<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%è¡¨é¡Œ  SPMODEL ãƒ¬ãƒ•ã‚¡ãƒ¬ãƒ³ã‚¹ãƒžãƒ‹ãƒ¥ã‚¢ãƒ«
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%      2 æ¬¡å…ƒæ°´è·¯é&nbsp;˜åŸŸã®æ‹¡æ•£æ–¹ç¨‹å¼ã®è§£æžè§£
%      ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°å±•é–‹ã‚’åˆ©ç”¨ã—ãŸè¨ˆç®—ãƒã‚§ãƒƒã‚¯ç”¨
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%å±¥æ­´  2006/02/06 ä½œæˆ  ç«¹åºƒçœŸä¸€
%      2006/02/13 æ”¹è¨‚
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\documentclass[a4j,12pt]{jarticle}
\usepackage{Dennou6}

\Dtitle[2 æ¬¡å…ƒæ°´è·¯ã§ã®æ‹¡æ•£æ–¹ç¨‹å¼è§£]
       {2 æ¬¡å…ƒæ°´è·¯é&nbsp;˜åŸŸã§ã®\\æ‹¡æ•£åž‹æ–¹ç¨‹å¼ã®è§£æžè§£}
\Dauthor{ç«¹åºƒçœŸä¸€}
\Ddate{å¹³æˆ 18 å¹´ 2 æœˆ 13 æ—¥}

\Dparskip

\begin{document}

\maketitle

ã“ã®æ–‡ç«&nbsp;ã§ã¯, ãƒ•ãƒ¼ãƒªã‚¨--ãƒã‚§ãƒ“ã‚·ã‚§ãƒ•é–¢æ•°å±•é–‹ã‚’åˆ©ç”¨ã—ãŸ
2 æ¬¡å…ƒæ°´è·¯é&nbsp;˜åŸŸã§ã®æµä½“è¨ˆç®—ã®ãŸã‚ã® SPMODEL ãƒ©ã‚¤ãƒ–ãƒ©ãƒª(spml)ã®
ãƒ¢ã‚¸ãƒ¥ãƒ¼ãƒ« et\_module ã®ãƒ†ã‚¹ãƒˆã®ãŸã‚ã«, 
2 æ¬¡å…ƒæ°´è·¯é&nbsp;˜åŸŸã§ã®æ‹¡æ•£åž‹æ–¹ç¨‹å¼ã®ã•ã¾ã–ã¾ãªå¢ƒç•Œæ¡ä»¶ã«å¯¾ã™ã‚‹
è§£ã‚’(ã§ãã‚‹ã&nbsp;ã‘)è§£æžçš„ã«æ±‚ã‚ã‚‹ã“ã¨ã‚’è¡Œã†. 


  \section{æ‹¡æ•£æ–¹ç¨‹å¼}

    å¹… $d$ ã®å¹³è¡Œå£ã§ã¯ã•ã¾ã‚ŒãŸ 2 æ¬¡å…ƒæ°´è·¯é&nbsp;˜åŸŸã§ã®æ‹¡æ•£å•é¡Œã‚’
    è€ƒãˆã‚‹. æ”¯é…æ–¹ç¨‹å¼ã¯ 2 æ¬¡å…ƒæ‹¡æ•£æ–¹ç¨‹å¼
    %
    \begin{equation}
      \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼}
      \DP{\phi}{t} = \kappa \left(\DP[2]{}{x}+\DP[2]{}{y}\right)\phi
    \end{equation}
    %
    $\phi$ ã¯æ‹¡æ•£ã™ã‚‹ç‰©ç†é‡ã§ã‚ã‚Š, ä¾‹ãˆã°æ¸©åº¦, çµ„æˆæ¿ƒåº¦, ç£å&nbsp;´ãªã©ã§ã‚ã‚‹.
    $x$, $y$ ã¯ãã‚Œãžã‚Œå£ã«å¹³è¡ŒãŠã‚ˆã³åž‚ç›´æ–¹å‘ã®åº§æ¨™, 
    $\kappa$ ã¯æ‹¡æ•£çŽ‡(æ­£æ•°)ã§ã‚ã‚‹. 
    ä¸€èˆ¬æ€§ã‚’å¤±ã†ã“ã¨ãªã, $y$ åº§æ¨™ã®åŽŸç‚¹ã‚’æ°´è·¯ã®ä¸­å¤®ã«é¸ã¶ã“ã¨ã«ã™ã‚‹. 

    ä¸€èˆ¬çš„ãªå¢ƒç•Œæ¡ä»¶ã¯, å¢ƒç•Œã§ã®ç‰©ç†é‡ã¨ãã® $y$ å¾®åˆ†ã®
    ç·šå½¢çµåˆã§ä¸Žãˆã‚‰ã‚Œã‚‹(æ··åˆåž‹å¢ƒç•Œæ¡ä»¶). 
    %
    \begin{eqnarray}
      \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸Š}
      &amp; &amp; \alpha_{1,d/2}\DP{\phi}{y} + \alpha_{0,d/2}\phi =0, 
          \quad {\rm at} \quad y=\frac{d}{2},\\
      \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸‹}
      &amp; &amp; \alpha_{1,-d/2}\DP{\phi}{y} + \alpha_{0,-d/2}\phi =0, 
          \quad {\rm at} \quad y=-\frac{d}{2},
    \end{eqnarray}
    %
    ã“ã“ã§ $\alpha_{i,j}$ ã¯å®šæ•°ä¿‚æ•°ã§ã‚ã‚Š, ä»¥ä¸‹ã®æ¡ä»¶ã‚’æº€ãŸã—ã¦ã„ã‚‹
    ã‚‚ã®ã¨ã™ã‚‹. 
    %
    \begin{equation}
      \alpha_{1,d/2}\cdot\alpha_{0,d/2} \ge 0, \quad
      \alpha_{1,-d/2}\cdot\alpha_{0,-d/2} \le 0, \quad
    \end{equation}
    %
    ãŸã&nbsp;ã— $\alpha_{1,d/2}$ ã¨ $\alpha_{0,d/2}$, 
    ãŠã‚ˆã³ $\alpha_{1,-d/2}$ ã¨ $\alpha_{0,-d/2}$ ã¯åŒæ™‚ã«ã¯
    0 ã«ã¯ãªã‚‰ãªã„ã‚‚ã®ã¨ã™ã‚‹. 

    \subsection{ãƒ•ãƒ¼ãƒªã‚¨é–¢æ•°å±•é–‹}

      $x$ æ–¹å‘ã«ã¯å‘¨æœŸçš„ã§ã‚ã‚‹ã‹ã‚‰ãƒ•ãƒ¼ãƒªã‚¨é–¢æ•°ç³»ã§å±•é–‹ã—ã¦è§£ãã“ã¨ãŒ
      ã§ãã‚‹. ä»¥ä¸‹, ãã® 1 æ³¢æ•°æˆåˆ†ã®ã¿å–ã‚Šã&nbsp;ã—ã¦æ‰±ã†. 
      æ™‚é–“ã«é–¢ã—ã¦ã‚‚æŒ‡æ•°åž‹ã®è§£ã‚’ä»®å®šã—, 
      %
      \begin{equation}
        \Deqlab{ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \phi(x,y,t)=\tilde{\phi}(y)e^{ikx+\sigma t},
      \end{equation}
      %
      ã“ã‚Œã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, 
      %
      \begin{equation}
        \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
          \sigma \tilde{\phi} = 
              \kappa \left(\DD[2]{}{y}-k^2\right) \tilde{\phi},
      \end{equation}
      %
      å¢ƒç•Œæ¡ä»¶ã¯, 
      %
      \begin{eqnarray}
        \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸Šãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        &amp; &amp; \alpha_{1,d/2}\DD{\tilde{\phi}}{y} 
             + \alpha_{0,d/2}\tilde{\phi} =0, 
               \quad {\rm at} \quad y=\frac{d}{2},\\
        \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸‹ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        &amp; &amp; \alpha_{1,-d/2}\DD{\phi}{y} 
              + \alpha_{0,-d/2}\tilde{\phi} =0, 
               \quad {\rm at} \quad y=-\frac{d}{2}.
      \end{eqnarray}

