
| |
| − | \begin{table}
| |
| − | \begin{tabular}{ccc}
| |
| − | \hline
| |
| − | <math>m</math> & <math>A_{m}^{-}</math> & <math>A_{m}^{+}</math> \\ \hline
| |
| − | <math>-2</math> & <math>-0.214-0.042i</math> & <math>0.232+0.023i</math> \\
| |
| − | <math>-1</math> & <math>0.266-0.268i</math> & <math>-0.185+0.349i</math> \\
| |
| − | <math>0</math> & <math>0.631-0.210i</math> & <math>-0.702-0.141i</math> \\
| |
| − | \end{tabular}
| |
| − | \caption[]{The coefficients <math>A_{m}^{\pm}</math> for the
| |
| − | case of a dock of geometry 1 with <math>{\lambda=4}</math>,
| |
| − | <math>{l=6}</math> and <math>{\theta=\pi/6}</math>.} (table_stiff)
| |
| − | \end{table}
| |
| − |
| |
| − | ==Scattering from Elastic Plates==
| |
| − |
| |
| − | We begin with a short table of numerical results. Table (flexible_table)
| |
| − | is equivalent to
| |
| − | Table (table_stiff) except that the plate is now elastic
| |
| − | with <math>\beta = 0.1</math> and <math>\gamma = 0</math>. As expected the reflected
| |
| − | energy is less because the waves can propagate under the elastic
| |
| − | plates.
| |
| − | \begin{table}
| |
| − | \begin{tabular}{ccc}
| |
| − | \hline
| |
| − | <math>m</math> & <math>A_{m}^{-}</math> & <math>A_{m}^{+} </math> \\ \hline
| |
| − | <math>-2</math> & <math>0.001+0.014i</math> & <math>-0.040-0.016i</math> \\
| |
| − | <math>-1</math> & <math>-0.016-0.008i</math> & <math>-0.070-0.099i</math> \\
| |
| − | <math>0</math> & <math>-0.058-0.072i</math> & <math>-0.209-0.582i</math> \\
| |
| − | \end{tabular}
| |
| − | \caption[]{The coefficients <math>A_{m}^{\pm}</math> for the
| |
| − | case of an elastic plate of geometry 1 with <math>{\beta=0.1}</math>,
| |
| − | <math>{\gamma=0}</math>, <math>{\lambda=4}</math>,
| |
| − | <math>{l=6}</math> and <math>{\theta=\pi/6}</math>.} (flexible_table)
| |
| − | \end{table}
| |
| − | Figures (fig_square_vartheta_beta0p1) and
| |
| − | (fig_triangle_vartheta_beta0p1) show the amplitudes of
| |
| − | the diffracted waves
| |
| − | due to the array as a function of the incident angle
| |
| − | for plates of geometry one and two respectively with <math>\beta=0.1</math>,
| |
| − | <math>\gamma=0</math>, <math>k=\pi/2</math>, and <math>l=6</math>. There are 3 pairs of diffracted
| |
| − | waves (including the reflected-transmitted pair) for any angle.
| |
| − | <math>G_{\bf P}</math> diverges if <math>\sigma_n = \pm k</math> which for our values
| |
| − | of <math>l</math> and <math>k</math> means that <math>\theta = \pm 0.3398</math>. As <math>\theta</math> moves
| |
| − | across these points one of the diffracted waves disappears
| |
| − | (at <math>\pm\pi/2</math>) and an other appears (at <math>\mp\pi/2</math>). In the plots
| |
| − | we have plotted <math>A^{\pm}_{-2}</math> and <math>A^{\pm}_{1}</math> with the same
| |
| − | line style and also <math>A^{\pm}_{-1}</math> and <math>A^{\pm}_{2}</math> since they represent
| |
| − | diffracted waves which appear and disappear together. Interestingly
| |
| − | the result of doing this is to produce smooth curves for
| |
| − | <math>-\pi/2<\theta<\pi/2</math>. Figures (fig_square_vartheta_beta0p1) and
| |
| − | (fig_triangle_vartheta_beta0p1) show that there is a very
| |
| − | strong dependence on the amount of reflected energy as a function
| |
| − | of incident angle with small angles giving the smallest reflection
| |
| − | and large angles giving the greatest reflection.
| |
| − |
| |
| − | Figures (fig_disp_p_square) to (fig_disp_p_trapezoid)
| |
| − | show the real part of the displacement for
| |
| − | five plates (<math>\Delta_{j}</math>, <math>j=-2,-1,0,1,2 </math>)
| |
| − | of the array for plates of geometry one to four
| |
| − | respectively, with <math>\beta=0.1</math>, <math>\gamma=0</math>, and <math>l=6</math>. The angle
| |
| − | of incidence is <math>\theta=\pi/6</math>.
| |
| − | We consider two values of the wavenumber, <math>k=\pi/2</math> (a) and
| |
| − | <math>k=\pi/4</math> (b). The complex response of the elastic plates is
| |
| − | apparent in these figures as is the coupling between
| |
| − | the water and the plate. It is also clear that there is a great
| |
| − | deal of difference in the individual behaviour of plates of
| |
| − | different geometries. This has practical implications for
| |
| − | experiments which might be performed on individual ice floes.
