%
% Chapter 2   Wave Motion in the Atmosphere  Part I
%             2-5 SURFACE GRAVITY WAVES
%
% 2002/09/24  ���� ����   ����
% 2002/09/30  ���� ����   �������
% 2002/10/31  ���� ����   ���߸�ν���
%
%
%
%
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\markboth{3-5 ɽ�̽�����}{3-5 ɽ�̽�����}
%\markright{2-5 ɽ�̽�����}
%\Ddate{2002/10/31}
%\Dfile{hw-02-05.tex}

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\section{ɽ�̽�����}

\begin{itembox}[l]{����}
 ή�Τζ����̤�ȯ������{\bfseries ɽ�̽�����}
 ({\itshape surface gravity waves})�ˤĤ��Ƶ�������.
 �Ѽ���ή���ؤ��ꤷ, ���ζ����̤�ȯ�������Ȥΰ���®�٤�Ƴ�Ф���.
 �ſ尵ʿ�դ��ꤹ����Ȥ��ʤ����������ΰۤʤ��Ȥ�������.
 �ſ尵���ꤹ������Ȥ�{\bfseries ������}
 ({\itshape shallow-water wave}),
 ���ꤷ�ʤ������Ȥ�{\bfseries ������}({\itshape deep-water wave})�ȸƤ�.
\end{itembox}

������Ǥ�, ��ήԡ��Ѽ����󰵽̤ʰ�Ĥ�ή���ؤȹͤ���.
����Ū, ʪ��Ū�˴�ñ�ʲ��ɽ�̳����Ȥ����ʣ�����絤��
�ץ�����������������.

