Interaction Theory for Infinite Arrays

Introduction

There are two approaches to solution for the Infinite Array, one is Infinite Array Green Function the other is based on Interaction Theory. We present here a solution based on the latter, using Kagemoto and Yue Interaction Theory to derive a system of equations for the infinite array. This is based on Peter, Meylan, and Linton 2006

System of equations

We start with the final system of equations of the Kagemoto and Yue Interaction Theory, namely

[math]\displaystyle{ A_{m\mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math], [math]\displaystyle{ l=1,\dots,N }[/math].

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by [math]\displaystyle{ R }[/math], we have [math]\displaystyle{ R_{jl} = |j-l| R }[/math] and

[math]\displaystyle{ \varphi_{n} = \begin{cases} \pi, & n\gt 0,\\ 0, & n\lt 0. \end{cases} }[/math]

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients [math]\displaystyle{ A_{m\mu}^l }[/math] can be written as [math]\displaystyle{ A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu} }[/math], where the phase factor [math]\displaystyle{ P_l }[/math] is given by

[math]\displaystyle{ \ P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi}, }[/math]

where [math]\displaystyle{ \chi }[/math] is the angle which the direction of the ambient waves makes with the [math]\displaystyle{ x }[/math]-axis. The same can be done for the coefficients of the ambient wave, i.e. [math]\displaystyle{ \tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu} }[/math].

Therefore, the system simplifies to

[math]\displaystyle{ A_{m\mu} = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu \sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_{\tau - \nu} (k_n |j|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big]. }[/math]

Introducing the constants

[math]\displaystyle{ \sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_\nu(k_n|j|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR), }[/math]

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

[math]\displaystyle{ A_{m\mu} = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu \sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math]. Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula [math]\displaystyle{ A_{m\mu}^l = P_l A_{m\mu} }[/math].

Far Field Waves

An algebraically exact solution to the problem of linear water-wave scattering by a periodic array of scatterers is presented in which the scatterers may be of arbitrary shape. The method of solution is based on an interaction theory

in which the incident wave on each body from all the other bodies in the array is expressed in the respective local cylindrical eigenfunction expansion. We show how to calculate the slowly convergent terms efficiently which arise in the formulation and how to calculate the scattered field far from the array. The application to the problem of linear acoustic scattering by cylinders with arbitrary cross-section is also discussed. Numerical calculations are presented to show that our results agree with previous calculations. We present some computations for the case of fixed, rigid and elastic floating bodies of negligible draft concentrating on presenting the amplitudes of the scattered waves as functions of the incident angle.

The far field

In this section, the far field is examined which describes the scattering far away from the array. The derivation is equivalent to that of Twersky 1962. First, we define the scattering angles which give the directions of propagation of plane scattered waves far away from the array. Letting [math]\displaystyle{ p=2\pi/R }[/math], define the scattering angles [math]\displaystyle{ \chi_m }[/math] by

[math]\displaystyle{ \chi_m = \cos^{-1} (\psi_m/k) \quad =where= \quad \psi_m = k \cos \chi + m p }[/math]

and write [math]\displaystyle{ \psi }[/math] for [math]\displaystyle{ \psi_0 }[/math]. Also note that [math]\displaystyle{ \chi_0 = \chi }[/math] by definition. If [math]\displaystyle{ \abs{\psi_m}\lt k }[/math], i.e.~if

[math]\displaystyle{ -1 \lt \cos \chi +\frac{mp}{k}\lt 1, }[/math]

we say that [math]\displaystyle{ m\in \mathcal{M} }[/math] and then [math]\displaystyle{ 0\lt \chi_m\lt \pi }[/math]. It turns out (see below) that these angles ([math]\displaystyle{ \pm \chi_m }[/math] for [math]\displaystyle{ m \in \mathcal{M} }[/math]) are the directions in which plane waves propagate away from the array. If [math]\displaystyle{ \abs{\psi_m}\gt k }[/math] then [math]\displaystyle{ \chi_m }[/math] is no longer real and the appropriate branch of the [math]\displaystyle{ \arccos }[/math] function is given by

[math]\displaystyle{ \arccos t = \begin{cases} \mathrm{i} \arccosh t, & t\gt 1,\\ \pi-\mathrm{i} \arccosh (-t) & t\lt -1, \end{cases} }[/math]

with [math]\displaystyle{ \arccosh t = \log \left(t+\sqrt{t^2-1}\right) }[/math] for [math]\displaystyle{ t\gt 1 }[/math].