    \subsection{æœ€å¤§æˆé•·çŽ‡ã®è¦‹ç©ã‚‚ã‚Š}

      æˆé•·çŽ‡ã®æœ€å¤§å€¤ã‚’é&nbsp;˜åŸŸç©åˆ†ã‹ã‚‰è¦‹ç©ã‚‚ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã« $\tilde{\phi}^*$ ã‚’ã‹ã‘ã¦
      $y$ æ–¹å‘ã«å…¨é&nbsp;˜åŸŸç©åˆ†ã‚’è¡Œã„, éƒ¨åˆ†ç©åˆ†ã‚’ç¹°ã‚Šè¿”ã™ã¨, 
      %
      \begin{eqnarray*}
        \sigma \int_{-d/2}^{d/2}|\tilde{\phi}|^2 dy &amp;=&amp;
              \int_{-d/2}^{d/2} dy \tilde{\phi}^*  
                \kappa \left(\DD[2]{}{y}-k^2\right) \tilde{\phi}\\
        &amp;=&amp; \kappa \int_{-d/2}^{d/2} dy \tilde{\phi}^*\DD[2]{\tilde{\phi}}{y}  
             -k^2 \kappa \int_{-d/2}^{d/2} |\tilde{\phi}|^2 dy\\
        &amp;=&amp;  \kappa\left[
                  \tilde{\phi}^*\DD{\tilde{\phi}}{y}\right]_{-d/2}^{d/2} 
            \kappa \int_{-d/2}^{d/2}\left|\DD{\tilde{\phi}}{y}\right|^2 dy 
             -\kappa k^2 \int_{-d/2}^{d/2} |\tilde{\phi}|^2 dy.
      \end{eqnarray*}
      %
      ã“ã“ã§, å¢ƒç•Œæ¡ä»¶ã‚’ç”¨ã„ã¦å³è¾ºç¬¬ 1 é&nbsp;…ç›®ã‚’è¦‹ç©ã‚‚ã‚‹ã¨
      \begin{eqnarray*}
        &amp; &amp; \left[\tilde{\phi}^*\DD{\tilde{\phi}}{y}\right]_{-d/2}^{d/2}\\
        &amp;=&amp; \left\{
        \begin{array}{cc}
          -\frac{\alpha_{1,d/2}}{\alpha_{0,d/2}}
               \left|\DD{\tilde{\phi}}{y}\right|^2_{y=d/2}
          &amp; (\alpha_{0,d/2}\neq0)\\
          -\frac{\alpha_{0,d/2}}{\alpha_{1,d/2}}|\tilde{\phi}|^2_{y=d/2}
          &amp; (\alpha_{1,d/2}\neq0)
        \end{array}
        \right\}
        - \left\{
        \begin{array}{cc}
          -\frac{\alpha_{1,-d/2}}{\alpha_{0,-d/2}}
               \left|\DD{\tilde{\phi}}{y}\right|^2_{y=-d/2}
          &amp; (\alpha_{0,-d/2}\neq0)\\
          -\frac{\alpha_{0,-d/2}}{\alpha_{1,-d/2}}|\tilde{\phi}|^2_{y=-d/2}
          &amp; (\alpha_{1,-d/2}\neq0)
        \end{array}
        \right\}\\
        &amp;=&amp; -\left\{
        \begin{array}{cc}
          \frac{\alpha_{1,d/2}}{\alpha_{0,d/2}}
               \left|\DD{\tilde{\phi}}{y}\right|^2_{y=d/2}
          &amp; (\alpha_{0,d/2}\neq0)\\
          \frac{\alpha_{0,d/2}}{\alpha_{1,d/2}}|\tilde{\phi}|^2_{y=d/2}
          &amp; (\alpha_{1,d/2}\neq0)
        \end{array}
        \right\}
        + \left\{
        \begin{array}{cc}
          \frac{\alpha_{1,-d/2}}{\alpha_{0,-d/2}}
               \left|\DD{\tilde{\phi}}{y}\right|^2_{y=-d/2}
          &amp; (\alpha_{0,-d/2}\neq0)\\
          \frac{\alpha_{0,-d/2}}{\alpha_{1,-d/2}}|\tilde{\phi}|^2_{y=-d/2}
          &amp; (\alpha_{1,-d/2}\neq0)
        \end{array}
        \right\}\\
        &amp; \le &amp; 0.
      \end{eqnarray*}
      %
      ã—ãŸãŒã£ã¦, 
      \begin{eqnarray*}
        \sigma \int_{-d/2}^{d/2}|\tilde{\phi}|^2 dy 
        &amp;\le&amp; 
            \kappa \int_{-d/2}^{d/2}\left|\DD{\tilde{\phi}}{y}\right|^2 dy 
             -\kappa k^2  \int_{-d/2}^{d/2} |\tilde{\phi}|^2 dy\\
        &amp;\le&amp; -\kappa k^2 \int_{-d/2}^{d/2} |\tilde{\phi}|^2 dy.
      \end{eqnarray*}
      %
      ã‚ˆã£ã¦æˆé•·çŽ‡ã®æœ€å¤§å€¤ãŒ
      %
      \begin{equation}
        \sigma &lt; -\kappa k^2 
      \end{equation}
      %
      ã¨è¦‹ç©ã‚‚ã‚‰ã‚Œã‚‹.

    \subsection{ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      ä¸¡å£ã§ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶
      %
      \begin{equation}
        \Deqlab{ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \tilde{\phi} = 0 \quad {\rm at} y=\pm \frac{d}{2},
      \end{equation}
      %
      ãŒä¸Žãˆã‚‰ã‚ŒãŸå&nbsp;´åˆã®è§£ã‚’æ±‚ã‚ã‚‹. 
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‹ã‚‰
      %
      \begin{equation}
        \left(\DD[2]{}{y}-k^2-\frac{\sigma}{\kappa}\right)\tilde{\phi}=0,
      \end{equation}
      %
      $\sigma \le -\kappa k^2$ ã‚ˆã‚Šã“ã®æ–¹ç¨‹å¼ã®ä¸€èˆ¬è§£ã¯
      %
      \begin{equation}
        \tilde{\phi} = A_1 \cos[m(\sigma,k)y] + A_2 \sin[m(\sigma,k)y],
      \end{equation}
      %
      ãŸã&nbsp;ã— $m(\sigma,k)$ ã¯ $y$ æ–¹å‘ã®æ³¢æ•°ã§ã‚ã‚Š
      %
      \begin{equation}
        m(\sigma,k)=\sqrt{-k^2-\frac{\sigma}{\kappa}}, \quad
        \sigma = -\kappa(k^2 + m^2)
      \end{equation}
      %
      ã‚’æº€ãŸã™. 

      å¢ƒç•Œæ¡ä»¶ãŒå¯¾ç§°ã§ã‚ã‚‹ã®ã§, è§£ã‚‚å¶é–¢æ•°ã¨å¥‡é–¢æ•°ã¨ã«ã‚ã‘ã¦
      åˆ¥ã€…ã«æ±‚ã‚ã‚‹ã“ã¨ãŒã§ãã‚‹. å¶é–¢æ•°ã®å&nbsp;´åˆã¯
      %
      \begin{displaymath}
        \tilde{\phi} = A_1 \cos[m(\sigma,k)y],
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶\Deqref{ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‚ˆã‚Š
      %
      \begin{displaymath}
        \cos\left[\frac{m(\sigma,k)d}{2}\right] = 0, \quad {\rm i.e.} \quad
        \frac{m(\sigma,k)d}{2} = \left(n+\frac{1}{2}\right)\pi, \quad
         m(\sigma,k) = \frac{(2n+1)\pi}{d}, 
      \end{displaymath}
      %
      ãŸã&nbsp;ã— $n=0,1,2,\ldots$ ã§ã‚ã‚‹. 
      ã—ãŸãŒã£ã¦
      %
      \begin{equation}
        \tilde{\phi} = A_1 \cos\left[\frac{(2n+1)\pi}{d}y\right], \quad
        \sigma = -\kappa\left(k^2 + \frac{(2n+1)^2\pi^2}{d^2}\right),
        \quad n=0,1,2,\ldots
      \end{equation}
      %
      å¥‡é–¢æ•°ã®å&nbsp;´åˆã¯
      %
      \begin{displaymath}
        \tilde{\phi} = A_1 \sin[m(\sigma,k)y],
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶\Deqref{ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‚ˆã‚Š
      %
      \begin{displaymath}
        \sin\left[\frac{m(\sigma,k)d}{2}\right] = 0, \quad {\rm i.e.} \quad
        \frac{m(\sigma,k)d}{2} = n\pi, \quad
         m(\sigma,k) = \frac{2n\pi}{d}, 
      \end{displaymath}
      %
      ãŸã&nbsp;ã— $n=1,2,\ldots$ ã§ã‚ã‚‹
      %
      ã—ãŸãŒã£ã¦
      %
      \begin{equation}
        \tilde{\phi} = A_2 \sin\left[\frac{2n\pi}{d}y\right], \quad
        \sigma = -\kappa\left(k^2 + \frac{4n^2\pi^2}{d^2}\right), 
        \quad n=1,2,\ldots
      \end{equation}