ή�Τ��󰵽̤ǶѼ�($\alpha' = 0$)�Ǥ���,
���������ϼ�ͳɽ�̤Ǥ���Ȥ���.
���ξ��, ����������(3-2)�ϰʲ��Τ褦�ˤʤ�.
�ʤ�, �ץ饤��($^{\prime}$)�ϵ��椫��������Ƥ���Τ����դ��뤳��
\footnote{
 �絤���󰵽̤ǶѼ��Ǥ���ȸ�����狼��$\alpha' = 0$,
 $\overline{\alpha} = $const�Ǥ���.
 ���� 2 ����(3-2)��1,2,4 �����������뤳�Ȥ���ʸ���(3-20a)��(3-20c)
 �μ���������.
}
:
%
\setcounter{equation}{0}
\def\theequation{3-20\alph{equation}}
%
\begin{eqnarray}
 \DP{u}{t} + U \DP{u}{x} + \frac{1}{\overline{\rho}} \DP{p}{x} &=& 0,
  \\
 \delta \left( \DP{w}{t} + U \DP{w}{x} \right)
  + \frac{1}{\overline{\rho}} \DP{p}{z} &=& 0,
  \\
 \DP{u}{x} + \DP{w}{z} &=& 0.
\end{eqnarray}
%
\setcounter{equation}{20}
\def\theequation{\arabic{chapter}-\arabic{equation}}
%
�����̵��ή��Ǥ��ſ尵ʿ�դμ��ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{\overline{p}}{z} = - g \overline{\rho}.
  \nonumber
\end{eqnarray}
%
���μ���$z=0$��������̵��ή���ĺ��$z=H$�ޤ���ʬ�����,
%
\begin{eqnarray}
 g \overline{\rho} H = p_0
\end{eqnarray}
%
�Ȥʤ�. $p_0$��$z=0$�Ǥΰ��ϤǤ���.
���˾����Ĵ�´ؿ����Ȳ��ꤹ���,
%
\begin{eqnarray}
 u &=& \Psi (z) e^{i \mu (x-ct)},
  \nonumber \\
 w &=& \Phi (z) e^{i \mu (x-ct)},
  \\
 \frac{p}{\overline{\rho}} &=& P (z) e^{i \mu (x-ct)}
  \nonumber 
\end{eqnarray}
%
�Ȥʤ�.
(3-22)��(3-20)��������, ��ñ�ʷ��ˤ���Ȱʲ��δط�����������.
%
\begin{eqnarray}
 (U - c) \Psi (z) + P (z) &=& 0,
  \nonumber \\
 i \mu \delta (U - c) \Phi (z) + \DP{P (z)}{z} &=& 0,
  \\
 i \mu \Psi (z) + \DP{\Phi (z)}{z} &=& 0.
  \nonumber
\end{eqnarray}
%
(3-23)�λϤ��2�Ĥμ����Ѥ���$P(z)$��ä�,
����ˤ��μ���(3-23)�κǸ�μ����Ѥ���$\Psi(z)$��ä��Ȱʲ��μ���
������.
%
\begin{eqnarray}
 \DP[2]{\Phi (z)}{z} - \mu^2 \delta \Phi (z) = 0.
\end{eqnarray}
%
������, 2�Ĥξ���ͤ���.
$\delta = 1$({\bfseries ��ľ��®�٤�����})�ξ���
$\delta = 0$({\bfseries �����ſ尵ʿ�դ�������})�ξ��Ǥ���.
�����β�Ϥ��줾��ʲ��Τ褦�ˤʤ�
\footnote{
 ���λؿ��β��(3-3)�β�η���Ʊ���Ȥʤ�.
 ���Τ��ᤳ�β��(3-5)���ü�ʾ��Ǥβ�Ȥ���������.
}.
%
\begin{eqnarray}
 \Phi (z) &=& a_1 e^{\mu z} + a_2 e^{- \mu z},
  \hspace{10mm} \delta = 1,
  \nonumber \\
 \Phi (z) &=& a_1^{\prime} z + a_2^{\prime}
  \hspace{21mm} \delta = 0.
  \nonumber
\end{eqnarray}
%
������$a$�϶�����狼�������Ǥ�դ�����Ǥ���.
�ſ尵���ꤹ���, ���������̤Ǥϱ�ľ®�٤Ͼä���.
���Τ����ľ��®�٤�¸�ߤ���(�Ĥޤ�$\delta =1$)�Ǥ�
$a_1 = - a_2 \equiv a$�Ȥʤ�. �Ĥޤ�$\Phi(z)$�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Phi (z) = a (e^{\mu z} - e^{- \mu z})
  \hspace{10mm}
  \delta = 1.
\end{eqnarray}
%
���λؿ��ι��٤�����Ȥ���Ÿ����
\footnote{
 �ؿ��ؿ��ϰʲ��Τ褦�ˤ٤����Ÿ�������.
 % 
 \begin{eqnarray}
  e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
   = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
   \qquad [| x | < \infty].
   \nonumber
 \end{eqnarray}
 %
 ��������Ѥ����,
 %
 \begin{eqnarray}
  e^{\mu z} = 1 + \mu z + (\mu z)^2 / 2 + \cdots, \qquad
   e^{- \mu z} = 1 - \mu z + (\mu z)^2 / 2 - \cdots
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. ��̤Ȥ���
 %
 \begin{eqnarray}
  e^{\mu z} - e^{- \mu z} = 2 \mu z + (\mu z)^3 / 3 + \cdots
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. �����Ǳ��դ���2��ʹߤ�̵�뤹�뤳�Ȥ���ʸ�β�ˤʤ�.
},
����˶���򤪤��ʤ����Ȥ�$\Phi (z)$�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Phi (z) &=& 2 a \mu z + \cdots
  \nonumber \\
 &\simeq&  2 a \mu z.
  \nonumber
\end{eqnarray}
%
�ſ尵ʿ��($\delta = 0$)�ξ���$a_2^{\prime} = 0$�Ȥʤ�.
$a_1^{\prime} = a$�Ȥ�����,
%
\begin{eqnarray}
 \Phi (z) = az
\end{eqnarray}
%
�Ȥʤ�.