For the total potential we have

[math]\displaystyle{ \begin{matrix} \notag \phi &=\phi^\mathrm{In}+ \sum_{m=0}^{\infty} f_m(z) \sum_{j=-\infty}^{\infty} P_j \sum_{\mu = -\infty}^{\infty} A_{m\mu} K_\mu(k_m r_j)\mathrm{e}^{\mathrm{i} \mu\theta_j} \\ &\sim \phi^\mathrm{In}+ \frac{\pi}{2} f_0(z) \sum_{j=-\infty}^{\infty} P_j \sum_{\mu = -\infty}^{\infty} A_{0\mu} \i^{\mu+1} H^{(1)}_\mu (kr_j) \mathrm{e}^{\mathrm{i} \mu\theta_j}, (26) \end{matrix} }[/math]

as [math]\displaystyle{ kr\to\infty }[/math], away from the array axis [math]\displaystyle{ y=0 }[/math], where we have used the identity \eqref{H_K}.

The far field can be determined as follows. If we insert the integral representation

[math]\displaystyle{ H^{(1)}_\mu (kr) \mathrm{e}^{\mathrm{i} \mu \theta}= \frac{(-\i)^{\mu+1}}{\pi} \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k\gamma(t)\abs{y}}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}\,\mathrm{e}^{\i \mu \sgn(y)\arccos t} \,\mathrm{d} t, }[/math]

in which [math]\displaystyle{ x=r\cos\theta }[/math], [math]\displaystyle{ y=r\sin\theta }[/math] and [math]\displaystyle{ \gamma(t) }[/math] is defined for real [math]\displaystyle{ t }[/math] by

[math]\displaystyle{ \gamma(t) = \begin{cases} -\mathrm{i} \sqrt{1-t^2} & \abs{t} \leq 1 \\ \sqrt{t^2-1} & \abs{t}\gt 1, \end{cases} }[/math]

into (26) we get

[math]\displaystyle{ \begin{matrix} \phi & \sim\phi^\mathrm{In}+ \frac{1}{2} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k \gamma(t)\abs{y}}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt} \,\mathrm{e}^{\i(\psi-kt) jR}\,\mathrm{e}^{\i \mu \sgn(y)\arccos t} \,\mathrm{d} t \\ & =\phi^\mathrm{In}+ \frac{\pi}{kR} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \frac{\mathrm{e}^{-k\gamma(\psi_j/k)\abs{y}}}{\gamma(\psi_j/k)} \,\mathrm{e}^{\mathrm{i} x\psi_j}\,\mathrm{e}^{\i \mu\sgn(y)\arccos \psi_j/k} \\ & =\phi^\mathrm{In}+ \frac{\pi\i}{kR} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \frac{1}{\sin\chi_j} \,\mathrm{e}^{\mathrm{i} kr\cos(\abs{\theta}-\chi_j)}\,\mathrm{e}^{\i \mu \sgn(\theta)\chi_j}, \end{matrix} }[/math]

in which we have used the Poisson summation formula,

[math]\displaystyle{ \sum_{m=-\infty}^\infty \int_{-\infty}^{\infty} f(u)\, \mathrm{e}^{-\mathrm{i} mu} \,\mathrm{d} u= 2\pi \sum_{m=-\infty}^\infty f(2m\pi). }[/math]

The only terms which contribute to the far field are those for which [math]\displaystyle{ \abs{\psi_m}\lt k }[/math]. Thus, as [math]\displaystyle{ y\to\pm\infty }[/math], the far field consists of a set of plane waves propagating in the directions [math]\displaystyle{ \theta=\pm\chi_m }[/math]:

[math]\displaystyle{ \phi\sim \phi^\mathrm{In}+ \frac{\pi \i}{kR} f_0(z) \sum_{m\in\mathcal{M}} \frac{1}{\sin\chi_m} \,\mathrm{e}^{\mathrm{i} kr\cos(\theta\mp\chi_m)} \sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}. (30) }[/math]

>From \eqref{eqn:inffar} the amplitudes of the scattered waves for each scattering angle [math]\displaystyle{ \pm \chi_m }[/math] are given in terms of the coefficients [math]\displaystyle{ A_{0\mu} }[/math] by

[math]\displaystyle{ (31) A^\pm_m = \frac{\pi \i}{kR} \frac{1}{\sin\chi_m} \sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}. }[/math]

Note that the primary reflection and transmission coefficients are recovered by [math]\displaystyle{ A^-_0 }[/math] and [math]\displaystyle{ 1 + A^+_0 }[/math], respectively.