| |
| − | For example, from these figures it appears that it will be very difficult
| |
| − | to make
| |
| − | measurements of wave spectra using an accelerometer
| |
| − | deployed on an ice floe if the floe size is comparable to the
| |
| − | wavelength.
| |
| − |
| |
| − | In Figure (energy_vs_k) we consider the total reflected energy
| |
| − | <math>E_R</math> given by
| |
| − | <center><math>
| |
| − | E_R = |R|^2 \cos\theta
| |
| − | + \sum_{\stackrel{\scriptstyle{m=-M} }{m ~ \neq 0}}^{N}
| |
| − | |A_{m}^{-}|^2 \cos\psi_{m}^{-}
| |
| − | </math></center>
| |
| − | as a function of <math>k</math> for <math>l= 6</math>, <math>\beta=0.1</math>, and <math>\gamma=0</math>
| |
| − | for floes of geometry 1 and 2 and for angles of <math>\theta = 0</math>, <math>\pi/6</math>,
| |
| − | and <math>\pi/3</math>. In Figure (energy_vs_k) we have
| |
| − | divided <math>E_R</math> by <math>\cos\theta</math> to normalise the curves,
| |
| − | since if all the energy is reflected
| |
| − | <math>E_R/\cos\theta=1</math>.
| |
| − | This figure shows the kind of important results which we
| |
| − | can produce from our model even with some simple calculations.
| |
| − | The results show that for <math>k<1</math> there is no scattering of energy
| |
| − | at all. While it is to be expected that long wavelength waves
| |
| − | will not be strongly scattered this figure gives us
| |
| − | a quantitative value for the <math>k</math> below which there
| |
| − | is negligible scattering. As <math>k</math> increases the reflection
| |
| − | increases and this increase is much more marked for waves incident
| |
| − | at a large angle. This is to be expected since waves which
| |
| − | are normally incident can be expected to pass through the plates
| |
| − | even for short wavelengths. However, the effect of angle appears to
| |
| − | be much stronger than might be expected on a simple geometric
| |
| − | argument (which is also what was found in Figures
| |
| − | (fig_square_vartheta_beta0p1) and (fig_triangle_vartheta_beta0p1)).
| |
| − | Interestingly, the effect of geometry is much less
| |
| − | significant than either <math>\theta</math> or <math>k</math>. This
| |
| − | implies that for practical purposes it
| |
| − | may be sufficient to take one or two representative geometries.
| |
| − | This seems surprising given the differences in the responses of
| |
| − | plates of different geometries shown in Figures (fig_disp_p_square)
| |
| − | to (fig_disp_p_trapezoid).
| |
| − | Figure (energy_vs_k) can be regarded as a preliminary figure,
| |
| − | showing the kinds of calculations which are required to develop
| |
| − | a model for wave scattering in the marginal ice zone from the present
| |
| − | results.
| |
| − |
| |
| − | =Concluding Remarks=
| |
| − |
| |
| − | Motivated by the problem of modelling wave propagation in the
| |
| − | marginal ice zone and by the general problem of scattering by
| |
| − | arrays of bodies we have presented a solution to the problem
| |
| − | of wave scattering by an infinite array of floating elastic
| |
| − | plates. The model is based on the linear theory and assumes that
| |
| − | the floe submergence is negligible. For this reason it is applicable
| |
| − | to low to moderate wave heights and to floes whose size is large compared
| |
| − | to the floe thickness.
| |
| − | The solution method is similar to that used to solve
| |
| − | for a single plate except that the periodic Green function must
| |
| − | be used. The periodic Green function is obtained by summing the free-surface
| |
| − | Green function for all <math>y</math>. However the periodic Green function is slowly
| |
| − | convergent and therefore a method, which is commonly used in Optics, is
| |
| − | devised to accelerate the convergence. The acceleration method involves
| |
| − | expressing the infinite sum of the Green function in its far-field form.
| |
| − | From the far-field representation of the periodic Green Function, we
| |
| − | calculate the diffracted wave
| |
| − | far from the array and the cut-off frequencies. We have checked our
| |
| − | numerical calculations for energy balance and against the
| |
| − | limiting case when the plates are rigid and joined where the
| |
| − | solution reduces to that of a rigid dock. We have also presented
| |
| − | solutions for a range of elastic plate geometries.
| |
| − |
| |
| − | The solution method could be extended to water of finite depth using
| |
| − | a similar periodic Green function, but in this case based on the free
| |
| − | surface Green function for finite depth water. The same problems of slow
| |
| − | convergence would arise and a similar method for convergence of the
| |
| − | series would be required. We believe that a method similar to the one
| |
| − | presented here could be used to accelerate the convergence of the periodic
| |
| − | Green function for water of finite depth. Another extension would be to consider
| |
| − | a doubly periodic lattice in which a doubly periodic Green function
| |
| − | would be required. This should also be able to be computed quickly by
| |
| − | methods similar to the ones presented here.