2���ܤζ�������, ���������̤Τ�����(�������϶����̤���Ф뤳�ȤϤǤ��ʤ�)
�ˤ�����������(���ܾ�Ⱦ���ι��)���Ѳ����ʤ����ȤǤ���
\footnote{
 ����Ū�ˤ��Τ褦�ʶ��������Ѥ��Ƥ����ΤȤ��Ƴ��Τ�����.
 ����ɽ�̤ζ������Ȥ���, �����ϤϾ���絤�������礬¿��.
}.
��äƼ�ͳɽ�̤�
%
\begin{eqnarray}
 d
  \left(
   \overline{p} + p
  \right)
 / dt = 0
 \nonumber
\end{eqnarray}
%
�Ȥʤ�. ������������ˤ�äƶ������,
�ʲ��μ��Ȥʤ�.
%
\begin{eqnarray}
 \DP{p}{t} + U \DP{p}{x} + w \DP{\overline{p}}{z} = 0
  \quad {\rm at} \quad z=H.
\end{eqnarray}
%

��(3-25), (3-26)�ȷ�(3-23)�����Ѥ��뤳�Ȥǰʲ���³���褦��
��̤�������.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      Case 1. Nonhydrostatic       %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�ޤ���, �ſ尵ʿ�դǤʤ�����ͤ���.
(3-25)�ȷ�(3-23)�ˤ�äưʲ��η�̤�������.
%
\begin{eqnarray}
 \begin{array}{l}
  \Psi (z) = ia (e^{\mu z} + e^{- \mu z}), 
  \\
  P (z) = - ia (U - c) (e^{\mu z} + e^{- \mu z}).
 \end{array}
\end{eqnarray}
%
(3-25)��(3-28)��(3-27)���������뤳�Ȥ������뿶ư��($\nu = \mu c$)
�μ���򤯤Ȱʲ��ΰ���®�٤�������
\footnote{
 ���β��Ƴ�Ф˴ؤ��Ƥ�{\bfseries 3-5-A}����.
}
:
%
\begin{eqnarray}
 c = U \pm
  \left(
   \frac{g \mathcal{L}}{2\pi} \tanh \frac{2 \pi H}{\mathcal{L}}
  \right)^{1/2}.
\end{eqnarray}
%
�ʤ�, $\mathcal{L}$���Ȥ���Ĺ���ȿ�$\mu$�Ȥ�$\mu = 2 \pi/\mathcal{L}$
�Ȥ����ط��ˤ���.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      Case 2. Hydrostatic          %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

����, �ſ尵ʿ�դξ���ͤ���.
(3-26)�ȷ�(3-23)�ˤ�äưʲ��η�̤�������.
%
\begin{eqnarray}
 \begin{array}{l}
  \Psi (z) = \dfrac{ia}{\mu},
  \\
  P (z) = - \dfrac{ia (U - c)}{\mu}.
 \end{array}
\end{eqnarray}
%
(3-26)��(3-30)��(3-27)���������ƴ�ñ�ˤ�����ſ尵ʿ�դȤ�������
����®�٤�������.
%
\begin{eqnarray}
 c = U \pm \sqrt{gH}.
\end{eqnarray}
%

