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% Chapter 2   Wave Motion in the Atmosphere  Part I
%             2-4 SOUND WAVES AND INTERNAL GRAVITY WAVES
%
% 2002/09/13  ���� ����   ����
% 2002/09/18  ���� ����   ����
% 2002/09/23  ���� ����   ����
% 2002/09/26  ���� ����   ���ߥʡ��ˤƽ���
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\section{���Ȥ�����������}

\begin{itembox}[l]{����}
 �����ȤˤĤ��ƹͤ���.
 ���������β��Ȥ��äǤϰ���ʤ��ä����Ϥ��θ����.
 ���Ϥ��θ������ư���2-2�������줿�������Ϥ���Ѥ���
 �����Ȥ˴ؤ���ط�����Ƴ��, �����ȤˤĤ��ƹͻ�����.
 �ޤ�, �絤��۴ĥ�ǥ�ˤ����벻�Ȥμ�갷���ˤĤ��Ƥ�ͻ�����.
\end{itembox}

{\bfseries ������}({\itshape gravity wave})��Ʊ�����٤ˤ���
��������̩�٤κ�������������Ϥκ��ˤ�ä�ȯ������.
���Ϥθ��̤�ͤ�����,
�⤵$z$�ˤΤ߰�¸����$\overline{p}$��$\overline{\alpha}$
\footnote{
 �������絤���������ؾ��֤Ǥ�������ͤ��̣����.
}
���ſ尵ʿ�դμ�, ������������ͤ���ɬ�פ�����:
%
\begin{eqnarray}
 0 &=& - \overline{\alpha} \DP{\overline{p}}{z} - g,
  \nonumber \\
 \bar{p} \bar{\alpha} &=& R \overline{T}.
  \nonumber
\end{eqnarray}
%
�����絤��ͤ�����, �����μ��β��
%
\begin{eqnarray}
 \bar{p} = \bar{p}(0) \ e^{-z/H},  \qquad
  \bar{\alpha} = \bar{\alpha} (0) \ e^{z/H}
\end{eqnarray}
%
�Ȥʤ�. $H = R \overline{T} / g$ �ϥ�������ϥ���
\footnote{
 {\bfseries ��������ϥ���}({\itshape scale height})�Ȥ�
 (3-11)�Ǥ��̣���Ƥ���褦��,
 �絤�ΰ��ϵڤ�̩�٤�$e^{-1}$�ܤȤʤ���٤Ǥ���.
}
�ȸƤФ���̤Ǥ���.
�����ξ�ﲼ�ǤΤ����Ĥ��Ȥ���ư��ϰʲ��Τ褦�ʷ��ˤʤ�
\footnote{
 ����$\overline{\alpha}^{1/2}$�Ͼ���ۤ�̩�٤�������, �������礭��
 �ʤ뤳�Ȥ��θ�ˤ���ƤĤ���줿��ΤǤ���.
 �㤨��$u'$�ˤ�륨�ͥ륮������ˤȤäƹͤ����,
 $\overline{\rho} = \overline{\rho}_0 e^{-z/H}$�ʤΤ�
 $\overline{\rho} {u'}^{2} = \overline{\rho}_0 e^{-z/H} S^2 
  \overline{\rho}_0^{-1} e^{z/H} e^{2i \theta}
  = S^2 e^{2i \theta}$
 �Ȥʤ�, ���٤˰�¸�����ѿ���̵���ʤ�.
 ����$\overline{\alpha}^{1/2}$�����٤ˤ�뿶�����Ѳ���ۼ����뤿��
 �ʹߤϹ��٤ˤ�뿶�����Ѳ���ͤ����˺Ѥ�, �׻����ñ�ˤʤ�.
}:
%
\begin{eqnarray}
 u' = S \overline{\alpha}^{1/2} e^{i(\mu x + k z + \nu t)}, \qquad & &
  w' = W \overline{\alpha}^{1/2} e^{i(\mu x + k z + \nu t)},
  \nonumber \\
 p' = P \overline{\alpha}^{1/2} e^{i(\mu x + k z + \nu t)}, \qquad & & 
  \alpha' = A \overline{\alpha}^{3/2} e^{i(\mu x + k z + \nu t)}.
\end{eqnarray}
%
������(3-2)�����������, ���֤���֤δؿ��ǤϤʤ�����
��������Ω�Ĺ��󼰤�������.
%
\begin{eqnarray}
 \left[
  \begin{array}{cccc}
   - \nu & 0 & \mu & 0 \\
   0 & - \delta_1 \nu &
    k + \frac{i}{2 \overline{\alpha}} \DP{\overline{\alpha}}{z} & i g\\
   0 & \left( - g + \frac{\gamma R \overline{T}}{\overline{\alpha}}
    \DP{\overline{\alpha}}{z} \right) i & \nu & \gamma R \overline{T} \nu \\
   \mu & k + \frac{i}{2 \overline{\alpha}} \DP{\overline{\alpha}}{z} &
    0 & \delta_2 \nu 
  \end{array}
 \right]
 \left[
  \begin{array}{c}
   S \\ W  \\ P \\ A
  \end{array}
 \right]
 = 0.
\end{eqnarray}
%
�����Ǥ�$U$��ñ��$x$�����ؤ����Ť�ä�������ʤΤǽ������Ƥ���.
����(3-11)��(3-13)��������, ���󼰤ϥ������������Ȥ����,
���Τ褦�ʿ�ư������������������:
%
\begin{eqnarray}
 \delta_1 \delta_2 \nu^{4} -
  \Big\{
   \gamma R \overline{T} \left( k^2 + \mu^2 \delta_1 \right)
   + \frac{g}{4H} \left[ (2 \delta_2 - 1) \gamma + 2 (1 - \delta_2 )\right]
   \nonumber \\
   + g (\gamma - 1)(\delta_2 - 1) i k
  \Big\} \nu^2
  + \mu^2 g^2 (\gamma - 1) = 0.
  \nonumber
\end{eqnarray}
%
����ξ��, ��ľ��������Ĺ$2 \pi / k$��$4 \pi H$����پ������Τ�,
$\nu^2$�η�����Τ���$g$�ι��$\gamma R \overline{T} k^2$�Ȥ���Ӥ���
��Ȥ����Ȥ��Ǥ���
\footnote{
 $2\pi / k \ll 4 \pi H$���$1/4H \ll k/2$�ʤΤ�$1/4H \ll k^2$�Ǥ��뤿��,
 $g/4H$�ι����Ȥ����Ȥ��Ǥ���.
 �ޤ�, $k$���礭�����Ȥ���$k \ll k^2$�ʤΤ�3���ܤ�
 $g(\gamma - 1) \cdots$�ι����Ȥ����Ȥ��Ǥ���.
}.
�ޤ�, �Ǹ�ι�ϲ��̤ǽ�ľ�����Ȥ��Ǥ���
\footnote{
 �ܤ����� {\bfseries 3-4-A}����. 
}.
��ľ���Ȱʲ��μ��ˤʤ�.
%
\begin{eqnarray}
 \delta_1 \delta_2 \nu^4 - \gamma R \overline{T} (k^2 + \mu^2 \delta_1) \nu^2
  + \mu^2 \frac{\gamma R \overline{T} g}{\overline{\theta}}
  \DP{\overline{\theta}}{z} = 0.
\end{eqnarray}
%