| |
| − |
| |
| − | The most important extension of this work concerns using it to construct
| |
| − | a scattering model for the marginal ice zone. Such a model would be based
| |
| − | on the solutions presented here but would be a far from trivial extension
| |
| − | of it. For a model to be realistic the effects which have been induced by
| |
| − | the assumption of periodicity would have to be removed. One method to do
| |
| − | this would be to consider random spacings of the ice floes and to average the
| |
| − | result over these. This should lead to a scattered wave field over
| |
| − | all directions rather than discrete scattering angles as well as a reflected
| |
| − | and transmitted wave. These solutions would then have to be combined, again with
| |
| − | some kind of averaging and wide spacing approximation, to compute the
| |
| − | scattering by multiple rows of ice floes.
| |
| − |
| |
| − |
| |
| − |
| |
| − | \bibliography{/home/groups/seaice/bibdata/mike,/home/groups/seaice/bibdata/others,/home/meylan/Papers/Periodic_Cynthia/ALMOSTALL,/home/meylan/Papers/Periodic_Cynthia/Cynthia}
| |
| − |
| |
| − |
| |
| − |
| |
| − | \pagebreak
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7]
| |
| − | {array}
| |
| − | \caption{A schematic diagram of the periodic array of floating elastic
| |
| − | plates.}
| |
| − | (fig_array)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}[ptb]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {floes4jfs}
| |
| − | \caption{Diagram of the four plate geometries for which we will
| |
| − | calculate solutions. }
| |
| − | (floes4jfs)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7]{stiff_compare}
| |
| − | \caption{The reflection coefficient <math>R</math> (solid line)
| |
| − | and the transmission coefficient <math>T</math> (dashed line) as a function
| |
| − | of <math>k</math> for a two-dimensional dock of length 4 for the incident
| |
| − | angles shown. The crosses are the same problem solved using
| |
| − | the three-dimensional array code with the dock boundary
| |
| − | condition and using plates of geometry 1 with <math>l=4</math>.}
| |
| − | (fig_RnT4stiff)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {diffracted_square}
| |
| − | \caption{The diffracted waves <math>A_m^\pm</math> for a periodic array
| |
| − | of geometry one plates with <math>k=\pi/2</math>, <math>l=6</math>, <math>\beta=0.1,</math> <math>\gamma=0</math>,
| |
| − | and <math>l=6</math>. The solid line
| |
| − | is <math>A_0^\pm</math>, the chained line is <math>A_{-2}^\pm</math> and <math>A_1^\pm</math> and the
| |
| − | dashed line is <math>A_{-1}^\pm</math> and <math>A_2^\pm</math>.}
| |
| − | (fig_square_vartheta_beta0p1)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {diffracted_triangle}
| |
| − | \caption{As for figure (fig_square_vartheta_beta0p1) except that
| |
| − | the pate has geometry 2. }
| |
| − | (fig_triangle_vartheta_beta0p1)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {disp_P_square}
| |
| − | \caption{The real part of the displacement <math>w</math> for five plates of geometry one
| |
| − | which are part of a periodic
| |
| − | array, <math>l= 6</math>, <math>\theta=\pi/6</math>, <math>\beta=0.1</math>, <math>\gamma=0</math> and
| |
| − | (a) <math>k=\pi/2</math> and (b) <math>k=\pi/4=8</math>.}
| |
| − | (fig_disp_p_square)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {disp_P_triangle}
| |
| − | \caption{As for figure (fig_disp_p_square) except that the
| |
| − | plate is of geometry two.}
| |
| − | (fig_disp_p_triangle)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {disp_P_parallelogram}
| |
| − | \caption{As for figure (fig_disp_p_square) except that the
| |
| − | plate is of geometry three.}
| |
| − | (fig_disp_p_parallelogram)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {disp_P_trapezoid}
| |
| − | \caption{As for figure (fig_disp_p_square) except that the
| |
| − | plate is of geometry four.}
| |
| − | (fig_disp_p_trapezoid)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − | \begin{figure}
| |
| − | [p]
| |
| − | \begin{center}
| |
| − | \includegraphics[scale = 0.7
| |
| − |
| |
| − |
| |
| − | ]
| |
| − | {energy_vs_k}
| |
| − | \caption{The total reflected energy <math>E_R</math> divided by
| |
| − | <math>\cos\theta</math> as a function of <math>k</math> for floes of geometry 1 (a)
| |
| − | and 2 (b). <math>l= 6</math>, <math>\beta=0.1</math>, <math>\gamma=0</math>, and
| |
| − | <math>\theta = 0</math> (solid line), <math>\pi/6</math> (dashed line), and <math>\pi/3</math>
| |
| − | (chained line).}
| |
| − | (energy_vs_k)
| |
| − | \end{center}
| |
| − | \end{figure}
| |
| − |
| |
| − |
| |
| | | | |
| | | | |
| | [[Category:Infinite Array]] | | [[Category:Infinite Array]] |