    \subsection{ä¸¡ç«¯ãƒŽã‚¤ãƒžãƒ³å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      ä¸¡å£ã§ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶
      %
      \begin{equation}
        \Deqlab{ä¸¡ç«¯ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \DD{\tilde{\phi}}{y} = 0 \quad {\rm at} y=\pm \frac{d}{2},
      \end{equation}
      %
      ãŒä¸Žãˆã‚‰ã‚ŒãŸå&nbsp;´åˆã®è§£ã‚’æ±‚ã‚ã‚‹. 
      å¢ƒç•Œæ¡ä»¶ãŒå¯¾ç§°ã§ã‚ã‚‹ã‹ã‚‰
      ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹ã®å&nbsp;´åˆã¨åŒæ§˜ã«ä¸€èˆ¬è§£ã‚’æ±‚ã‚, 
      ãã‚Œã‚’å¶é–¢æ•°ã¨å¥‡é–¢æ•°åˆ¥ã€…ã«æ±‚ã‚ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      å¶é–¢æ•°ã®å&nbsp;´åˆã¯
      %
      \begin{displaymath}
        \tilde{\phi} = A_1 \cos[m(\sigma,k)y],
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶\Deqref{ä¸¡ç«¯ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‚ˆã‚Š
      %
      \begin{displaymath}
        m\sin\left[\frac{m(\sigma,k)d}{2}\right] = 0, \quad {\rm i.e.} \quad
        \frac{m(\sigma,k)d}{2} = n\pi, \quad
         m(\sigma,k) = \frac{2n\pi}{d}, 
      \end{displaymath}
      %
      ãŸã&nbsp;ã— $n=0,1,2,\ldots$ ã§ã‚ã‚‹
      ã—ãŸãŒã£ã¦
      %
      \begin{equation}
        \tilde{\phi} = A_1 \cos\left[\frac{2n\pi}{d}y\right], \quad
        \sigma = -\kappa\left(k^2 + \frac{4n^2\pi^2}{d^2}\right), 
        \quad n=0,1,2,\ldots
      \end{equation}
      %
      å¥‡é–¢æ•°ã®å&nbsp;´åˆã¯
      %
      \begin{displaymath}
        \tilde{\phi} = A_1 \sin[m(\sigma,k)y],
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶\Deqref{ä¸¡ç«¯ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‚ˆã‚Š
      %
      \begin{displaymath}
        m\cos\left[\frac{m(\sigma,k)d}{2}\right] = 0, \quad {\rm i.e.} \quad
        \frac{m(\sigma,k)d}{2} = \left(n+\frac{1}{2}\right)\pi, \quad
         m(\sigma,k) = \frac{(2n+1)\pi}{d}, 
      \end{displaymath}
      %
      ãŸã&nbsp;ã— $n=1,2,\ldots$ ã§ã‚ã‚‹. 
      ã—ãŸãŒã£ã¦
      %
      \begin{equation}
        \tilde{\phi} = A_1 \sin\left[\frac{(2n+1)\pi}{d}y\right], \quad
        \sigma = -\kappa\left[k^2 + \frac{(2n+1)^2\pi^2}{d^2}\right],
        \quad n=1,2,\ldots
      \end{equation}

    \subsection{ç‰‡å´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬æ¡ä»¶ç‰‡å´ãƒŽã‚¤ãƒžãƒ³å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      ç‰‡å´ã§ãƒ‡ã‚£ãƒªã‚¯ãƒ¬æ¡ä»¶, ã‚‚ã†ä¸€æ–¹ã§ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ãŒ
      ä¸Žãˆã‚‰ã‚ŒãŸæ¬¡ã®å¢ƒç•Œæ¡ä»¶ã§ã®è§£ã‚’æ±‚ã‚ã‚‹. 
      %
      \begin{equation}
        \Deqlab{ä¸Šå´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬ä¸‹å´ãƒŽã‚¤ãƒžãƒ³å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \tilde{\phi} = 0 \quad {\rm at} \quad y=\frac{d}{2},\qquad
        \DD{\tilde{\phi}}{y} = 0 \quad {\rm at} \quad y=-\frac{d}{2},
      \end{equation}
      %
      ã“ã®å¢ƒç•Œæ¡ä»¶ã‚’æº€ãŸã™è§£ã¯, ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬å•é¡Œã®å¶æ•°è§£ã® 
      $0 \le y \le d/2$ ã®éƒ¨åˆ†ãŒç›¸å½“ã™ã‚‹. 
      åº§æ¨™å¤‰æ›
      %
      \begin{displaymath}
        \tilde{y} = -\frac{d}{2} + 2y, \quad 
        y = \frac{1}{2}\left(\tilde{y} +\frac{d}{2}\right)
      \end{displaymath}
      %
      ã‚’ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬å•é¡Œã®å¶é–¢æ•°è§£ã«é©ç”¨ã™ã‚‹ã¨
      %
      \begin{equation}
        \tilde{\phi} 
          = A_1 \cos\left[\frac{(2n+1)\pi}{d}
                          \frac{1}{2}\left(y+\frac{d}{2}\right)
                    \right]
          = A_1 \cos\left[\left(n + \frac{1}{2}\right)\frac{\pi}{d}
                         \left(y + \frac{d}{2}\right)
                       \right]
      \end{equation}
      %
      æˆé•·çŽ‡ã¯
      %
      \begin{equation}
        \sigma = -\kappa\left[k^2 
          + \left(n+\frac{1}{2}\right)^2\frac{\pi^2}{d^2}\right], 
      \end{equation}
      ã¨ãªã‚‹. 

      å¢ƒç•Œæ¡ä»¶ãŒå…¥ã‚Œæ›¿ã‚ã£ãŸ
      ç‰‡å´ã§ãƒŽã‚¤ãƒžãƒ³æ¡ä»¶, ã‚‚ã†ä¸€æ–¹ã§ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶
      %
      \begin{equation}
        \Deqlab{ä¸Šå´ãƒ‡ã‚£ãƒªã‚¯ãƒ¬ä¸‹å´ãƒŽã‚¤ãƒžãƒ³å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \DD{\tilde{\phi}}{y} = 0 \quad {\rm at} \quad y=\frac{d}{2},\qquad
        \tilde{\phi} = 0 \quad {\rm at} \quad y=-\frac{d}{2},
      \end{equation}
      %
      ã®è§£ã¯, å…ˆã®è§£ã‚’ $y\rightarrow -y$ ã¨åº§æ¨™å¤‰æ›ã™ã‚‹
      %
      \begin{equation}
        \tilde{\phi} 
          = A_1 \cos\left[\left(n + \frac{1}{2}\right)\frac{\pi}{d}
                         \left(-y + \frac{d}{2}\right)
                       \right]
          = A_1 \cos\left[\left(n + \frac{1}{2}\right)\frac{\pi}{d}
                         \left(y - \frac{d}{2}\right)
                       \right].
      \end{equation}
      %
      æˆé•·çŽ‡ã¯å…ˆã®ã‚‚ã®ã¨ãŠãªã˜ã§ã‚ã‚‹. 