It is implicit in all the above that [math]\displaystyle{ \sin\chi_m\neq 0 }[/math] for any [math]\displaystyle{ m }[/math]. If [math]\displaystyle{ \sin\chi_m=0 }[/math] then we have the situation where one of the scattered plane waves propagates along the array. We will not consider this resonant case here except for stating that then, the scattered field is dominated by waves travelling along the array, either towards </math>x = \infty</math> (if [math]\displaystyle{ \chi_m = 0 }[/math]) or towards [math]\displaystyle{ x=-\infty }[/math] (if [math]\displaystyle{ \chi_m = \pi }[/math]). Also, we will not consider the excitation of Rayleigh-Bloch waves, which are waves which travel along the array with a phase difference between adjacent bodies greater than [math]\displaystyle{ Rk }[/math] (include refs). Both the resonant and Rayleigh-Bloch case are important but beyond the scope of the present work.

The efficient computation of the [math]\displaystyle{ \sigma_{\nu}^0 }[/math]

The constants [math]\displaystyle{ \sigma_{\nu}^0 }[/math] (cf.~\eqref{eq_op_sigma}) appearing in the system of equations for the coefficients of the scattered wavefield of the bodies cannot be computed straightforwardly. This is due to the slow decay of the modified Bessel function of the second kind for large imaginary argument as was discussed in \S 14. First, note that

[math]\displaystyle{ \sigma_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) K_{\nu} (-\mathrm{i} k j R) = \frac{\pi \i^{\nu+1}}{2} \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR), }[/math]

where we have used \eqref{H_K}. Therefore, it suffices to discuss the computation of the constants [math]\displaystyle{ \tilde{\sigma}^0_\nu }[/math] defined via

[math]\displaystyle{ \tilde{\sigma}_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR) }[/math]

as the [math]\displaystyle{ \sigma^0_\nu }[/math] are then determined by </math>\sigma^0_\nu = \pi/2 \,\, \i^{\nu+1} \, \tilde{\sigma}^0_\nu[math]\displaystyle{ . An efficient way of computing the \lt math\gt \tilde{\sigma}_{\nu}^0 }[/math] is given in linton98 and the results are briefly outlined in our notation. Noting that [math]\displaystyle{ H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu} H^{(1)}_{\nu} (\,\cdot\,) }[/math], it suffices to discuss the computation of the [math]\displaystyle{ \sigma_{\nu}^0 }[/math] for non-negative [math]\displaystyle{ \nu }[/math].

Referring to Linton 1998, the constants [math]\displaystyle{ \tilde{\sigma}_{\nu}^0 }[/math] can be written as

[math]\displaystyle{ \tilde{\sigma}_{0}^0 &= -1 -\frac{2\i}{\pi} \left( C + \log \frac{k}{2p} \right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2 \psi^2)}{p^3 R} \zeta(3)\\ &\quad + \frac{2}{R} \sum_{m=1}^\infty \left( \frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} + \frac{2 \i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right) }[/math]

where [math]\displaystyle{ C \approx 0.5772 }[/math] is Euler's constant and [math]\displaystyle{ \zeta }[/math] is the Riemann zeta function and the terms in the sum converge like [math]\displaystyle{ O(m^{-4}) }[/math] as [math]\displaystyle{ m\rightarrow\infty }[/math] (by which we mean that the error in the sum is proportional to [math]\displaystyle{ m^{-4} }[/math] for large values of [math]\displaystyle{ m }[/math]) as well as

