<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«
%
%      2 æ¬¡å…ƒå††ç›¤é&nbsp;˜åŸŸã®æ‹¡æ•£æ–¹ç¨‹å¼ã®è§£æžè§£
%      æ–¹ä½è§’ãƒ•ãƒ¼ãƒªã‚¨å‹•å¾„å¤šé&nbsp;…å¼å±•é–‹ã‚’åˆ©ç”¨ã—ãŸè¨ˆç®—ãƒã‚§ãƒƒã‚¯ç”¨
%
%å±¥æ­´  2008/04/13 ä½œæˆ  ç«¹åºƒçœŸä¸€
%
\documentclass[a4j,12pt]{jarticle}
\usepackage{Dennou6}

\Dtitle[å††ç›¤é&nbsp;˜åŸŸã§ã®æ‹¡æ•£æ–¹ç¨‹å¼è§£]
       {2 æ¬¡å…ƒå††ç›¤é&nbsp;˜åŸŸã§ã®\\æ‹¡æ•£åž‹æ–¹ç¨‹å¼ã®è§£æžè§£}
\Dauthor{ç«¹åºƒçœŸä¸€}
\Ddate{å¹³æˆ 20 å¹´ 4 æœˆ 13 æ—¥}

\Dparskip

\begin{document}

\maketitle

ã“ã®æ–‡ç«&nbsp;ã§ã¯, ãƒ•ãƒ¼ãƒªã‚¨ç´šæ•°--å¤šé&nbsp;…å¼é–¢æ•°å±•é–‹ã‚’åˆ©ç”¨ã—ãŸ
2 æ¬¡å…ƒå††ç›¤é&nbsp;˜åŸŸã§ã®æµä½“è¨ˆç®—ã®ãŸã‚ã® SPMODEL ãƒ©ã‚¤ãƒ–ãƒ©ãƒª(spml)ã®
ãƒ¢ã‚¸ãƒ¥ãƒ¼ãƒ« eq\_module ã®ãƒ†ã‚¹ãƒˆã®ãŸã‚ã«, 
2 æ¬¡å…ƒå††ç›¤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}

    åŠå¾„ $ao$ ã®å††ç›¤å†…ã® 2 æ¬¡å…ƒé&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{\psi}{t} = \kappa \Dlapla \psi
    \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‚‹. 
    $\Dlapla$ ã¯ 3 æ¬¡å…ƒãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã§ã‚ã‚Š, 
    2 æ¬¡å…ƒæ¥µåº§æ¨™ç³»$ (r,\phi)$ ã§ã®è¡¨ç¾ã¯
    %
    \begin{equation}
      \Dlapla = \frac{1}{r}\DP{}{r}\left(r\DP{}{r}\right)
             + \frac{1}{r^2}\DP[2]{}{\phi}.
    \end{equation}
    %

    ä¸€èˆ¬çš„ãªå¢ƒç•Œæ¡ä»¶ã¯, å¤–å´å¢ƒç•Œå††å‘¨ã§ã®ç‰©ç†é‡ã¨ãã® $r$ 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»¶}
      &amp; &amp; \alpha_{1}\DP{\psi}{r} + \alpha_{0}\psi =0, 
          \quad {\rm at} \quad r=a,\\
    \end{eqnarray}
    %
    ã“ã“ã§ $\alpha_{i}$ 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}\cdot\alpha_{0} \le 0, \quad
    \end{equation}
    %
    ãŸã&nbsp;ã— $\alpha_1$ ã¨ $\alpha_0$ ã¯åŒæ™‚ã«ã¯ 
    0 ã«ã¯ãªã‚‰ãªã„ã‚‚ã®ã¨ã™ã‚‹. 

