# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 4 Dissipation Radiation Field Analysis of Antenna Array of Limited Dimensions
### Solution to Distraction Problem
\[\underline{\phi}_{q}(x,y)=\sum_{k}\underline{\phi}_{qk}\underline{\Psi}_{k}(x,y), \tag{164}\]  
\[\underline{I}(x,y)=\sum_{q,k}\underline{\dot{j}_{q}\underline{\phi}_{qk}} \overline{H}_{mnk}\epsilon_{k}(x,y), \tag{165}\]  
\[\underline{E}_{l}^{i}(x,y,0)=\sum_{k}v_{k}^{i}\underline{\Psi}_{k}(x,y), \tag{166}\]  
\[\underline{E}_{l}^{\prime}(x,y,0)=\sum_{k}\left(1\pm I_{k}\right)v_{k}^{i} \underline{\Psi}_{k}(x,y), \tag{167}\]  
\[\underline{E}_{l}^{s}(x,y,0)=\sum_{k}v_{k}^{s}\underline{\Psi}_{k}(x,y). \tag{168}\]  
By relating the scattered field to the induced currents using [PERSON]'s function,  
\[\underline{E}_{tmn}^{s}(x,y,0)=\sum_{k}\overline{\underline{\alpha}}_{k} \overline{I}_{k}\overline{H}_{mnk}\epsilon_{k}(x,y) \tag{169}\]  
and imposing, using the Galerkin method, the condition of zeroing of the total field on the surface of the antenna's metal elements on the representation of the field and induced currents:  
\[\underline{E}_{tmn}(x,y,0)A_{MN}=(\underline{E}_{l}^{t}(x,y,0)+\underline{E}_ {tmn}^{s}(x,y,0))A_{MN}=0 \tag{170}\]  
As a result, we obtain a system of two matrix algebraic equations:  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]0  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]1  
where, respectively,  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]2  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]3  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]4  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]5  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]6  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]7  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]8  
\[\underline{E}^{s}(x,y,0)=\left(\underline{G}*\underline{l}*H_{mn}\right)(x,y )=\sum_{k}\overline{\underline{G}}_{k}\overline{l}_{k}\overline{H}_{mnk}e_{ k}(x,y) \tag{162}\]9  
in the cell defined by the indices of the window function \(H_{mn}\). By solving the above system of algebraic equations for subsequent window functions \(H_{mn}\), we obtain the distribution of current \(\underline{J}_{mn}\) on subsequent cells (\(m\),\(n\)) of the limited periodic antenna array, i.e., we obtain \(M\times N\) solutions of the above system of equations for \(M\times N\) waveforms of the function \(H_{mn}(x\),\(y)\). The total distribution of currents on all cells simultaneously can be presented in the following form:  
\[\overline{\underline{l}}_{k}=\sum_{q}\,\underline{\oint}_{q}\,\underline{ \oint}_{qk} \tag{163}\]0  
where  
\[\overline{\underline{l}}_{k}=\sum_{q}\,\underline{\oint}_{q}\,\underline{ \oint}_{qk} \tag{163}\]1The final solution, however, requires an unambiguous connection of the current distribution with the incident and scattered fields:  
\[\overline{\underline{l}}_{k}=\sum_{q}\,\underline{\oint}_{q}\,\underline{ \oint}_{qk} \tag{163}\]2  
That is, it requires a single, unique solution to the above system of equations, considering the current distributions on all antenna cells simultaneously.