(3-31)��Ϳ����줿����®�٤ǿʤ��Ȥ�����,
{\bfseries ������}({\itshape shallow-water wave})�⤷����
{\bfseries Ĺ��}({\itshape long wave})�ȸƤФ��.
(3-29)�ˤ�������$H/\mathcal{L}$�����Ū�礭�����(���褽 0.5�ʾ�)
\footnote{
 $H/\mathcal{L} \gtrsim 0.5$�ξ��,
 $e^{2\pi H / \mathcal{L}} \gtrsim 23.1407$, 
 $e^{- 2\pi H / \mathcal{L}} \lesssim 0.0432$�Ȥʤ�, 
 $e^{2\pi H / \mathcal{L}} \gg e^{- 2\pi H / \mathcal{L}}$�Ǥ���.
 ��ä�
 %
 \begin{eqnarray}
  \tanh (2 \pi H / \mathcal{L})
   = \frac{e^{2\pi H / \mathcal{L}} - e^{- 2\pi H / \mathcal{L}}}
   {e^{2\pi H / \mathcal{L}} + e^{-2\pi H / \mathcal{L}}}
   \simeq
   \frac{e^{2\pi H / \mathcal{L}}}{e^{2\pi H / \mathcal{L}}}
   = 1
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�.
},
����®�٤϶��Ū��
%
\begin{eqnarray}
 c \doteq U \pm \sqrt{\frac{g \mathcal{L}}{2 \pi}}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. �����Ȥ�{\bfseries ������}({\itshape deep-water wave})
\footnote{
 �ֿ����ȡפ�$H \gg \mathcal{L} $�ξ��, �Ĥޤ�ή�Το���(�⤵)��
 ��Ĺ����٤ƽ�ʬ���礭�����ο����ȤθƾΤǤ���.
 �դ�$H \ll \mathcal{L} $�ξ��ˤ�ή�Το�������Ĺ����٤ƽ�ʬ��
 ����������, ���ξ����Ȥ�������ȡפȸƤ�.
}
�ȸƤФ��.
���ξ��, ή��γ�Ҥε�ƻ�Ϥܱۤߤ�����
\footnote{
 ��¼(1983)�򻲾ȤΤ���.
}.
����, $H/\mathcal{L}$��������(0.04�ʲ�)�ξ���
(3-29)��(3-31)�η��ˤʤ�
\footnote{
 $2 \pi H/ \mathcal{L} = x$�Ȥ���, $e^x$, $e^{-x}$��ƥ��顼Ÿ�������,
 $e^x = 1 + x + x^2 + \cdots$, $e^{-x} = 1 - x + x^2 - \cdots$
 �Ȥʤ�. $2 \pi H / \mathcal{L} = x$�������Ǥ��뤳�Ȥ���$x$��2��ʾ�
 ��̵�뤹���,
 %
 \begin{eqnarray}
  \tanh x =
   \left(
    \frac{e^x - e^{-x}}{e^x + e^{-x}}
   \right)
   \simeq \frac{2 x}{2} = x = 2 \pi H/ \mathcal{L}
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. �����(3-29)�����������, (3-31)��Ʊ���ˤʤ�.
}.
���ξ���γ�Ҥε��פϤ褯���Ӥ��ʱ߾��ˤʤ�,
��̤Ȥ��Ƥۤܿ�ʿ���ˤʤ�.
�Ѽ��絤���ꤹ����, (3-21)�Ⱦ������������
$gH = p_0 / \rho = RT$�Ȥʤ�
\footnote{
 �����Ǥ�$\rho$, $T$�϶Ѽ��絤��̩��, ���٤�ɽ��.
}.
�����(3-31)�ϰʲ��Τ褦�˽񤯤��Ȥ�Ǥ���.
%
\begin{eqnarray}
 c = U \pm \sqrt{RT}.
\end{eqnarray}
%
���Τ褦�ˤ����, ��Ĺ�׽����Ȥ�®�٤�(3-10)������줿���Ȥ����Ȥ�
®�٤ȤۤȤ��Ʊ���ˤʤ뤳�Ȥ��狼��.

(3-31)��Ϳ����줿Ĺ�����ȤȸƤФ���Ȥΰ���®�٤ϱ�ľ������®�٤�
�����������, ���ʤ�������ſ尵ʿ�դʾ���������.
���η�̤ϰʲ��Τ褦�ˤ⤦����ľ��Ū����ˡ�Ǥ����뤳�Ȥ��Ǥ���.
�ſ尵ʿ�դ��ꤹ���, ή�ΤΤ����ʤ����Ǥ�
$g \overline{\rho} (h - z) = p$������Ω��
\footnote{
 ��ͳɽ�̾�$z=h$�Ǥ�$p=0$�Ȥ��Ƥ���.
}.
$h$�ϼ�ͳɽ�̤ι⤵�Ǥ���.
$h = H + h'$, $p = \overline{p} + p'$�Ȥ���,
���ܾ���ſ尵ʿ�դ�ߤ����Ȥ����,
$g \overline{\rho} (H - z) = \overline{p}$���,
$g \overline{\rho} h' = p'$��������
\footnote{
 $g \overline{\rho} (H - z) = \overline{p}$���,
 %
 \begin{eqnarray}
  g \overline{\rho} (h - z) &=& p
   \nonumber \\
  g \overline{\rho} (H + h' - z) &=& \overline{p} + p'
   \nonumber \\
  g \overline{\rho} h' &=& p'.
   \nonumber
 \end{eqnarray}
}.
���Τ褦�ˤ����, �ʲ��μ�������Ω��. (�ץ饤��($^{\prime}$)�ϵ��椫��
�����Ƥ���Τ����դ��뤳��).
%
\begin{eqnarray}
 g \DP{h}{x} = \frac{1}{\overline{\rho}} \DP{p}{x}.
  \nonumber
\end{eqnarray}
%