[math]\displaystyle{ \begin{matrix} &\quad \tilde{\sigma}_{2\nu}^0 &= 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi} }{R k \sin \chi} - \frac{\i}{\pi} \left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) + \frac{\i}{\nu \pi} \\ &\quad + 2 (-1)^\nu \sum_{m=1}^\infty \left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} + \frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} + \frac{\i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu} \right)\\ &\quad + \frac{\i}{\pi} \sum_{m=1}^\nu \frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m} B_{2m}(\psi/p), \\ & \tilde{\sigma}_{2\nu-1}^0 &= - 2 (-1)^\nu \left( \frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1) \chi}}{R k \sin \chi} - \frac{ \psi R \nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1} \zeta(2\nu +1) \right)\\ &\quad - 2 (-1)^\nu \sum_{m=1}^\infty \left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} + \frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} + \frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2 m p} \right)^{2\nu-1} \right)\\ &\quad - \frac{2}{\pi} \sum_{m=0}^{\nu-1} \frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k} \right)^{2m+1} B_{2m+1}(\psi/p), \end{matrix} }[/math]

for [math]\displaystyle{ \nu\gt 0 }[/math] where [math]\displaystyle{ B_m }[/math] is the [math]\displaystyle{ m }[/math]th Bernoulli polynomial. The slowest convergence in this representation occurs in [math]\displaystyle{ \tilde{\sigma}^0_1 }[/math] and [math]\displaystyle{ \tilde{\sigma}^0_2 }[/math] in which the terms converge like [math]\displaystyle{ O(m^{-5}) }[/math] as [math]\displaystyle{ m\rightarrow\infty }[/math].

Note that since [math]\displaystyle{ \sin \chi_m }[/math] is purely imaginary for </math>m \notin \mathcal{M}[math]\displaystyle{ , the computation of the real part of \lt math\gt \tilde{\sigma}_{2\nu}^0 }[/math] and the imaginary part of [math]\displaystyle{ \tilde{\sigma}_{2\nu-1}^0 }[/math] is particularly simple. For [math]\displaystyle{ \nu \geq 0 }[/math], they are given by

[math]\displaystyle{ \begin{matrix} \Re \tilde{\sigma}_{2\nu}^0 &= -\delta_{\nu 0} + 2(-1)^\nu \sum_{m\in\mathcal{M}} \frac{\cos 2 \nu \chi_m}{R k \sin \chi_m}, \\ \Im \tilde{\sigma}_{2\nu+1}^0 &= 2\mathrm{i} (-1)^\nu \sum_{m\in\mathcal{M}} \frac{\cos (2 \nu-1) \chi_m}{R k \sin \chi_m}, \end{matrix} }[/math]

where [math]\displaystyle{ \delta_{mn} }[/math] is the Kronecker delta.

Acoustic scattering by an infinite array of identical generalized cylinders

The theory above has so far been developed for water-wave scattering of a plane wave by an infinite array of identical arbitrary bodies. It can easily be adjusted to the (simpler) two-dimensional problem of acoustic scattering. Namely, we consider the problem that arises when a plane sound wave is incident upon an infinite array of identical generalized cylinders (i.e.~bodies which have arbitrary cross-section in the [math]\displaystyle{ (x,y) }[/math]-plane but the cross-sections at any height are identical) in an acoustic medium.

For this problem, the [math]\displaystyle{ z }[/math]-dependence can be omitted and the above theory applies with the following modifications:

1. The dispersion relation \eqref{eq_k} is replaced by </math>k=\omega /

c[math]\displaystyle{ where }[/math]c[math]\displaystyle{ is the speed of sound in the medium under consideration and the dispersion relation is \eqref{eq_k_m} omitted. #All factors \lt math\gt \cos k_m(z+d) }[/math], [math]\displaystyle{ \cos k_m(c+d) }[/math], [math]\displaystyle{ \cos k_m d }[/math] and [math]\displaystyle{ f_0 }[/math] are replaced by 1.

