Interaction Theory for Infinite Arrays

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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

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

$m \in \mathbb{N}$, $\mu \in \mathbb{Z}$, $l=1,\dots,N$.

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 $R$, we have $R_{jl} = |j-l| R$ and

$\varphi_{n} = \begin{cases} \pi, & n>0,\\ 0, & n<0. \end{cases}$

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

$\ P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},$

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

Therefore, the system simplifies to

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

Introducing the constants

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

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

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

$m \in \mathbb{N}$, $\mu \in \mathbb{Z}$. 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 $A_{m\mu}^l = P_l A_{m\mu}$.

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 $p=2\pi/R$, define the scattering angles $\chi_m$ by

$\chi_m = \cos^{-1} (\psi_m/k) \quad \mathrm{where} \quad \psi_m = k \cos \chi + m p$

and write $\psi$ for $\psi_0$. Also note that $\chi_0 = \chi$ by definition. If $|\psi_m|<k$, i.e. if

$-1 < \cos \chi +\frac{mp}{k}<1,$

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

$\cos^{-1} t = \begin{cases} \mathrm{i} \cosh^{-1} t, & t> 1,\\ \pi-\mathrm{i} \cosh^{-1} (-t) & t<-1, \end{cases}$

with $\cosh^{-1} t = \log \left(t+\sqrt{t^2-1}\right)$ for $t>1$.

For the total potential we have

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

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

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

$H^{(1)}_\mu (kr) \mathrm{e}^{\mathrm{i} \mu \theta}= \frac{(-i)^{\mu+1}}{\pi} \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k\gamma(t)|y|}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}\,\mathrm{e}^{i \mu \sgn(y)\cos^{-1} t} \,\mathrm{d} t,$

in which $x=r\cos\theta$, $y=r\sin\theta$ and $\gamma(t)$ is defined for real $t$ by

$\gamma(t) = \begin{cases} -\mathrm{i} \sqrt{1-t^2} & |t| \leq 1 \\ \sqrt{t^2-1} & |t|>1, \end{cases}$

we get

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

in which we have used the Poisson summation formula,

$\sum_{m=-\infty}^\infty \int_{-\infty}^{\infty} f(u)\, \mathrm{e}^{-\mathrm{i} mu} \,\mathrm{d} u= 2\pi \sum_{m=-\infty}^\infty f(2m\pi).$

The only terms which contribute to the far field are those for which $|\psi_m|<k$. Thus, as $y\to\pm\infty$, the far field consists of a set of plane waves propagating in the directions $\theta=\pm\chi_m$:

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

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

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

Note that the primary reflection and transmission coefficients are recovered by $A^-_0$ and $1 + A^+_0$, respectively.

It is implicit in all the above that $\sin\chi_m\neq 0$ for any $m$. If $\sin\chi_m=0$ 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 $x = \infty$ (if $\chi_m = 0$) or towards $x=-\infty$ (if $\chi_m = \pi$). 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 $Rk$.

The efficient computation of the $\sigma_{\nu}^0$

The constants $\sigma_{\nu}^0$ (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

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

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

$\tilde{\sigma}_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR)$

as the $\sigma^0_\nu$ are then determined by $\sigma^0_\nu = \pi/2 \,\, i^{\nu+1} \, \tilde{\sigma}^0_\nu$.

An efficient way of computing the $\tilde{\sigma}_{\nu}^0$ is given in Linton 1998 and the results are briefly outlined in our notation. Noting that $H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu} H^{(1)}_{\nu} (\,\cdot\,)$, it suffices to discuss the computation of the $\sigma_{\nu}^0$ for non-negative $\nu$.

Referring to Linton 1998, the constants $\tilde{\sigma}_{\nu}^0$ can be written as

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

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

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

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

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

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

where $\delta_{mn}$ 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 $(x,y)$-plane but the cross-sections at any height are identical) in an acoustic medium.

For this problem, the $z$-dependence can be omitted and the above theory applies with the following modifications:

1. The Dispersion Relation for a Free Surface is replaced by $k=\omega / c$ where $c$ is the speed of sound in the medium under consideration

and the Dispersion Relation for a Free Surface is omitted.

1. All factors $\cos k_m(z+d)$, $\cos k_m(c+d)$, $\cos k_m d$

and $f_0$ are replaced by 1.

1. The factor $N_0$ in \eqref{green_d} is $k/\pi$.

Note that there are no evanescent modes in this problem, i.e. the sums over $m$ and $n$ in the eigenfunction expansions only contain the terms for $m=0$ and $n=0$. Moreover, we have $k_0 = - \mathrm{i} \, \omega /c$.

For circular cylinders, i.e. cylinders which have a circular cross-section, this problem has been considered by Linton and Evans 1993.