�嵭�μ������Ѥ����, (3-20)�κǽ���������ϰʲ��Τ褦�˽񤯤��Ȥ��Ǥ���.
%
\begin{eqnarray}
 \DP{u}{t} + U \DP{u}{x} + g \DP{h}{x} = 0.
\end{eqnarray}
%
$u$��$h$�Ͼ���Ǥ���.

$u$��$h$�˴ؤ���2���ܤμ���Ϣ³�μ�(3-20c)���ľ��������ʬ���뤳�Ȥ�����
���. ������, ������¸�μ��ϰʲ��Τ褦�˴�ñ��ľ�����뤳�Ȥ�Ǥ���
(��2-2�򸫤�):
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/2-2.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize ���̤�ή����
  (Haltiner, G. J., and Williams, R. T., 1979:
      Numerical Prediction and Dynamic Meteorology.
      {\itshape John Wiley \& Sons,} 477pp)}
  }
 \Dfiglab{hwfig-02-02}
 \end{quote}
\end{figure}
%
\begin{eqnarray}
 \DP{(\overline{\rho} h)}{t} = - \DP{(\overline{\rho} u h)}{x}.
  \nonumber
\end{eqnarray}
%
��������, ��ñ�ˤ��뤳�Ȥǰʲ��μ���������
\footnote{
 $h = H + h'$, $u = U + u'$�Ǥ���, $\overline{\rho}$�ϰ���ʤΤ�,
 %
 \begin{eqnarray}
  \partial (\overline{\rho} h) / \partial t
   &=& - \partial (\overline{\rho} u h) / \partial x
   \nonumber \\
  \partial (H + h') / \nonumber t &=&
   - (U + u') \partial (H + h') / \partial x
   - (H + h') \partial (U + u') / \partial t
   \nonumber 
 \end{eqnarray}
 %
 �Ȥʤ�. ���μ������������뤳�Ȥ�(3-34)��������.
}.
%
\begin{eqnarray}
 \DP{h}{t} + U \DP{h}{x} + H \DP{u}{x} = 0.
\end{eqnarray}
%
������$h$�����$u$�Ͼ���Ǥ���.
$h$��$z$�δؿ��ǤϤʤ�
\footnote{
 $h$�ϼ�ͳɽ�̤ξ���ʤΤ�����$z$�˰�¸���ʤ�.
}
�Τ�$u$��$z$�δؿ��ǤϤʤ��ʤ�
\footnote{
 �Ĥޤ�, �ſ尵��Ƴ���������Ȥ�$u(x,z,t)$�Ǥ��ä���Τ�$u(x,t)$�Ȥʤ뤳
 �Ȥ��狼��.
}.
��ä�$h$��$x$�ϰʲ��η��ǽ񤯤��Ȥ��Ǥ���.
%
\begin{eqnarray}
 u = u_0 e^{i \mu (x - ct)}, \quad
  h = h_0 e^{i \mu (x - ct)}.
  \nonumber
\end{eqnarray}
%
$u_0, h_0$������Ǥ���. ������(3-33)�����(3-34)��������,
����줿Ʊ���Ϥι���
\footnote{
 \begin{eqnarray}
  \left[
   \begin{array}{cc}
    U - c & g \\ H & U - c \\
   \end{array}
  \right]
  \left[
   \begin{array}{c}
    u_0 \\ h_0 \\
   \end{array}
  \right] = 0.
  \nonumber
 \end{eqnarray} 
}
�򥼥��Ȥ��뤳�Ȥ������Ȥΰ���®�٤�������.
%
\begin{eqnarray}
 c = U \pm \sqrt{gH}.
  \nonumber
\end{eqnarray}
%
�����(3-31)��������.