1. The factor [math]\displaystyle{ N_0 }[/math] in \eqref{green_d} is [math]\displaystyle{ k/\pi }[/math].

Note that point [math]\displaystyle{ (a) }[/math] implies that there are no evanescent modes in this problem, i.e.~the sums over [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] in the eigenfunction expansions \eqref{basisrep_out_d} and \eqref{basisrep_in_d}, respectively, only contain the terms for [math]\displaystyle{ m=0 }[/math] and [math]\displaystyle{ n=0 }[/math]. Moreover, we have </math>k_0 = - \mathrm{i} \, \omega /c[math]\displaystyle{ . For circular cylinders, i.e.~cylinders which have a circular cross-section, this problem has been considered by [[linton93]]. In \S 52 we numerically compare our results for this problem with theirs. \section{Wave forcing of a fixed, rigid and flexible body of shallow draft}(37) The theory which has been developed so far has been for arbitrary bodies. No assumption has been made about the body geometry or its equations of motion. However, we want use this theory to make calculations for the specific case of bodies of shallow draft which may be fixed (which we shall refer to as a dock), rigid, or elastic (modelled as a thin plate). In the formulation, we concentrate on the elastic case of which the other two situations are subcases. This allows us to present a range of results while focusing on the geophysical problem which motivates our work, namely the wave scattering by a field of ice floes. ==Mathematical model for an elastic plate.== We briefly describe the mathematical model of a floating elastic plate. A more detailed account can be found in [[JGR02,JFM04]]. We assume that the elastic plate is sufficiently thin that we may apply the shallow-draft approximation, which essentially applies the boundary conditions underneath the plate at the water surface. Assuming the elastic plate to be in contact with the water surface at all times, its displacement \lt math\gt W }[/math] is that of the water surface and [math]\displaystyle{ W }[/math] is required to satisfy the linear plate equation in the area occupied by the elastic plate [math]\displaystyle{ \Delta }[/math]. In analogy to \eqref{time}, denoting the time-independent surface displacement (with the same radian frequency as the water velocity potential due to linearity) by [math]\displaystyle{ w }[/math] ([math]\displaystyle{ W=\Re\{w \exp(-\mathrm{i}\omega t)\} }[/math]), the plate equation becomes

[math]\displaystyle{ (38) D \, \nabla^4 w - \omega^2 \, \rho_\Delta \, h \, w = \mathrm{i} \, \omega \, \rho \, \phi - \rho \, g \, w, \quad {\mathbf{x}} \in \Delta, }[/math]

with the density of the water [math]\displaystyle{ \rho }[/math], the modulus of rigidity of the elastic plate [math]\displaystyle{ D }[/math], its density [math]\displaystyle{ \rho_\Delta }[/math] and its thickness [math]\displaystyle{ h }[/math]. The right-hand side of \eqref{plate_non} arises from the linearized Bernoulli equation. It needs to be recalled that [math]\displaystyle{ \mathbf{x} }[/math] always denotes a point of the undisturbed water surface. Free-edge boundary conditions apply, namely

[math]\displaystyle{ (39) \left[ \nabla^2 - (1-\nu) \left(\frac{\partial^2}{\partial s^2} + \kappa(s) \frac{\partial}{\partial n} \right) \right] w = 0, }[/math] [math]\displaystyle{ (40) \left[ \frac{\partial}{\partial n} \nabla^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]

where [math]\displaystyle{ \nu }[/math] is Poisson's ratio and \begin{gather} \nabla^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}. \end{gather} Here, [math]\displaystyle{ \kappa(s) }[/math] is the curvature of the boundary, [math]\displaystyle{ \partial \Delta }[/math], as a function of arclength [math]\displaystyle{ s }[/math] along [math]\displaystyle{ \partial \Delta }[/math]; [math]\displaystyle{ \partial/\partial s }[/math] and [math]\displaystyle{ \partial/\partial n }[/math] represent derivatives tangential and normal to the boundary [math]\displaystyle{ \partial \Delta }[/math], respectively.

Non-dimensional variables (denoted with an overbar) are introduced,

[math]\displaystyle{ (\bar{x},\bar{y},\bar{z}) = \frac{1}{L} (x,y,z), \quad \bar{w} = \frac{w}{L}, \quad \bar{\alpha} = L\, \alpha, \quad \bar{\omega} = \omega \sqrt{\frac{L}{g}} \quad =and= \quad \bar{\phi} = \frac{\phi}{L \sqrt{L g}}, }[/math]

where [math]\displaystyle{ L }[/math] is a length parameter associated with the plate. In non-dimensional variables, the equation for the elastic plate \eqref{plate_non} reduces to

[math]\displaystyle{ (41) \beta \nabla^4 \bar{w} - \bar{\alpha} \gamma \bar{w} = \i \sqrt{\bar{\alpha}} \bar{\phi} - \bar{w}, \quad \bar{\mathbf{x}} \in \bar{\Delta}, }[/math]

with

[math]\displaystyle{ \beta = \frac{D}{g \rho L^4} \quad =and= \quad \gamma = \frac{\rho_\Delta h}{ \rho L}. }[/math]

The constants [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] represent the stiffness and the mass of the plate, respectively. For convenience, the overbars are dropped and non-dimensional variables are assumed in what follows.