    \subsection{é&nbsp;˜åŸŸå¤–éƒ¨ã®å¢ƒç•Œé&nbsp;˜åŸŸã§ãƒ©ãƒ—ãƒ©ã‚¹æ–¹ç¨‹å¼ã«å¾“ã†å&nbsp;´åˆ}

      æœ‰é™ã®ç†±ä¼å°ŽçŽ‡ã‚’æŒã¤ç‰©è³ªã§æ°´è·¯ä¸¡å´ãŒæº€ãŸã•ã‚Œã¦ã„ã‚‹å&nbsp;´åˆã®
      ç†±æ‹¡æ•£å•é¡Œ, ã‚ã‚‹ã„ã¯ãƒãƒ­ã‚¤ãƒ€ãƒ«ç£å&nbsp;´ã®æ‹¡æ•£ã®å•é¡Œã®å&nbsp;´åˆã«ã¯
      æ°´è·¯é&nbsp;˜åŸŸã®æ‹¡æ•£æ–¹ç¨‹å¼ã‚’æ¬¡ã®ã‚ˆã†ãªæ··åˆåž‹ã®å¢ƒç•Œæ¡ä»¶ã®ä¸‹ã§
      è§£ãã“ã¨ã«ãªã‚‹
      \footnote{
        é&nbsp;˜åŸŸå¤–éƒ¨ã®æ”¯é…æ–¹ç¨‹å¼ã¯
        %
        \begin{displaymath}
          \Dlapla \phi =0, 
        \end{displaymath}
        %
        $x$ ã«é–¢ã—ã¦ãƒ•ãƒ¼ãƒªã‚¨å¤‰æ›ã—, $y=\pm\infty$ ã§ç™ºæ•£ã—ãªã„è§£ã‚’
        æ±‚ã‚ã‚‹ã¨
        \begin{displaymath}
          \tilde{\phi} = C_1 e^{-|k|y} 
              \quad \left(x \ge \frac{d}{2} \right),\qquad
          \tilde{\phi} = C_2 e^{|k|y} 
              \quad \left(x \le -\frac{d}{2} \right),
        \end{displaymath}
        %
        å¢ƒç•Œ $y=d/2$ ã§å€¤ã¨ 1 éšŽå¾®åˆ†ãŒé€£ç¶šã§ã‚ã‚‹ã“ã¨ã‹ã‚‰
        \begin{eqnarray*}
          \tilde{\phi}|_{d/2} =  C_1 e^{-|k|d/2}, \quad
          \left.\DD{\tilde{\phi}}{y}\right|_{d/2} =  -|k| C_1 e^{-|k|d/2}
        \end{eqnarray*}
        %
        ã“ã‚Œã‚‰ã‚ˆã‚Š $C_1$ ã‚’æ¶ˆåŽ»ã™ã‚‹ã¨ $y=d/2$ ã§ã®å¢ƒç•Œæ¡ä»¶
        %
        \begin{displaymath}
          \left.\DD{\tilde{\phi}}{y}\right|_{d/2}+|k|\tilde{\phi}|_{d/2}=0
        \end{displaymath}
        %
        ãŒå¾—ã‚‰ã‚Œã‚‹. $y=-d/2$ ã®å&nbsp;´åˆã‚‚åŒæ§˜ã«è¨ˆç®—ã™ã‚‹ã¨, 
        %
        \begin{displaymath}
          \left.\DD{\tilde{\phi}}{y}\right|_{-d/2}-|k|\tilde{\phi}|_{-d/2}=0.
        \end{displaymath}
      }.
      %
      \begin{equation}
        \Deqlab{ä¸¡ç«¯æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \DD{\tilde{\phi}}{y} + |k|\tilde{\phi} = 0, 
          \quad {\rm at} \quad y=\frac{d}{2}, \qquad
        \DD{\tilde{\phi}}{y} - |k|\tilde{\phi} = 0, 
           \quad {\rm at} \quad y=-\frac{d}{2}, 
      \end{equation}
      %
      å¢ƒç•Œæ¡ä»¶ãŒå¯¾ç§°ã§ã‚ã‚‹ã‹ã‚‰
      ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹ã®å&nbsp;´åˆã¨åŒæ§˜ã«ä¸€èˆ¬è§£ã‚’æ±‚ã‚, 
      ãã‚Œã‚’å¶é–¢æ•°ã¨å¥‡é–¢æ•°åˆ¥ã€…ã«æ±‚ã‚ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      å¶é–¢æ•°ã®å&nbsp;´åˆã¯
      %
      \begin{displaymath}
        \tilde{\phi} = A_1 \cos[m(\sigma,k)y],
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶\Deqref{ä¸¡ç«¯æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‚ˆã‚Š
      %
      \begin{displaymath}
        -m(\sigma,k)\sin\left[\frac{m(\sigma,k)d}{2}\right]
            + |k| \cos\left[\frac{m(\sigma,k)d}{2}\right] = 0,
      \end{displaymath}
      %
      ã™ãªã‚ã¡ $m$ ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ãŒ
      %
      \begin{equation}
        \frac{md}{2}\tan\left[\frac{md}{2}\right] 
        = \frac{|k|d}{2},
      \end{equation}
      %
      ã¨ä¸Žãˆã‚‰ã‚Œã‚‹. ã“ã®è§£ã¯ $x\tan x$ ã®é–¢æ•°å½¢ã‹ã‚‰
      %
      \begin{displaymath}
        n\pi &lt; \frac{md}{2} &lt; \left(n+\frac{1}{2}\right)\pi, 
        \quad n=0,1,2,\ldots
      \end{displaymath}
      %
      ã«å­˜åœ¨ã™ã‚‹. 
      
      å¥‡é–¢æ•°ã®å&nbsp;´åˆã¯
      %
      \begin{displaymath}
        \tilde{\phi} = A_2 \sin[m(\sigma,k)y],
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶\Deqref{ä¸¡ç«¯æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã‚ˆã‚Š
      %
      \begin{displaymath}
        m(\sigma,k)\cos\left[\frac{m(\sigma,k)d}{2}\right]
            + |k| \sin\left[\frac{m(\sigma,k)d}{2}\right] = 0, \\
      \end{displaymath}
      %
      ã™ãªã‚ã¡ $m$ ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ãŒ
      %
      \begin{equation}
        \frac{md}{2}\cot\left[\frac{md}{2}\right] 
        = -\frac{|k|d}{2},
      \end{equation}
      %
      ã¨ä¸Žãˆã‚‰ã‚Œã‚‹. ã“ã®è§£ã¯ $x\cot x$ ã®é–¢æ•°å½¢ã‹ã‚‰
      %
      \begin{displaymath}
        \left(n+\frac{1}{2}\right)\pi &lt; \frac{md}{2} &lt; (n+1)\pi, 
        \quad n=0,1,2,\ldots
      \end{displaymath}
      %
      ã«å­˜åœ¨ã™ã‚‹. 


  \section{éžåœ§ç¸®æµä½“ã®ç²˜æ€§æ‹¡æ•£è§£}

    å¹… $d$ ã®å¹³è¡Œå£ã§ã¯ã•ã¾ã‚ŒãŸ 2 æ¬¡å…ƒæ°´è·¯é&nbsp;˜åŸŸã§ã®éžåœ§ç¸®æµä½“ã®é‹å‹•ã‚’è€ƒãˆã‚‹. 
    ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹è¿‘ä¼¼ã—ãŸ 2 æ¬¡å…ƒéžåœ§ç¸®æµä½“ã®æµç·šé–¢æ•° $\psi$ ã®
    å¾“ã†æ–¹ç¨‹å¼ã¯
    %
    \begin{equation}
        \Deqlab{ç²˜æ€§æµæ–¹ç¨‹å¼}
      \DP{}{t}\Dlapla\psi = \nu\Dlapla\Dlapla\psi,
    \end{equation}
    %
    ã“ã“ã§ $\Dlapla$ ã¯ 2 æ¬¡å…ƒã®ãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³
    \begin{displaymath}
      \Dlapla = \DP[2]{}{x}+ \DP[2]{}{y},
    \end{displaymath}
    %
    $x$, $y$ ã¯ãã‚Œãžã‚Œå£ã«å¹³è¡ŒãŠã‚ˆã³åž‚ç›´æ–¹å‘ã®åº§æ¨™, 
    $\nu$ ã¯å‹•ç²˜æ€§çŽ‡ã§ã‚ã‚‹. 
    ä¸€èˆ¬æ€§ã‚’å¤±ã†ã“ã¨ãªã, $y$ åº§æ¨™ã®åŽŸç‚¹ã‚’æ°´è·¯ã®ä¸­å¤®ã«é¸ã¶ã“ã¨ã«ã™ã‚‹. 