    \subsection{ãƒ•ãƒ¼ãƒªã‚¨ç´šæ•°å±•é–‹}

      æ–¹ä½è§’æ–¹å‘ã«ã¯ãƒ•ãƒ¼ãƒªã‚¨ç´šæ•°å±•é–‹ã—ã¦è§£ãã“ã¨ãŒã§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‚¨ç´šæ•°å±•é–‹}
        \psi(x,y,t)=\tilde{\psi}_{m}(r)e^{im\phi+\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‚¨ç´šæ•°å±•é–‹}
          \sigma \tilde{\psi}_{m} = 
              \kappa \left[
                \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\phi}_{m}}{r}\right)
                  -\frac{m^2}{r^2}\tilde{\phi}_{m}\right],
      \end{equation}
      %
      å¢ƒç•Œæ¡ä»¶ã¯, 
      %
      \begin{equation}
        \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ä¸Šæ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨ç´šæ•°å±•é–‹}
        \alpha_{1}\DD{\tilde{\psi}_{m}}{r} 
             + \alpha_{0}\tilde{\psi}_{m} =0 
               \quad {\rm at} \quad r=a.
      \end{equation}
      %

    \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–‹} $r\tilde{\psi}_{m}^*$ ã‚’ã‹ã‘ã¦
      $r$ 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_0^a r|\tilde{\psi}_m|^2 dr &amp;=&amp;
               \int_0^a dr r \tilde{\psi}_m^*  
                \kappa \left[
                    \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
                   -\frac{m^2}{r^2}\tilde{\psi}_m\right]\\
        &amp;=&amp; \kappa \int_0^a dr \tilde{\psi}_m^*
                     \DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
             -\kappa m^2 \int_0^a \frac{1}{r}|\tilde{\psi}_m|^2 dr\\
        &amp;=&amp;  \kappa\left[
                  r\tilde{\psi}_m^*\DD{\tilde{\psi}_m}{r}\right]_0^a 
             -\kappa \int_0^a 
                  r\left|\DD{\tilde{\psi}_m}{r}\right|^2 dr
             -\kappa m^2 \int_0^a \frac{1}{r}|\tilde{\psi}_m|^2 dr\\
      \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{displaymath}
         \left[r\tilde{\psi}_m^*\DD{\tilde{\psi}_m}{r}\right]_0^a
        =  \left\{
        \begin{array}{cc}
          \Ddsty -\frac{\alpha_{1}}{\alpha_{0}}
               a^2\left|\DD{\tilde{\psi}_m}{r}\right|^2_{r=a}
          &amp; (\alpha_{0}\neq0)\\
          \Ddsty -\frac{\alpha_{0}}{\alpha_{1}}
               a^2|\tilde{\psi}_m|^2_{r=a}
          &amp; (\alpha_{1}\neq0)
        \end{array}
        \right\}
         \le  0.
      \end{displaymath}
      %
      ã—ãŸãŒã£ã¦, 
      %
      \begin{eqnarray*}
        \sigma \int_0^a r|\tilde{\psi}_m|^2 dr
        &amp;\le&amp; 
             -\kappa \int_0^a 
                  r\left|\DD{\tilde{\psi}_m}{r}\right|^2 dr
             -\kappa m^2 \int_0^a \frac{1}{r}|\tilde{\psi}_m|^2 dr
         &lt; -\kappa m^2 \int_0^a |\tilde{\psi}_m|^2 dr,\\
        \sigma &amp;&lt;&amp; 
            \frac{\Ddsty -\kappa m^2\int_0^a \frac{1}{r}|\tilde{\psi}_m|^2 dr}
                 {\Ddsty \int_0^a r|\tilde{\psi}_m|^2 dr}. 
      \end{eqnarray*}
      %
      ã‚ˆã£ã¦æˆé•·çŽ‡ã®æœ€å¤§å€¤ã¯è²&nbsp;ã§ã‚ã‚‹ã“ã¨ãŒã‚ã‹ã‚‹. 

