Interaction Theory for Infinite Arrays

Introduction

We want to use the Kagemoto and Yue Interaction Theory to derive a system of equations for the infinite array.

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 l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n |j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big], }[/math]

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

Introducing the constants

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