The fixed and rigid body cases

The fixed and rigid body cases can easily be solved by the method outlined above since they can be considered special cases. In the problem of a fixed body (dock), the displacement is always zero, [math]\displaystyle{ w=0 }[/math], so we simply need to solve equation (46) for [math]\displaystyle{ \phi=\phi^0 }[/math]. For the case of a rigid body, we need to truncate the sums in \eqref{expanded} to include the first three modes only (which correspond to the three modes of rigid motion of the plate, namely the heave, pitch and roll). Note that for these modes the eigenvalue is [math]\displaystyle{ \lambda_k=0 }[/math] so that the term involving the stiffness [math]\displaystyle{ \beta }[/math] does not appear in equation (50).

Numerical calculations

In this section, we present some numerical computations using the theory developed in the previous sections. We are particularly interested in comparisons with results from other methods as well as using our method to compare the behaviour of different bodies. Besides comparisons with results from other works, one way to check the correctness of the implementation is to verify that energy is conserved, i.e.~the energy of the incoming wave must be equal to the sum of the energies of all outgoing waves. In terms of the amplitudes of the scattered waves for each scattering angle </math>\pm

\chi_m[math]\displaystyle{ , }[/math]A^\pm_m[math]\displaystyle{ , (cf.~\eqref{scat_ampl}) this can be written as \lt center\gt \lt math\gt (51) \sin \chi = \sum_{m\in \mathcal{M}} \left(\abs{A^-_m}^2 + \abs{A^+_m + \delta_{0m}}^2 \right) \, \sin \chi_m }[/math]

where we have assumed an ambient incident potential of unit amplitude.

In all calculations presented below, the absolute value of the difference of both sides in \eqref{energy_cons} is at most [math]\displaystyle{ 10^{-3} }[/math].

==Comparison with results from linton93== (52) We first compare our results to those of linton93 who considered the acoustic scattering of a plane sound wave incident upon a periodic array of identical rigid circular cylinders of radius [math]\displaystyle{ a }[/math]. It can be noted that they also discussed the application of their theory to the water-wave scattering by an infinite row of rigid vertical circular cylinders extending throughout the water depth. Their method of solution was based on a multipole expansion but they also included a separation of variables method which can be viewed as a special case of our method.

As \citeauthor{linton93} considered circular cylinders, we need to obtain the diffraction transfer matrix of rigid circular cylinders. Due to the axisymmetry, they are particularly simple. In fact, they are diagonal with diagonal elements

[math]\displaystyle{ (\mathbf{B})_{pp} = -I_p'(k_0 a) / K_p'(k_0 a) }[/math]

\cite[cf.][p.~177, for example]{linton01}. Also, there are no evanescent modes if the ambient incident wave does not contain evanescent modes (which is the case in their considerations as well as ours).