    \subsection{ä¸€èˆ¬è§£}

      æ”¯é…æ–¹ç¨‹å¼ã‚’å¤‰å½¢ã™ã‚‹ã¨
      %
      \begin{displaymath}
        \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
          +\left[\frac{-\sigma}{\kappa} -\frac{m^2}{r^2}\right]\tilde{\psi}_m=0.
      \end{displaymath}
      %
      ã•ã‚‰ã«å‹•å¾„åº§æ¨™ã‚’
      \begin{equation}
        \tilde{r} = \sqrt{\frac{-\sigma}{\kappa}}r, \quad
        r  = \sqrt{\frac{\kappa}{-\sigma}}\tilde{r},
      \end{equation}
      %
      ã¨å¤‰å½¢ã™ã‚‹ã¨,
      %
      \begin{displaymath}
        \frac{1}{\tilde{r}}\DD{}{\tilde{r}}
             \left(\tilde{r}\DD{\tilde{\psi}_m}{\tilde{r}}\right)
          +\left[1 -\frac{m^2}{\tilde{r}^2}\right]\tilde{\psi}_m,
      \end{displaymath}
      %
      ã“ã®å¾®åˆ†æ–¹ç¨‹å¼ã¯ãƒ™ãƒƒã‚»ãƒ«ã®å¾®åˆ†æ–¹ç¨‹å¼ã§ã‚ã‚Š, 
      ãã®åŽŸç‚¹ã§ç‰¹ç•°æ€§ã®ãªã„ä¸€èˆ¬è§£ã¯ $m$ æ¬¡ã®ç¬¬ 1 ç¨®ãƒ™ãƒƒã‚»ãƒ«å‡½æ•° 
      $J_m(\tilde{r})$ã§è¡¨ã•ã‚Œã‚‹.
      %
      \begin{equation}
        \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}
        \tilde{\psi}_m = A_1 J_m(\tilde{r}) 
               = A_1 J_m\left(\sqrt{\frac{-\sigma}{\kappa}}r\right) 
      \end{equation}


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

      å¤–å´ã§ã®å¢ƒç•Œæ¡ä»¶ãŒãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹ã®
      %
      \begin{equation}
        \Deqlab{ãƒ‡ã‚£ãƒªã‚¯ãƒ¬åž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
        \tilde{\psi}_m =0 \quad {\rm at} \quad r=a.
      \end{equation}
      %
      å&nbsp;´åˆã‚’è€ƒãˆã‚‹. ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã™ã‚‹ã¨
      %
      \begin{equation}
           J_l\left(a \alpha\right) =0.
      \end{equation}
      % 
      ãŸã&nbsp;ã— $\Ddsty \alpha = \sqrt{\frac{-\sigma}{\kappa}}$ ã¨ãŠã„ãŸ. 
      ãƒ™ãƒƒã‚»ãƒ«å‡½æ•°ã®é›¶ç‚¹ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\kappa \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 
      ä¿‚æ•° $A_1$ ã¯åˆæœŸæ¡ä»¶ã‹ã‚‰å®šã¾ã‚‹. 

    \subsection{å¤–å´ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆ}

      å¤–å´ã§ã®å¢ƒç•Œæ¡ä»¶ãŒãƒŽã‚¤ãƒžãƒ³åž‹ã®
      %
      \begin{equation}
        \Deqlab{ãƒŽã‚¤ãƒžãƒ³åž‹å¢ƒç•Œæ¡ä»¶çƒé¢èª¿å’Œå‡½æ•°å±•é–‹}
        \DD{\tilde{\psi}_m}{r} =0 \quad {\rm at} \quad y=a, 
      \end{equation}
      %
      ã®å&nbsp;´åˆã‚’è€ƒãˆã‚‹. ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
           \alpha J'_m\left(a \alpha\right)  = 0. 
      \end{equation}
      % 
      ãŸã&nbsp;ã— $\Ddsty \alpha = \sqrt{\frac{-\sigma}{\kappa}}$ ã¨ãŠã„ãŸ. 
      %
      ä¿‚æ•°è¡Œåˆ—ã®è¡Œåˆ—å¼ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰, æˆé•·çŽ‡ã‚’å®šã‚ã‚‹åˆ†æ•£é–¢ä¿‚ã«ç›¸å½“ã™ã‚‹
      å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
      ãƒ™ãƒƒã‚»ãƒ«å‡½æ•°ã®å¾®åˆ†ã®é›¶ç‚¹ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\kappa \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 
      ä¿‚æ•° $A_1$ ã¯åˆæœŸæ¡ä»¶ã‹ã‚‰å®šã‚ã‚‰ã‚Œã‚‹
      \footnote{
        å¾®åˆ†è¨ˆç®—ã«ã¯ãƒ™ãƒƒã‚»ãƒ«é–¢æ•°ã®æ¼¸åŒ–å¼
        \begin{displaymath}
          J'_n(x)   ï¼  \frac{n}{x}J_n(x)- J_{n+1}(x)ã€€
        \end{displaymath}
        ã‚’ç”¨ã„ã‚Œã°ã‚ˆã„.
      }. 