    é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯, å£ã‚’é€šã—ã¦ã®è³ªé‡æµæŸãŒ 0 ã¨ãªã‚‹ã“ã¨ã§ã‚ã‚‹:
    %
    \begin{equation}
      \Deqlab{é‹å‹•å­¦å¢ƒç•Œæ¡ä»¶}
      \psi = {\rm const.} \quad {\rm at} \quad y=\pm \frac{d}{2}
    \end{equation}
    %
    åˆæœŸã«ä¸Žãˆã‚‹ $x$ æ–¹å‘ã®å…¨è³ªé‡æµæŸãŒ 0 ã§ã‚ã‚‹å&nbsp;´åˆã&nbsp;ã‘ã‚’è€ƒãˆã‚‹ã¨, 
    ã“ã®ä¸€å®šå€¤ã‚’ä¸¡å£ã§ 0 ã¨é¸ã¶ã“ã¨ãŒã§ãã‚‹. 

    åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯ 2 ç¨®é¡žè€ƒãˆã‚‹. è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶(å¿œåŠ›ãªã—æ¡ä»¶)ã®å&nbsp;´åˆ
    %
    \begin{equation}
      \Deqlab{è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
      \DP[2]{\psi}{y} = {\rm const.} \quad {\rm at \ boundaries}.
    \end{equation}
    %
    ä¸€æ–¹, ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆ
    %
    \begin{equation}
      \Deqlab{ç²˜ç€æ¡ä»¶}
      \DP{\psi}{y} = {\rm const.} \quad {\rm at \ boundaries}.
    \end{equation}
    %
    ã§ã‚ã‚‹. 

    \subsection{ãƒ•ãƒ¼ãƒªã‚¨é–¢æ•°å±•é–‹}

      $x$ æ–¹å‘ã«ã¯å‘¨æœŸçš„ã§ã‚ã‚‹ã‹ã‚‰ãƒ•ãƒ¼ãƒªã‚¨é–¢æ•°ç³»ã§å±•é–‹ã—ã¦è§£ãã“ã¨ãŒ
      ã§ãã‚‹. ä»¥ä¸‹, ãã® 1 æ³¢æ•°æˆåˆ†ã®ã¿å–ã‚Šã&nbsp;ã—ã¦æ‰±ã†. 
      æ™‚é–“ã«é–¢ã—ã¦ã‚‚æŒ‡æ•°åž‹ã®è§£ã‚’ä»®å®šã—, 
      %
      \begin{equation}
        \Deqlab{æµç·šé–¢æ•°ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \psi(x,y,t)=\tilde{\psi}(y)e^{ikx+\sigma t},
      \end{equation}
      %
      ã“ã‚Œã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, 
      \begin{equation}
        \Deqlab{ç²˜æ€§æµæ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
          \sigma \left(\DD[2]{\tilde{\psi}}{y}-k^2\right) = 
              \nu \left(\DD[2]{}{y}-k^2\right)^2 \tilde{\psi},
      \end{equation}

      å¢ƒç•Œæ¡ä»¶ã¯, è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶(å¿œåŠ›ãªã—æ¡ä»¶)ã®å&nbsp;´åˆ
      %
      \begin{equation}
        \Deqlab{è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \tilde{\psi}=\DD[2]{\tilde{\psi}}{y} = 0 \quad 
        \quad {\rm at \ boundaries}.
      \end{equation}
      %
      ä¸€æ–¹, ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆ
      % 
      \begin{equation}
        \Deqlab{ç²˜ç€æ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \tilde{\psi}=\DD{\tilde{\psi}}{y} = 0 \quad 
        \quad {\rm at \ boundaries}.
      \end{equation}
      % 
      ã§ã‚ã‚‹. 

    \subsection{æœ€å¤§æˆé•·çŽ‡ã®è¦‹ç©ã‚‚ã‚Š}

      æˆé•·çŽ‡ã®æœ€å¤§å€¤ã‚’é&nbsp;˜åŸŸç©åˆ†ã‹ã‚‰è¦‹ç©ã‚‚ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      \Deqref{ç²˜æ€§æµæ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã« $\tilde{\psi}^*$ ã‚’ã‹ã‘ã¦
      $y$ æ–¹å‘ã«å…¨é&nbsp;˜åŸŸç©åˆ†ã‚’è¡Œã„, éƒ¨åˆ†ç©åˆ†ã‚’ç¹°ã‚Šè¿”ã™ã¨, 
      %
      \begin{eqnarray*}
         &amp; &amp; 
           \int_{-d/2}^{d/2} dy
             \sigma \tilde{\psi}^*\left(\DD[2]{\tilde{\psi}}{y}-k^2\right) 
         = \int_{-d/2}^{d/2} dy
             \nu \tilde{\psi}^* \left(\DD[2]{}{y}-k^2\right)^2 \tilde{\psi},\\
         &amp; &amp; 
           \sigma \int_{-d/2}^{d/2} dy\tilde{\psi}^*\DD[2]{\tilde{\psi}}{y}
            -\sigma k^2 \int_{-d/2}^{d/2}dy |\tilde{\psi}|^2 \\
         &amp; &amp;=  \nu\int_{-d/2}^{d/2} dy \tilde{\psi}^* \DD[4]{\tilde{\psi}}{y}
           -2\nu k^2\int_{-d/2}^{d/2} dy \tilde{\psi}^* \DD[2]{\tilde{\psi}}{y}
           + \nu k^4\int_{-d/2}^{d/2} dy |\tilde{\psi}|^2,\\
         &amp; &amp; 
           \sigma\left[\tilde{\psi}^*\DD[]{\tilde{\psi}}{y}\right]_{-d/2}^{d/2}
           -\sigma \int_{-d/2}^{d/2} dy \left|\DD[]{\tilde{\psi}}{y}\right|^2
           -\sigma k^2 \int_{-d/2}^{d/2}dy |\tilde{\psi}|^2 \\
         &amp; &amp;
         = \nu\left[\tilde{\psi}^* \DD[3]{\tilde{\psi}}{y}\right]_{-d/2}^{d/2} 
          - \nu\int_{-d/2}^{d/2} dy
               \DD{\tilde{\psi}^*}{y}\DD[3]{\tilde{\psi}}{y} \\&amp;&amp;
          -2\nu k^2\left[\tilde{\psi}^*\DD{\tilde{\psi}}{y}\right]_{-d/2}^{d/2}
          +2\nu k^2\int_{-d/2}^{d/2} dy \left|\DD{\tilde{\psi}}{y}\right|^2
           + \nu k^4\int_{-d/2}^{d/2} dy |\tilde{\psi}|^2,\\
         &amp; &amp; 
           -\sigma\int_{-d/2}^{d/2} dy\left[
               \left|\DD[]{\tilde{\psi}}{y}\right|^2
                 + k^2|\tilde{\psi}|^2 \right]\\ &amp; &amp;
         = - \nu\left[\DD{\tilde{\psi}^*}{y}
                          \DD[2]{\tilde{\psi}}{y}\right]_{-d/2}^{d/2} 
           + \nu\int_{-d/2}^{d/2} dy \left|\DD[2]{\tilde{\psi}}{y}\right|^2
           +2\nu k^2\int_{-d/2}^{d/2} dy \left|\DD{\tilde{\psi}}{y}\right|^2
           + \nu k^4\int_{-d/2}^{d/2} dy |\tilde{\psi}|^2,\\
         &amp; &amp; 
           -\sigma\int_{-d/2}^{d/2} dy\left[
               \left|\DD[]{\tilde{\psi}}{y}\right|^2
                 + k^2  |\tilde{\psi}|^2 \right]
         = \nu\int_{-d/2}^{d/2} dy \left[
                \left|\DD[2]{\tilde{\psi}}{y}\right|^2
                +2 k^2 \left|\DD{\tilde{\psi}}{y}\right|^2
               + k^4|\tilde{\psi}|^2 \right],\\
      \end{eqnarray*}
      %
      $\tilde{\psi}$ ãŒæ’ç­‰çš„ã« 0 ã§ãªã„ã®ã§, å·¦è¾ºã‚’ä¸ç­‰å¼ã§æŠ¼ãˆã‚‹ã“ã¨ãŒã§ãã¦
      %
      \begin{eqnarray*}
           -\sigma\int_{-d/2}^{d/2} dy\left[
               \left|\DD[]{\tilde{\psi}}{y}\right|^2
                 + k^2  |\tilde{\psi}|^2 \right]
          &amp;&gt;&amp; \nu\int_{-d/2}^{d/2} dy \left[
                 k^2 \left|\DD{\tilde{\psi}}{y}\right|^2
               + k^4|\tilde{\psi}|^2 \right]\\
          &amp;&amp;= \nu k^2\int_{-d/2}^{d/2} dy \left[
                   \left|\DD{\tilde{\psi}}{y}\right|^2
                 + k^2|\tilde{\psi}|^2 \right]
      \end{eqnarray*}
      %
      ã‚ˆã£ã¦æˆé•·çŽ‡ã®æœ€å¤§å€¤ãŒ
      %
      \begin{equation}
        \sigma &lt; - \nu k^2,
      \end{equation}
      %
      ã¨è¦‹ç©ã‚‚ã‚‰ã‚Œã‚‹. 