We compare \citeauthor{linton93}' results to ours in terms of the amplitudes of the scattered waves, [math]\displaystyle{ A^\pm_m }[/math]. In particular, we reproduce their figures 1 (a) and (b) (corresponding to our figure 53) which show the absolute values of the amplitudes of the scattered waves plotted against [math]\displaystyle{ ka }[/math] when </math>a/R = 0.2[math]\displaystyle{ for the cases }[/math]\chi = \pi/2[math]\displaystyle{ and }[/math]\pi/3[math]\displaystyle{ , respectively. Note that they use different choices for defining the incident angle and spacing of the cylinders. Since \citeauthor{linton93} only give plotted data we also plot our results (shown in figure 53). A visual comparison of the plots shows that they are in good agreement with \citeauthor{linton93}' results. \begin{figure} \begin{tabular}{p{.46\columnwidth}p{.02\columnwidth}p{.46\columnwidth}} \includegraphics[width=.38\columnwidth]{linton_chi2} && \includegraphics[width=.38\columnwidth]{linton_chi3} \end{tabular} \caption{Absolute values of the amplitudes of the scattered waves plotted against \lt math\gt ka }[/math] when [math]\displaystyle{ a/R = 0.2 }[/math] for the two cases </math>\chi = \pi/2[math]\displaystyle{ (left) and }[/math]\chi = \pi/3[math]\displaystyle{ (right).}(53) \end{figure} It is worth noting that for an ambient incident angle of }[/math]\chi = \pi/2[math]\displaystyle{ (normal incidence) the scattered waves appear in pairs of two corresponding to \lt math\gt \pm m }[/math], i.e.~they travel in the directions [math]\displaystyle{ \pm \chi_m }[/math] with respect to the array axis. Note that this is generally true for normal incidence upon arrays of arbitrary bodies and easily follows from the considerations at the beginning of \S 4. Moreover, as [math]\displaystyle{ ka }[/math] is increased, more scattered waves appear. From the plots in figure 53, it seems that the amplitudes of the scattered waves have poles in these points of appearance but a careful consideration shows that they are actually continuous at these points \cite[cf.][]{linton93}.

==Comparison with results from JEM05== (54)

Next, we compare our results to those of JEM05 who considered the water-wave scattering by an infinite array of floating elastic plates in water of infinite depth. The plates were modelled in exactly the same way as our elastic plates in \S 37. Their method of solution was based on the use of a special periodic Green's function in \eqref{int_eq_hs}. As a way of testing their method, \citeauthor{JEM05} also considered the scattering from an array of docks (fixed bodies). Therefore, we reproduce their results for the dock \cite[table 1 in][]{JEM05} and for elastic plates \cite[table 2 in][]{JEM05} in tables 55 and 56, respectively. In both cases, the plates are of square geometry with sidelength 4 and spacing [math]\displaystyle{ R=6 }[/math]. The ambient wave is of the same wavelength as the sidelength of the bodies and the incident angle is [math]\displaystyle{ \chi = \pi/3 }[/math] (in our notation). In table 56, the elastic plates have non-dimensionalized stiffness [math]\displaystyle{ \beta = 0.1 }[/math] and mass [math]\displaystyle{ \gamma = 0 }[/math]. We choose [math]\displaystyle{ d=4 }[/math] in order to simulate infinite depth. Since the elastic plates tend to lengthen the wave it is necessary to choose a water depth greater than the standard choice of half the ambient wavelength \cite[cf.][]{FoxandSquire}.

As can be seen, each amplitude in table 55 has a relative difference of less than [math]\displaystyle{ 6 \cdot 10^{-2} }[/math] with respect to the values obtained by \citeauthor{JEM05}. The analogue is true for table 56 with a relative error of less than [math]\displaystyle{ 9 \cdot 10^{-2} }[/math] except for [math]\displaystyle{ A^-_{-1} }[/math] where the relative error is [math]\displaystyle{ \approx 0.34 }[/math] (note, however, that the values in \citeauthor{JEM05} are only given up to the third decimal place). The results given in tables 55 and 56 were obtained using 23 angular propagating modes, three roots of the dispersion relation \eqref{eq_k_m} (not counting the zeroth root) and seven corresponding angular evanescent modes each. Note that fewer modes also yield reasonably good approximations. For example, taking 15 angular propagating modes, one root of the dispersion relation and three corresponding angular evanescent modes yields answers differing from those in tables 55 and 56 only in the fourth decimal place.