    \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§å††ç›¤å†…å¤–é&nbsp;˜åŸŸãŒæº€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¯
      å††ç›¤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 \psi =0, 
        \end{displaymath}
        %
        æ–¹ä½è§’æ–¹å‘ã«é–¢ã—ã¦ãƒ•ãƒ¼ãƒªã‚¨å¤‰æ›ã™ã‚‹ã¨
        %
        \begin{displaymath}
          \Deqlab{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
          \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
                    -\frac{m^2}{r^2}\tilde{\psi}_m=0
        \end{displaymath}
        %
        ã“ã®æ–¹ç¨‹å¼ã®ç‹¬ç«‹ãª 2 ã¤ã®è§£ã¯ $r^{-m}, r^m$ ã§ã‚ã‚‹. 
         $\infty$ ã§ç™ºæ•£ã—ãªã„è§£ã‚’æ±‚ã‚ã‚‹ã¨
        \begin{displaymath}
          \tilde{\psi}_m = C_1 r^{-m}
              \quad \left(r \ge a \right).
        \end{displaymath}
        %
        å¢ƒç•Œ $r=a$ ã§å€¤ã¨ 1 éšŽå¾®åˆ†ãŒé€£ç¶šã§ã‚ã‚‹ã“ã¨ã‹ã‚‰
        \begin{eqnarray*}
          \tilde{\psi}_m|_{r=a} =  C_1 r^{-ma}, \quad
          \left.\DD{\tilde{\psi}_m}{r}\right|_a =  -m C_1 r^{-(m+1)a}
        \end{eqnarray*}
        %
        ã“ã‚Œã‚‰ã‚ˆã‚Š $C_1$ ã‚’æ¶ˆåŽ»ã™ã‚‹ã¨ $r=a$ ã§ã®å¢ƒç•Œæ¡ä»¶
        %
        \begin{displaymath}
          \left.\DD{\tilde{\psi}_m}{r}\right|_a
              +\frac{m}{a}\tilde{\psi}_m|_a=0,
        \end{displaymath}
        %
        ãŒå¾—ã‚‰ã‚Œã‚‹. 
      }.
      %
      \begin{equation}
        \Deqlab{æ··åˆåž‹å¢ƒç•Œæ¡ä»¶ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
         \left.\DD{\tilde{\psi}_m}{r}\right|_a
              +\frac{m}{a}\tilde{\psi}_m|_a=0,\quad
      \end{equation}
      %
      ã“ã‚Œã‚’ä¸€èˆ¬è§£
      \Deqref{æ‹¡æ•£æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}ã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨, 
      %
      \begin{equation}
           J'_m\left(a \alpha\right) 
             +\frac{m}{a\alpha} J_m\left(a \alpha\right) =0.
      \end{equation}
      %
      ã“ã“ã§ãƒ™ãƒƒã‚»ãƒ«é–¢æ•°ã®æ€§è³ª
      \begin{displaymath}
        xJ'_n(x)   ï¼  xJ_{n-1}(x) - nJ_n(x)ã€€
      \end{displaymath}
      %
      ã‚’é©ç”¨ã™ã‚‹ã¨, 
      %
      \begin{equation}
        J_{l-1}\left(a \alpha\right)  = 0.
      \end{equation}
      %
      ãƒ™ãƒƒã‚»ãƒ«å‡½æ•°ã®é›¶ç‚¹ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