�����ǽҤ٤�줿�����Ȥ�����{\bfseries ����������}
({\itshape external gravity wave})
\footnote{
 ������;�̤���, ��ľ�����ˤ��ȤȤ������Ť���, ������ñĴ�˸�������
 ����褦���Ȥ����Τ���{\bfseries ������}({\itshape external wave})
 �ȸƤ�.
 �㤨��, ��ʿ�����ˤϽ����Ȥޤ��ϲ��ȤȤ������Ť��뤬, �⤵�����ˤ�
 ���Ť��ʤ��Ȥ�����. �����ޤǤ˾Ҳ𤷤�{\bfseries ����������}
 ({\itshape external gravity wave})��{\bfseries �����}
 ({\itshape Lamb wave})�Ǥ���.

 ����;�̤���, ���̤ˤ����볰�������Ȥ�{\bfseries ������}
 ({\itshape water wave})�ȸƤФ��.
 �ܤ����ϵ��ݲʳػ�ŵ(1998)����.
}
�ȸƤФ��.
����Ϥ����Ȥκ��翶�������礦��ή�Τζ����Ȥʤ뤫��Ǥ���.
�ޤ�, 2-4��Ǥ��о줷��{\bfseries ����}�����ȤȤ�����Τ⤢��.
�����̩�٤ΰۤʤ�ή��Ʊ�Τζ�����ȯ��������, ̩�٤�Ϣ³Ū�˰ۤʤ����
ȯ������. ���Τ褦�����������Ȥ�����®�٤Ϥ����ǽҤ٤�줿�����ȤȤ�
�����ۤʤ���⤢��.
��ñ����Ȥ���, 2�Ĥ�Ⱦ̵������ؤ�̩�٤�®�٤Υ��������Ϣ³�̤�
ʬ�����Ƥ���Ȥ���.
���ξ����Ȥ�®�٤ϰʲ��μ���Ϳ������.
(Haltiner and Martin, 1957
\footnote{
 �ܤ�����Drazin(1981)��Haltiner(1957)�򻲾ȤΤ���.
}
�򸫤�).
%
\begin{eqnarray}
 c = \frac{\rho U + \rho ' U'}{\rho + \rho '}
  \pm
  \left[
   \frac{g L (\rho - \rho ')}{2 \pi (\rho + \rho ')}
   - \frac{\rho \rho ' ( U - U')^2}{(\rho + \rho ')^2}
  \right]^{1/2}.
  \nonumber
\end{eqnarray}
%
�����ǤΥץ饤��($^{\prime}$)�Ͼ��ؤǤ��ѿ���ɽ��.
���η�̤����뤿��ˤϾ���$\pm \infty$(��Ȳ�)�ˤ����ƾä��뤳�Ȥ�
���ꤷ�ʤ���Фʤ�ʤ�.
�����˵��������ֿ�����Ȥΰ���®�٤Ϥ��θ�����$\rho ' = 0$��
�������Ȥ��Ǥ������̤ʾ��ˤΤ����뤳�Ȥ��Ǥ���.
����¾�ˤ⥷������̵���Ȥξ���$U = U'$, ���˥������Ȥξ���
$\rho = \rho '$�Ȥ��뤳�ȤǤ������Ȥΰ���®�٤��Τ뤳�Ȥ��Ǥ���.
���������դ��ʤ���Фʤ�ʤ��Τ�, ��������̤���ˤʤä����Ǥ���.
���ξ��, ����®�٤�ʣ�ǿ��Ȥʤ�, �԰�����Ȥ�������.
���Τ褦������������{\bfseries ����ӥ�-�إ��ۥ����}
({\itshape Kelvin-Helmholtz wave})�ȸƤФ��.