    \subsection{ä¸€èˆ¬è§£}

      æ”¯é…æ–¹ç¨‹å¼ã‚’å¤‰å½¢ã™ã‚‹ã¨
      %
      \begin{displaymath}
        \left(\DD[2]{}{y}-k^2\right)
           \left(\DD[2]{}{y}- k^2 - \frac{\sigma}{\nu}\right)
          \tilde{\psi}=0,
      \end{displaymath}
      %
      ã“ã®å¾®åˆ†æ–¹ç¨‹å¼ã®ä¸€èˆ¬è§£ã¯
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸€èˆ¬è§£}
        \tilde{\psi} = C_1 \cosh(ky) + C_2 \sinh(ky) + 
                       C_3 \cos(m(\sigma,k)y) +  C_4\sin(m(\sigma,k)y)
      \end{equation}
      %
      ãŸã&nbsp;ã— $m(\sigma,k)$ ã¯ $\sigma &lt; -\nu k^2$ ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµåˆ†æ•£é–¢ä¿‚}
        m(\sigma,k) = \sqrt{-\frac{\sigma}{\nu}-k^2}, \quad
        \sigma = -\nu (k^2 + m^2),
      \end{equation}
      %
      ã¨ä¸Žãˆã‚‰ã‚Œã‚‹. 

      ç‰¹ã«å¢ƒç•Œæ¡ä»¶ã®å¯¾ç§°æ€§ã‹ã‚‰å¥‡é–¢æ•°ã¨å¶é–¢æ•°ã«ã‚ã‘ã¦æ‰±ãˆã‚‹å&nbsp;´åˆã«ã¯, 
      å¶é–¢æ•°ã¯
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸€èˆ¬è§£å¶é–¢æ•°}
        \tilde{\psi} = A_1 \cosh(ky) + A_2 \cos[m(\sigma,k)y],
      \end{equation}
      %
      å¥‡é–¢æ•°ã¯
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸€èˆ¬è§£å¥‡é–¢æ•°}
        \tilde{\psi} = A_3 \sinh(ky) + A_4 \sin[m(\sigma,k)y],
      \end{equation}
      %
      ã¨ãªã‚‹. 

    \subsection{ä¸¡ç«¯è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ã®å&nbsp;´åˆ}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯
      %
      \begin{equation}
        \Deqlab{ä¸¡ç«¯è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
        \tilde{\psi}=\DD[2]{\tilde{\psi}}{y} = 0 \quad 
         {\rm at} \quad y=\pm \frac{d}{2}.
      \end{equation}
      %
      å¢ƒç•Œæ¡ä»¶ãŒå¯¾ç§°ãªã®ã§, è§£ã‚’å¶é–¢æ•°ã¨å¥‡é–¢æ•°ã«ã‚ã‘ã¦æ‰±ã†ã“ã¨ãŒã§ãã‚‹. 
      å¶é–¢æ•°ã®å&nbsp;´åˆ, 
      \Deqref{ç²˜æ€§æµä¸€èˆ¬è§£å¶é–¢æ•°}ã‚’\Deqref{ä¸¡ç«¯è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã«ä»£å…¥ã—ã¦, 
      %
      %
      \begin{eqnarray*}
        &amp; &amp; A_1 \cosh(kd/2) + A_2 \cos[m(\sigma,k)d/2] = 0, \\
        &amp; &amp; A_1 k^2 \cosh(kd/2) - A_2 m^2\cos[m(\sigma,k)d/2] = 0, 
      \end{eqnarray*}
      %
      è¡Œåˆ—ã®å½¢ã§è¡¨ã›ã°,
      \begin{displaymath}
        \left(
          \begin{array}[]{cc}
             \cosh(kd/2)   &amp; \cos[m(\sigma,k)d/2]   \\
             k^2 \cosh(kd/2) &amp;  -m^2\cos[m(\sigma,k)d/2]
          \end{array}
        \right)
        \left(
          \begin{array}[]{c}
             A_1 \\ A_2
          \end{array}
        \right)
        = 0,
      \end{displaymath}
      %
      $A_1 = A_2 = 0$ ä»¥å¤–ã®è§£ãŒå­˜åœ¨ã™ã‚‹ãŸã‚ã«ã¯, 
      ã“ã®ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ãªã‘ã‚Œã°ãªã‚‰ãªã„. 
      ã—ãŸãŒã£ã¦
      %
      \begin{eqnarray*}
       &amp; &amp;  -m^2\cosh(kd/2)\cos[m(\sigma,k)d/2]
         -k^2 \cosh(kd/2)\cos[m(\sigma,k)d/2]\\
       &amp;=&amp; -(m^2 + k^2)\cosh(kd/2)\cos[m(\sigma,k)d/2] \\
       &amp;=&amp; \frac{\sigma}{\nu}\cosh(kd/2)\cos[m(\sigma,k)d/2] =0.
      \end{eqnarray*}
      %
      ã—ãŸãŒã£ã¦
      \begin{equation}
        \frac{m(\sigma,k)d}{2} = \left(n+\frac{1}{2}\right)\pi, \quad
        n=0,1,2,\ldots \quad {\rm i.e.} \quad
        m(\sigma,k) = \frac{(2n+1)\pi}{d}
      \end{equation}
      %
      æˆé•·çŽ‡ã¯
      %
      \begin{equation}
        \sigma = -\nu ( k^2 + m^2 )
               = -\nu \left\{k^2 + \left[\frac{(2n+1)\pi}{d}\right]^2 \right\},
               \quad n=0,1,2,\ldots.
      \end{equation}
      %
      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯
      %
      \begin{displaymath}
        \left(
          \begin{array}[]{c}
             A_1 \\ A_2
          \end{array}
        \right)
        = 
        \left(
          \begin{array}[]{c}
            \cos[m(\sigma,k)d/2]\\
             \cosh(kd/2)  
          \end{array}
        \right)
        =
        \left(
          \begin{array}[]{c}
            0 \\
            \cosh(kd/2)  
          \end{array}
        \right)
      \end{displaymath}
      %
      ã—ãŸãŒã£ã¦è§£ã¯
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸¡ç«¯ã™ã¹ã‚Šå¶é–¢æ•°}
        \tilde{\psi} = A_2 \cos(my), \quad
        m = \frac{(2n+1)\pi}{d}, \quad
        \sigma = -\nu \left\{k^2 + \left[\frac{(2n+1)\pi}{d}\right]^2 \right\},
               \quad n=0,1,2,\ldots.
      \end{equation}