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

    åŠå¾„ $a$ ã® 2 æ¬¡å…ƒå††ç›¤é&nbsp;˜åŸŸã§ã®éžåœ§ç¸®æµä½“ã® 2 æ¬¡å…ƒé‹å‹•ã‚’è€ƒãˆã‚‹. 
    éžåœ§ç¸®é€Ÿåº¦å&nbsp;´ã¯æµç·šé–¢æ•°ã‚’ç”¨ã„ã¦æ¬¡ã®ã‚ˆã†ã«è¡¨ã™ã“ã¨ãŒã§ãã‚‹. 
    %
    \begin{eqnarray}
      \Deqlab{é€Ÿåº¦åˆ†è§£}
      \Dvect{u} &amp;=&amp; \Dvect{e_z}\times\Dgrad{\psi}, \\
      &amp; &amp; u_r     = - \frac{1}{r}\DP{\psi}{\phi}, \quad
          u_\phi  =  \DP{\psi}{r}.
    \end{eqnarray}
    %
    ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹è¿‘ä¼¼ã®ä¸‹ã§ã“ã‚Œã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å¾“ã†æ–¹ç¨‹å¼ã¯
    % 
    \begin{equation}
      \Deqlab{æ¸¦åº¦æ–¹ç¨‹å¼}
          \DP{}{t} \Dlapla \psi = \nu \Dlapla\Dlapla \psi.
    \end{equation}
    %
    é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯, å¤–å´å††å‘¨ã«ã¦æ³•ç·šæ–¹å‘ã®é€Ÿåº¦æˆåˆ†ãŒ 0 ã¨ãªã‚‹ã“ã¨ã§ã‚ã‚‹. 
    æµç·šé–¢æ•°ã§è¡¨ç¾ã™ã‚Œã°,
    \begin{equation}
      \psi = 0, \quad \mbox{at} \quad r=a
    \end{equation}
    %
    åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒè‡ªç”±ã™ã¹ã‚Šã®å&nbsp;´åˆã¯
    \begin{equation}
      \Deqlab{æµç·šé–¢æ•°è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
      r\DP[]{}{r}\left(\frac{1}{r}\DP{\psi}{r}\right)  = 0, 
      \quad \mbox{at} \quad r=a.
    \end{equation}
    %
    ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, 
    \begin{eqnarray*}
      \Deqlab{æµç·šé–¢æ•°ç²˜ç€æ¡ä»¶}
         \DP{\psi}{r} = 0, \quad \mbox{at} \quad r=a.
    \end{eqnarray*}

    \subsection{ãƒ•ãƒ¼ãƒªã‚¨ç´šæ•°å±•é–‹}

      æ–¹ä½è§’æ–¹å‘ã«ã¯ãƒ•ãƒ¼ãƒªã‚¨ç´šæ•°ã§å±•é–‹ã—ã¦è§£ãã“ã¨ãŒã§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}_m(r)e^{im\phi+\sigma t},
      \end{equation}
      %
      ã“ã‚Œã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, 
      %
      \begin{displaymath}
        \Deqlab{æ¸¦åº¦æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}
          \sigma \left[
                     \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
                  -\frac{m^2}{r^2}\right]\tilde{\psi}_m =
              \nu \left[
                     \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
                  -\frac{m^2}{r^2}\right]
                  \left[
                     \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
                  -\frac{m^2}{r^2}\right]
              \tilde{\psi}_m
      \end{displaymath}
      %
      å¢ƒç•Œæ¡ä»¶ã¯è‡ªç”±ã™ã¹ã‚Šã®å&nbsp;´åˆ, 
      %
      åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒ
      \begin{equation}
        \Deqlab{æµç·šé–¢æ•°è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}
        \tilde{\psi}_m = 
        r\DD[]{}{r}\left(\frac{1}{r}\DD{\tilde{\psi}_m}{r}\right)  = 0, \quad
        \quad \mbox{at} \quad r=a.
      \end{equation}
      %
      ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, 
      \begin{equation}
        \Deqlab{æµç·šé–¢æ•°ç²˜ç€æ¡ä»¶}
           \tilde{\psi}_m = \DD{\tilde{\psi}_m}{r} = 0, \quad
           \quad \mbox{at} \quad r=a.
      \end{equation}

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

      å¢ƒç•Œæ¡ä»¶ãŒç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, æœ€å¤§æˆé•·çŽ‡ãŒè²&nbsp;ã¨ãªã‚‹ã“ã¨ã‚’
      æ”¯é…æ–¹ç¨‹å¼ã‚’å…¨ç©åˆ†ã™ã‚‹ã“ã¨ã§æ±‚ã‚ã‚‰ã‚Œã‚‹. 
      ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«é–¢ã—ã¦ã¯, æ‹¡æ•£æ–¹ç¨‹å¼ã®ä¸¡ç«¯ãƒ‡ã‚£ãƒªã‚¯ãƒ¬
      å¢ƒç•Œæ¡ä»¶ã®å&nbsp;´åˆã¨åŒã˜ã§ã‚ã‚‹ã‹ã‚‰æœ€å¤§æˆé•·çŽ‡ãŒè²&nbsp;ã§ã‚ã‚‹ã“ã¨ãŒåˆ†ã‹ã‚‹. 

      ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«ã¤ã„ã¦ã¯, æ¼”ç®—å­ ${\cal D}_l$ ã‚’
      %
      \begin{displaymath}
        {\cal D}_l \equiv \frac{1}{r}\DD{}{r}\left(r\DD{}{r}\right)
                        -\frac{m^2}{r^2}
      \end{displaymath}
      %
      ã¨å®šç¾©ã™ã‚‹ã¨, ä»»æ„ã® 2 ã¤ã®é–¢æ•° $f(r),g(r)$ ã«å¯¾ã—ã¦
      %
      \begin{eqnarray*}
        \int_0^a f(r) {\cal D}_l g(r) r dr
        &amp;=&amp; \int_0^a f
             \left[\frac{1}{r}\DD{}{r}\left(r\DD{}{r}\right)
                        -\frac{m^2}{r^2}\right]g r dr\\
        &amp;=&amp; \int_0^a f \DD{}{r}\left(r\DD{g}{r}\right)dr
             - \int_0^a m^2 \frac{fg}{r} dr\\
        &amp;=&amp; \left[ f r\DD{g}{r} \right]_0^a
             - \int_0^a \DD{f}{r}\left(r\DD{g}{r}\right) dr
             - \int_0^a m^2 \frac{fg}{r} dr\\
        &amp;=&amp; \left[ f r\DD{g}{r} \right]_0^a
             - \left[ r\DD{f}{r} g \right]_0^a
             + \int_0^a g \DD{}{r}\left(r\DD{f}{r}\right)
             - \int_0^a m^2 \frac{fg}{r} dr\\
        &amp;=&amp; \left[ f r\DD{g}{r} \right]_0^a
             - \left[ r\DD{f}{r} g \right]_0^a
             + \int_0^a g(r) {\cal D}_l f(r) r dr
      \end{eqnarray*}
      %
      $f=\psi^*_m(r), g={\cal D}_l\psi_m(r)$ ã¨é¸ã¶ã¨
      å¢ƒç•Œæ¡ä»¶\Deqref{æµç·šé–¢æ•°ç²˜ç€æ¡ä»¶}ã‚’ç”¨ã„ã¦
      \begin{eqnarray*}
        \int_0^a \psi^*_m(r)
                 {\cal D}_l{\cal D}_l\psi_m(r) r dr        
        &amp;=&amp; \left[ \psi^*_m r\DD{{\cal D}_l\psi_m}{r} 
            \right]_0^a
             - \left[ r\DD{\psi^*_m}{r}
                     {\cal D}_l\psi_m \right]_0^a
             + \int_0^a {\cal D}_l\psi^*_m{\cal D}_l\psi_m r dr\\
        &amp;=&amp; \int_0^a |{\cal D}_l\psi_m|^2 r dr
      \end{eqnarray*}
      