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\subsection*{3-5-A\qquad ���ſ尵ʿ�դǤ�ɽ���Ȥΰ���®�٤�Ƴ��}
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(3-29)��Ƴ�Ф򤪤��ʤ�.
(3-25)��(3-28)��갵�Ϥξ���$p$�ȱ�ľ®�٤ξ���$w$�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Deqlab{hw-02-05-���ſ尵���Ǥΰ��Ϥξ���}
  p &=& - i \overline{\rho} a (U - c) (e^{\mu z} + e^{- \mu z})
  e^{i \mu (x - ct)},
  \\
 \Deqlab{hw-02-05-���ſ尵���Ǥα�ľ®�٤ξ���}
  w &=& a (e^{\mu z} - e^{- \mu z}) e^{i \mu (x - ct)}.
\end{eqnarray}
%
\Deqref{hw-02-05-���ſ尵���Ǥΰ��Ϥξ���}���
%
\begin{eqnarray}
 \partial p /\partial t &=& - \overline{\rho} a c \mu
  (U - c) (e^{\mu z} + e^{- \mu z}) e^{i \mu (x - ct)},
  \nonumber \\
 \Deqlab{hw-02-05-���ſ尵���Ǥΰ��Ϥξ���ο�ʿ��ʬ}
 U \partial p / \partial x &=& U \overline{\rho} a \mu
  (U - c) (e^{\mu z} + e^{- \mu z}) e^{i \mu (x - ct)}
\end{eqnarray}
%
\Deqref{hw-02-05-���ſ尵���Ǥα�ľ®�٤ξ���}��
\Deqref{hw-02-05-���ſ尵���Ǥΰ��Ϥξ���ο�ʿ��ʬ}��
����̵�����ſ尵�μ�
\footnote{
 \begin{eqnarray}
  \DP{\overline{p}}{z} = -g \overline{\rho}.
   \nonumber
 \end{eqnarray}
}
��(3-27)���������Ʒ׻������,
%
\begin{eqnarray}
 - \overline{\rho} a c \mu (U - c)
  (e^{\mu H} + e^{- \mu H}) e^{i \mu (x - ct)}
  + U \overline{\rho} a \mu (U - c)
  (e^{\mu H} + e^{- \mu H}) e^{i \mu (x - ct)}
  & &
  \nonumber \\
  - g \overline{\rho} a
  (e^{\mu H} - e^{- \mu H}) e^{i \mu (x - ct)}
  &=& 0
  \nonumber \\
 - c \mu (U - c)
  (e^{\mu H} + e^{- \mu H})
  + U \mu (U - c)
  (e^{\mu H} + e^{- \mu H})
  - g 
  (e^{\mu H} - e^{- \mu H})
  &=& 0
  \nonumber \\
 - c (U - c)
  + U (U - c)
  - \frac{g}{\mu}
  \left(
   \frac{e^{\mu H} - e^{- \mu H}}{e^{\mu H} + e^{- \mu H}}
  \right)
  &=& 0
  \nonumber \\
 c^2 - 2 c U + U^2
  =
  \frac{g}{\mu} \tanh (\mu H)
  & &
  \nonumber \\
 (c - U)^2
  =
  \frac{g}{\mu} \tanh (\mu H)
  & &
  \nonumber \\
 c = U \pm
  \left\{
   \frac{g}{\mu} \tanh (\mu H)
  \right\}^{1/2}
  & &
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. �Ȥ�$x$�����ؤΤ����Ť���Τ�(�Ĥޤ�$z$�����Ϲͤ��ʤ��Τ�),
��Ĺ$\mathcal{L}$��$\mathcal{L} = 2 \pi / \mu$�Ǥ���.
���������®�٤�
%
\begin{eqnarray}
 c = U \pm
  \left\{
   \frac{g \mathcal{L}}{2\pi}
   \tanh \left( \frac{2 \pi H}{\mathcal{L}} \right)
  \right\}^{1/2}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�.


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%\subsection*{Appendix \qquad }

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