      å¥‡é–¢æ•°ã®å&nbsp;´åˆã‚‚åŒæ§˜ã«è¨ˆç®—ã‚’è¡Œã†ã¨, 
      %
      \begin{displaymath}
        \left(
          \begin{array}[]{cc}
             \sinh(kd/2)   &amp; \sin[m(\sigma,k)d/2]   \\
             k^2 \sinh(kd/2) &amp;  -m^2\sin[m(\sigma,k)d/2]
          \end{array}
        \right)
        \left(
          \begin{array}[]{c}
             A_3 \\ A_4
          \end{array}
        \right)
        = 0,
      \end{displaymath}
      %
      ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰
      \begin{displaymath}
         \frac{\sigma}{\nu}\sinh(kd/2)\sin[m(\sigma,k)d/2] =0.
      \end{displaymath}
      %
      ã—ãŸãŒã£ã¦, 
      %
      \begin{equation}
        \frac{m(\sigma,k)d}{2} = n\pi, \quad
        n=1,2,\ldots \quad {\rm i.e.} \quad
        m(\sigma,k) = \frac{2n\pi}{d}
      \end{equation}
      %
      æˆé•·çŽ‡ã¯
      %
      \begin{equation}
        \sigma = -\nu ( k^2 + m^2 )
               = -\nu \left[k^2 + \left(\frac{2n\pi}{d}\right)^2 \right],
               \quad n=1,2,\ldots.
      \end{equation}
      %
      ä¿‚æ•° $A_1, A_2$ ã®çµ„åˆã›ã¯
      %
      \begin{displaymath}
        \left(
          \begin{array}[]{c}
             A_1 \\ A_2
          \end{array}
        \right)
        = 
        \left(
          \begin{array}[]{c}
            \sin[m(\sigma,k)d/2] \\ \sinh(kd/2)
          \end{array}
        \right)
        =
        \left(
          \begin{array}[]{c}
            0 \\ \sinh(kd/2)
          \end{array}
        \right)
      \end{displaymath}
      %
      ã—ãŸãŒã£ã¦è§£ã¯
      %
       \begin{equation}
         \Deqlab{ç²˜æ€§æµä¸¡ç«¯ã™ã¹ã‚Šå¥‡é–¢æ•°}
         \tilde{\psi} = A_4 \sin(my), \quad
         m(\sigma,k) = \frac{2n\pi}{d}, \quad
         \sigma = -\nu \left[k^2 + \left(\frac{2n\pi}{d}\right)^2 \right],
                \quad n=1,2,\ldots.
       \end{equation}

    \subsection{ä¸¡ç«¯ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆ}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯
      %
      \begin{equation}
        \Deqlab{ä¸¡ç«¯ç²˜ç€æ¡ä»¶}
        \tilde{\psi}=\DD{\tilde{\psi}}{y} = 0 \quad 
         {\rm at} \quad y=\pm \frac{d}{2}.
      \end{equation}
      %
      å¢ƒç•Œæ¡ä»¶ãŒå¯¾ç§°ãªã®ã§, è§£ã‚’å¶é–¢æ•°ã¨å¥‡é–¢æ•°ã«ã‚ã‘ã¦æ‰±ã†ã“ã¨ãŒã§ãã‚‹. 
      å¶é–¢æ•°ã®å&nbsp;´åˆ, 
      \Deqref{ç²˜æ€§æµä¸€èˆ¬è§£å¶é–¢æ•°}ã‚’\Deqref{ä¸¡ç«¯ç²˜ç€æ¡ä»¶}ã«ä»£å…¥ã—ã¦, 
      %
      \begin{eqnarray*}
        &amp; &amp; A_1 \cosh(kd/2) + A_2 \cos(m(\sigma,k)d/2) = 0, \\
        &amp; &amp; A_1 k \sinh(kd/2) - A_2 m\sin(m(\sigma,k)d/2) = 0, 
      \end{eqnarray*}
      %
      è¡Œåˆ—ã®å½¢ã§è¡¨ã›ã°,
      \begin{displaymath}
        \left(
          \begin{array}[]{cc}
             \cosh(kd/2)   &amp; \cos(m(\sigma,k)d/2)   \\
             k \sinh(kd/2) &amp;  -m\sin(m(\sigma,k)d/2) 
          \end{array}
        \right)
        \left(
          \begin{array}[]{c}
             A_1 \\ A_2
          \end{array}
        \right)
        = 0,
      \end{displaymath}
      %
      $A_1 = A_2 = 0$ ä»¥å¤–ã®è§£ãŒå­˜åœ¨ã™ã‚‹ãŸã‚ã«ã¯, 
      ã“ã®ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ãªã‘ã‚Œã°ãªã‚‰ãªã„. 
      ã—ãŸãŒã£ã¦
      %
      \begin{displaymath}
        -m\cosh(kd/2)\sin(m(\sigma,k)d/2) -k \sinh(kd/2)\cos(m(\sigma,k)d/2)=0,
      \end{displaymath}
      %
      ã™ãªã‚ã¡, æˆé•·çŽ‡ $\sigma$ ã¨ $y$ æ–¹å‘ã®æ³¢æ•° $m(\sigma,k)$ ã‚’
      å®šã‚ã‚‹ãŸã‚ã®åˆ†æ•£é–¢ä¿‚ãŒ
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸¡ç«¯ç²˜ç€æ¡ä»¶å¶é–¢æ•°åˆ†æ•£é–¢ä¿‚}
        \frac{m(\sigma,k)d}{2}\tan\left(\frac{m(\sigma,k)d}{2}\right)
        = -\frac{kd}{2}\tanh\left(\frac{kd}{2}\right),
      \end{equation}
      %
      ã¨å¾—ã‚‰ã‚Œã‚‹. 
      ä¸Žãˆã‚‰ã‚ŒãŸ $k,d$ ã‹ã‚‰ã“ã®å¼ã‚’æº€ãŸã™ $m$ ã‚’æ•°å€¤çš„ã«æ±‚ã‚ã‚‹ã“ã¨ã«ã‚ˆã‚Š, 
      \Deqref{ç²˜æ€§æµåˆ†æ•£é–¢ä¿‚}ã‹ã‚‰
      æˆé•·çŽ‡ $\sigma$ ã¨ $y$ æ–¹å‘ã®æ³¢æ•° $m$ ã®çµ„åˆã›ã‚’å¾—ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      ã“ã‚Œã‚’æº€ãŸã™è§£ã¯ $x\tan x $ ã®é–¢æ•°å½¢ã‹ã‚‰
      %
      \begin{displaymath}
          \left(n+\frac{1}{2}\right)\pi &lt; \frac{m(\sigma,k)d}{2}   
          &lt; (n+1)\pi, \quad n=0,1,2,\ldots
      \end{displaymath}
      %
      ã«å­˜åœ¨ã™ã‚‹ã“ã¨ãŒã‚ã‹ã‚‹. 
      \Deqref{ç²˜æ€§æµä¸€èˆ¬è§£å¶é–¢æ•°}ã®å®šæ•°ä¿‚æ•°ã¯
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸¡ç«¯ç²˜ç€æ¡ä»¶å¶é–¢æ•°ä¿‚æ•°é–¢ä¿‚}
        \left(
          \begin{array}[]{c}
             A_1 \\ A_2
          \end{array}
        \right)
        =
        {\rm const.}\cdot
        \left(
          \begin{array}[]{c}
            \cos\left[\frac{m(\sigma,k)d}{2}\right]\\ 
            -\cosh\left(\frac{kd}{2}\right)
          \end{array}
        \right).
      \end{equation}