      æµç·šé–¢æ•°ã®å¼
      \Deqref{æ¸¦åº¦æ–¹ç¨‹å¼ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹}ã«
      $\psi^*_mr$ ã‚’ã‹ã‘ã¦ç©åˆ†ã™ã‚‹ã¨
      \begin{eqnarray*}
        \sigma \int_0^a
             \tilde{\psi}^*_m{\cal D}_l\tilde{\psi}_m r dr 
        &amp;=&amp; \nu \int_0^a
             \tilde{\psi}^*_m{\cal D}_l{\cal D}_l\tilde{\psi}_m r dr,\\
      \end{eqnarray*}
      %
      å·¦è¾ºã¯æ‹¡æ•£æ–¹ç¨‹å¼ã®å³è¾ºã®å¼å¤‰å½¢ã¨åŒæ§˜ã«è¨ˆç®—ã§ãã¦, 
      \begin{displaymath}
        \sigma \int_0^a
             \tilde{\psi}^*_m{\cal D}_l\tilde{\psi}_m r dr 
        =-\sigma \int_0^a \left[
              r\left|\DD{\tilde{\psi}_m}{r}\right|^2 
             + m^2 \int_0^a \frac{1}{r}|\tilde{\psi}_m|^2 \right]dr
      \end{displaymath}
      %
      ã§ã‚ã‚‹ã‹ã‚‰, 
      \begin{displaymath}
        -\sigma \int_0^a \left[
              r^2\left|\DD{\tilde{\psi}_m}{r}\right|^2 
                       + \frac{m^2}{r} |\tilde{\psi}_m|^2 \right]dr
        = \nu\int_0^a |{\cal D}_l\psi_m|^2 r dr, 
      \end{displaymath}
      %
      ã‚ˆã£ã¦
      \begin{displaymath}
        \sigma = - \nu
           \frac{\Ddsty \int_0^a |{\cal D}_l\psi_m|^2 r dr}
                {\Ddsty \int_0^a \left[
                        r\left|\DD{\tilde{\psi}_m}{r}\right|^2 
                        + \frac{m^2}{r} |\tilde{\psi}_m|^2 \right]dr}.
      \end{displaymath}


    \subsection{ä¸€èˆ¬è§£}

      æ‹¡æ•£æ–¹ç¨‹å¼ã®å&nbsp;´åˆã¨åŒæ§˜ã«æ”¯é…æ–¹ç¨‹å¼ã‚’å¤‰å½¢ã™ã‚‹ã¨
      %
      \begin{displaymath}
        \left\{
          \frac{1}{r}\DD{}{r}\left(r\DD{\tilde{\psi}_m}{r}\right)
            +\left[\frac{-\sigma}{\kappa} -\frac{m^2}{r^2}\right]
        \right\}
        \left[
          \frac{1}{r}\DD{}{r}\left(r\DD{}{r}\right)-\frac{m^2}{r^2}
        \right]
             \tilde{\psi}_m=0.
      \end{displaymath}
      %
      æˆé•·çŽ‡ $\sigma &lt;0$ ã«å¯¾ã—ã¦ã®ã“ã®ä¸€èˆ¬è§£ã¯
      %
      \begin{equation}
        \Deqlab{æµç·šé–¢æ•°ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}
        \tilde{\psi}_m 
               = B_1 r^m
               + B_2 J_m\left(\sqrt{\frac{-\sigma}{\kappa}}r\right) 
      \end{equation}
      %
      æµç·šé–¢æ•°ã«ä½œç”¨ã™ã‚‹ 2 ã¤ã®ã‚«ãƒƒã‚³ã§ããã‚‰ã‚ŒãŸæ¼”ç®—å­ã¯
      äº¤æ›å¯èƒ½ã§ã‚ã‚‹ã“ã¨ã«æ³¨æ„ã•ã‚ŒãŸã„.


    \subsection{å¤–å´ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã®æµç·šé–¢æ•°}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯\Deqref{æµç·šé–¢æ•°è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚ˆã‚Š
      %
      \begin{displaymath}
        \tilde{\psi}_m= \DD{\tilde{\psi}_m}{r}=0, 
        \quad {\rm at} \quad r=a. 
      \end{displaymath}
      %
      ä¸€èˆ¬è§£\Deqref{æµç·šé–¢æ•°ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ã“ã‚Œã«ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      \begin{equation}
        \left(
        \begin{array}{cc}
           a^l &amp; J_m\left(a \alpha\right) \\
           ma^{m-1} &amp; \alpha  J'_m\left(a \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          B_1 \\ B_2 
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      %
      \begin{equation}
        \left|
        \begin{array}{cc}
           a^m      &amp;  J_m\left(a\alpha\right) \\
           ma^{m-1} &amp; \alpha J'_m\left(a\alpha\right)
        \end{array}
        \right| = 0.
      \end{equation}
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $B_1, B_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          B_1 \\ B_2 
        \end{array}
        \right)
        = 
        \left(
        \begin{array}{c}
          J_m(a\alpha) \\
          - a^m
        \end{array}
        \right)
      \end{equation}
      %
      ã¨ä¸Žãˆã‚‰ã‚Œã‚‹. 