\begin{table} \begin{center} \begin{tabular} [math]\displaystyle{ m }[/math] & [math]\displaystyle{ A^-_m }[/math] & [math]\displaystyle{ A^+_m }[/math]\\ [math]\displaystyle{ -2 }[/math] & [math]\displaystyle{ -0.2212 - 0.0493\i }[/math] & [math]\displaystyle{ +0.2367 + 0.0268\i }[/math] \\ [math]\displaystyle{ -1 }[/math] & [math]\displaystyle{ +0.2862 - 0.2627\i }[/math] & [math]\displaystyle{ -0.2029 + 0.3601\i }[/math]\\ [math]\displaystyle{ 0 }[/math] & [math]\displaystyle{ +0.6608 - 0.1889\i }[/math] & [math]\displaystyle{ -0.7203 - 0.1237\i }[/math] \end{tabular} \caption{Amplitudes of the scattered waves for the case of a dock.} (55) \end{center} \end{table}

\begin{table} \begin{center} \begin{tabular} [math]\displaystyle{ m }[/math] & [math]\displaystyle{ A^-_m }[/math] & [math]\displaystyle{ A^+_m }[/math]\\ [math]\displaystyle{ -2 }[/math] & [math]\displaystyle{ +0.0005 + 0.0149\i }[/math] & [math]\displaystyle{ -0.0405 - 0.0138\i }[/math] \\ [math]\displaystyle{ -1 }[/math] & [math]\displaystyle{ -0.0202 - 0.0125\i }[/math] & [math]\displaystyle{ -0.0712 - 0.1004\i }[/math] \\ [math]\displaystyle{ 0 }[/math] & [math]\displaystyle{ -0.0627 - 0.0790\i }[/math] & [math]\displaystyle{ -0.2106 - 0.5896\i }[/math] \end{tabular} \caption{Amplitudes of the scattered waves for the case of a elastic plates.} (56) \end{center} \end{table}

\subsection{Comparison of the scattering by an array of docks, rigid plates and elastic plates} In this section, we use our method to compare the behaviour of arrays of docks, rigid plates and elastic plates. The equations describing the different bodies have been derived in \S 37. In order to have a common setting, we choose all bodies to be square with sidelength 2 and a body spacing of [math]\displaystyle{ R=4 }[/math]. The ambient wavelength is [math]\displaystyle{ \lambda = 1.5 }[/math] and the water depth is [math]\displaystyle{ d=0.5 }[/math].

In figures 58, 59 and 60 we show the absolute values of the amplitudes of the scattering angles as functions of incident angle as well as the solution of the scattering problem for [math]\displaystyle{ \chi = \pi/5 }[/math] for an array of docks, rigid plates and elastic plates, respectively. The elastic plates are chosen to have non-dimensional stiffness and mass [math]\displaystyle{ \beta=\gamma=0.02 }[/math] while the rigid plates have the same mass. In the plots of the amplitudes of the scattered waves, we plot [math]\displaystyle{ \abs{A^-_0} }[/math] and [math]\displaystyle{ \abs{1+A^+_0} }[/math] as solid lines and the additional scattered waves with symbols as listed in table 57. Note that the calculation of the amplitudes of the scattered waves is fairly fast since the most difficult task -- the calculation of the diffraction transfer matrix -- only needs to be performed once for each type of body.

>From figures 58, 59 and 60, it can be seen that docks generally reflect the energy much more than the flexible plates. From this point of view, the rigid plates can be seen as a kind of intermediate setting.

For [math]\displaystyle{ \chi=\pi/5\approx 0.628 }[/math], the scattering angles are </math>\chi_{-4} \approx 2.33[math]\displaystyle{ , }[/math]\chi_{-3} \approx 1.89[math]\displaystyle{ , }[/math]\chi_{-2} \approx 1.51[math]\displaystyle{ , }[/math]\chi_{-1} \approx 1.12[math]\displaystyle{ (and their negative values). The docks particularly reflect in the direction \lt math\gt -\chi_{-1} }[/math] (beside [math]\displaystyle{ -\chi }[/math]). It can also be seen that the flexible plates already transmit most of the energy for this incident angle. The strong decrease in the amplitudes of their reflected waves appears at about [math]\displaystyle{ \chi\approx 0.58 }[/math]. The decrease of the amplitudes of the reflected waves for the rigid plates does not appear until a larger incident angle and is also not as strong. For the docks, such a strong decrease is not observed at all. Moreover, note that all three types of bodies reflect in the direction [math]\displaystyle{ -\chi_1 }[/math] fairly strongly for incident angles around [math]\displaystyle{ 0.91 }[/math] (where we have </math>-\chi_1 \approx -0.150[math]\displaystyle{ for }[/math]\chi = 0.91[math]\displaystyle{ ). [[Category:Infinite Array]] }[/math]