      å¥‡é–¢æ•°ã®å&nbsp;´åˆã‚‚åŒæ§˜ã«è¨ˆç®—ã™ã‚‹ã“ã¨ãŒã§ãã¦, 
      %
      \begin{displaymath}
        \left(
          \begin{array}[]{cc}
             \sinh(kd/2)   &amp; \sin(m(\sigma,k)d/2)   \\
             k \cosh(kd/2) &amp;  m\cos(m(\sigma,k)d/2) 
          \end{array}
        \right)
        \left(
          \begin{array}[]{c}
             A_3 \\ A_4
          \end{array}
        \right)
        = 0,
      \end{displaymath}
      %
      ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰,
      %
      \begin{displaymath}
        m\sinh(kd/2)\cos(m(\sigma,k)d/2) -k \cosh(kd/2)\sin(m(\sigma,k)d/2)=0,
      \end{displaymath}
      %
      ã—ãŸãŒã£ã¦åˆ†æ•£é–¢ä¿‚ã¯
      %
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸¡ç«¯ç²˜ç€æ¡ä»¶å¥‡é–¢æ•°åˆ†æ•£é–¢ä¿‚}
        \frac{m(\sigma,k)d}{2}\cot\left(\frac{m(\sigma,k)d}{2}\right)
        = \frac{kd}{2}\coth\left(\frac{kd}{2}\right),
      \end{equation}
      %
      ã¨å¾—ã‚‰ã‚Œã‚‹. 
      ä¸Žãˆã‚‰ã‚ŒãŸ $k,d$ ã‹ã‚‰ã“ã®å¼ã‚’æº€ãŸã™ $m$ ã‚’æ•°å€¤çš„ã«æ±‚ã‚ã‚‹ã“ã¨ã«ã‚ˆã‚Š, 
      \Deqref{ç²˜æ€§æµåˆ†æ•£é–¢ä¿‚}ã‹ã‚‰
      æˆé•·çŽ‡ $\sigma$ ã¨ $y$ æ–¹å‘ã®æ³¢æ•° $m$ ã®çµ„åˆã›ã‚’å¾—ã‚‹ã“ã¨ãŒã§ãã‚‹. 
      ã“ã‚Œã‚’æº€ãŸã™è§£ã¯ $x\cot x $ ã®é–¢æ•°å½¢ã‹ã‚‰
      %
      \begin{displaymath}
          n\pi &lt; \frac{m(\sigma,k)d}{2}   
          &lt; \left(n+\frac{1}{2}\right)\pi, \quad n=0,1,2,\ldots
      \end{displaymath}
      %
      ã«å­˜åœ¨ã™ã‚‹ã“ã¨ãŒã‚ã‹ã‚‹. 
      \Deqref{ç²˜æ€§æµä¸€èˆ¬è§£å¥‡é–¢æ•°}ã®å®šæ•°ä¿‚æ•°ã¯
      \begin{equation}
        \Deqlab{ç²˜æ€§æµä¸¡ç«¯ç²˜ç€æ¡ä»¶å¥‡é–¢æ•°ä¿‚æ•°é–¢ä¿‚}
        \left(
          \begin{array}[]{c}
             A_3 \\ A_4
          \end{array}
        \right)
        =
        {\rm const.}\cdot
        \left(
          \begin{array}[]{c}
            \sin\left[\frac{m(\sigma,k)d}{2}\right]\\ 
            -\sinh\left(\frac{kd}{2}\right)
          \end{array}
        \right).
      \end{equation}


    \subsection{ç‰‡å´ç²˜ç€æ¡ä»¶ç‰‡å´è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ã®å&nbsp;´åˆ}

      ã¾ãš $y=d/2$ ã§ç²˜ç€æ¡ä»¶, $y=-d/2$ ã§è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶
      ã®å&nbsp;´åˆã‚’è€ƒãˆã‚‹. ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯, 
      %
      \begin{equation}
        \Deqlab{ç‰‡å´ç²˜ç€æ¡ä»¶ç‰‡å´è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
        \tilde{\psi}=0 \quad {\rm at} \quad y=\pm\frac{d}{2}, \quad
        \DD{\tilde{\psi}}{y} = 0 \quad 
         {\rm at} \quad y=\frac{d}{2}, \quad
        \tilde{\psi}=\DD[2]{\tilde{\psi}}{y} = 0 \quad 
         {\rm at} \quad y=-\frac{d}{2}.
      \end{equation}
      %
      ä¸¡ç«¯ç²˜ç€æ¡ä»¶ã®å¥‡é–¢æ•°è§£ã® $0 \le y \le d/2$ ã®éƒ¨åˆ†ãŒ
      ã¡ã‚‡ã†ã©ã“ã®å¢ƒç•Œæ¡ä»¶ã‚’æº€ãŸã™è§£ã¨ãªã£ã¦ã„ã‚‹. 
      ãã“ã§, é&nbsp;˜åŸŸ $-d\le y \le d$ ã§ã®å¥‡é–¢æ•°è§£ã‚’æ±‚ã‚ã‚‹ã¨, 
      ä¸€èˆ¬è§£
      %
      \begin{equation}
        \tilde{\psi} 
           = A_3 \sinh(ky)+ A_4 \sin\left[ m(\sigma,k) y \right]
      \end{equation}
      %
      ã«å¯¾ã™ã‚‹åˆ†æ•£é–¢ä¿‚ã‚’\Deqref{ç²˜æ€§æµä¸¡ç«¯ç²˜ç€æ¡ä»¶å¥‡é–¢æ•°åˆ†æ•£é–¢ä¿‚}
      ã«ã¦ $d/2 \rightarrow d$ ã¨ã—ã¦
      % 
      \begin{equation}
        m(\sigma,k)d\cot[m(\sigma,k)d]=kd\coth(kd),
      \end{equation}
      %
      ã§ã‚ã‚‹. 
      \Deqref{ç²˜æ€§æµä¸€èˆ¬è§£å¥‡é–¢æ•°}ã®å®šæ•°ä¿‚æ•°ã¯
      \begin{equation}
        \left(
          \begin{array}[]{c}
             A_3 \\ A_4
          \end{array}
        \right)
        =
        {\rm const.}\cdot
        \left(
          \begin{array}[]{c}
            \sin(md) \\ -\sinh(kd)
          \end{array}
        \right).
      \end{equation}
      %
      ã“ã®ã‚ˆã†ã«ã—ã¦æ±‚ã¾ã£ãŸè§£ã‚’ $y$ æ–¹å‘ã« $-d/2$ å¹³è¡Œç§»å‹•ã•ã›ãŸ
      ã‚‚ã®ãŒæ±‚ã‚ã‚‹è§£ã§ã‚ã‚‹. 
      %
      \begin{equation}
        \tilde{\psi} 
           = A_3 \sinh\left[k\left(y+\frac{d}{2}\right)\right]
             + A_4 \sin\left[m\left(y+\frac{d}{2}\right)\right]
      \end{equation}
      %
      æˆé•·çŽ‡ã¯\Deqref{ç²˜æ€§æµåˆ†æ•£é–¢ä¿‚}ã‚ˆã‚Š, 
      å®šã‚ã‚‰ã‚ŒãŸ $m$ ã‹ã‚‰ $\sigma = -\nu(k^2+m^2)$ ã§è¨ˆç®—ã•ã‚Œã‚‹. 

      å¢ƒç•Œæ¡ä»¶ãŒé€†ã«ãªã£ãŸ,  $y=d/2$ ã§è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶, $y=-d/2$ ã§ç²˜ç€æ¡ä»¶
      ã®å&nbsp;´åˆ, ã™ãªã‚ã¡
      %
      \begin{equation}
        \Deqlab{ç‰‡å´è‡ªç”±ã™ã¹ã‚Šç‰‡å´ç²˜ç€æ¡ä»¶}
        \tilde{\psi}=0 \quad {\rm at} \quad y=\pm\frac{d}{2}, \quad
        \tilde{\psi}=\DD[2]{\tilde{\psi}}{y} = 0 \quad 
         {\rm at} \quad y=\frac{d}{2}, \quad
        \DD{\tilde{\psi}}{y} = 0 \quad 
         {\rm at} \quad y=-\frac{d}{2}.
      \end{equation}
      %
      ã®è§£ã¯, ä¸Šã®è§£ã® $y$ æ–¹å‘ã®å¹³è¡Œç§»å‹•ã‚’ $d/2$ ã«å¤‰ãˆã‚‹ã&nbsp;ã‘ã§å¾—ã‚‰ã‚Œã‚‹.
      %
      \begin{equation}
        \tilde{\psi} 
           = A_3 \sinh\left[k\left(y-\frac{d}{2}\right)\right]
             + A_4 \sin\left[m\left(y-\frac{d}{2}\right)\right]
      \end{equation}

\end{document}

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