    \subsection{ä¸Šå´è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶ã®å&nbsp;´åˆã®æµç·šé–¢æ•°}

      ã“ã®å&nbsp;´åˆã®å¢ƒç•Œæ¡ä»¶ã¯\Deqref{æµç·šé–¢æ•°è‡ªç”±ã™ã¹ã‚Šæ¡ä»¶}ã‚ˆã‚Š
      %
      \begin{displaymath}
        \tilde{\psi}_m= r\DD{}{r}\left(\frac{1}{r}\DP{\tilde{\psi}_m}{r}\right)=0, 
        \quad {\rm at} \quad r=a
      \end{displaymath}
      %
      2 ã¤ã‚ã®å¢ƒç•Œæ¡ä»¶ã‚’æ›¸ãæ›ãˆã‚‹ã¨
      %
      \begin{displaymath}
        r\DD{}{r}\left(\frac{1}{r}\DP{\tilde{\psi}_m}{r}\right)
        = \DD[2]{\tilde{\psi}_m}{r}-\frac{1}{r}\DD{\tilde{\psi}_m}{r}=0.
        \quad {\rm at} \quad r=a
      \end{displaymath}
      %
      ã“ã‚Œã«ä¸€èˆ¬è§£\Deqref{æµç·šé–¢æ•°ãƒ•ãƒ¼ãƒªã‚¨å±•é–‹ä¸€èˆ¬è§£}ã‚’
      ä»£å…¥ã—ã¦æ•´ç†ã™ã‚‹ã¨
      %
      \begin{equation}
        \left(
        \begin{array}{cc}
           a^m &amp;  J_m\left(a\alpha\right) \\
           m(m-2)a^{m-2} &amp; \alpha^2  J''_m\left(a \alpha\right) 
                           -\frac{\alpha}{a} J'_m\left(a \alpha\right)
        \end{array}
        \right)
        \left(
        \begin{array}{c}
          B_1 \\ B_2 
        \end{array}
        \right)
        =0,
      \end{equation}
      %
      å·¦è¾ºã®ä¿‚æ•°è¡Œåˆ—å¼ãŒ 0 ãªã‚‹æ¡ä»¶ãŒå›ºæœ‰å€¤ $\alpha$ ã‚’å®šã‚ã‚‹
      åˆ†æ•£é–¢ä¿‚ã¨ãªã‚‹. 
      \footnote{
        ãƒ™ãƒƒã‚»ãƒ«å‡½æ•°ã® 2 éšŽå¾®åˆ†ã¯ãƒ™ãƒƒã‚»ãƒ«ã®æº€ãŸã™å¾®åˆ†æ–¹ç¨‹å¼ã‹ã‚‰
        \begin{displaymath}
          J''= -\frac{1}{r}J'
                      -\left[1-\frac{m^2}{r^2}\right] J, \quad
        \end{displaymath}
        ã¨è¨ˆç®—ã•ã‚Œã‚‹. 
      }
      %
      \begin{equation}
        \left|
        \begin{array}{cc}
           a^m &amp;  J_m\left(a\alpha\right) \\
           m(m-2)a^{m-2} &amp; \alpha^2  J''_m\left(a \alpha\right) 
                           -\frac{\alpha}{a} J'_m\left(a \alpha\right)
        \end{array}
        \right| = 0.
      \end{equation}
      ã“ã®åˆ†æ•£é–¢ä¿‚ã‹ã‚‰å‹•å¾„æ³¢æ•° $\alpha$ ã¨
      æˆé•·çŽ‡ $\sigma = -\nu \alpha^2$ ãŒå®šã‚ã‚‰ã‚Œã‚‹. 

      ä¿‚æ•° $B_1, B_2$ ã®çµ„åˆã›ã¯, æ±‚ã¾ã£ãŸ $\alpha$ ã‚’ç”¨ã„ã¦
      %
      \begin{equation}
        \left(
        \begin{array}{c}
          B_1 \\ B_2 
        \end{array}
        \right)
        = 
        \left(
        \begin{array}{c}
           J_m\left(a\alpha\right)\\
           -a^m 
        \end{array}
        \right)
      \end{equation}
      %
      ã§å®šã‚ã‚‰ã‚Œã‚‹. 
      

\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 
</pre></body></html>