Difference between revisions of "Interaction Theory for Infinite Arrays"

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\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
 
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
 
\Big[ \tilde{D}_{n\nu} + (-1)^\nu  
 
\Big[ \tilde{D}_{n\nu} + (-1)^\nu  
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=1,j \neq  l}^{\infty} P_{j-l} K_{\tau - \nu}  (k_n
+
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq  l}^{\infty} P_{j-l} K_{\tau - \nu}  (k_n
 
R |j-l|) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],  
 
R |j-l|) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],  
 
</math></center>
 
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.
+
<math>m \in \mathbb{N}</math>, <math>\mu,l \in \mathbb{Z}</math>.
  
 
[[Category:Infinite Array]]
 
[[Category:Infinite Array]]

Revision as of 14:45, 18 July 2006

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 R |j-l|) \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].