����4�Ĥκ��Ϥ��줾�첻�ȤΥڥ���
{\bfseries ����������}({\itshape internal gravity wave})�Υڥ��Ǥ���.
$g$�򥼥��ˤ��뤳�Ȥǽ����Ȥϼ��������Ȥ��Ǥ���.
���η�̤�2-3�������줿��ΤȰ��Ǥ���.
�����Ȥ����ˤ���������$\delta_2 = 0$(�󰵽�), $\delta_1 = 1$�Ȥ���.
����Ȱʲ��μ���������.
%
\begin{eqnarray}
 \nu^2 = \frac{\displaystyle \mu^2 \frac{g}{\overline{\theta}}\DP{\overline{\theta}}{z}}{k^2 + \mu^2}.
\end{eqnarray}
%
(3-9)
\footnote{
 $c = \nu (k^2 + \mu^2)^{-1/2}$.
}
��Ȥ���, ����®�٤�
%
\begin{eqnarray}
 c = \pm \frac{\mu}{\mu^2 + k^2}
  \left(
   \frac{g}{\overline{\theta}} \DP{\overline{\theta}}{z}
  \right)^{1/2}
\end{eqnarray}
%
�Ȥʤ�
\footnote{
 ����®�٤��ȿ�$\mu$, $k$�˰�¸���Ƥ��뤳�Ȥ������������Ȥ�
 ʬ���ȤǤ��뤳�Ȥ��狼��.
 �����ʬ���ط��μ�(3-15)�����狼��.

 ;�̤���, ���Ȥ�2-3���ʬ���ط��μ�(3-7), ����®��(3-10)����
 ʬ���ȤǤϤʤ����Ȥ��狼��.
}.
���θ�����������ﲼ�Ǥʤ��Ƥ�Ŭ���ʶ���򤪤��ʤ����Ȥ�
���뤳�Ȥ��Ǥ���
\footnote{
 �����Ȥ�������Ȥ�ʤ����Ϥޤ�(3-11)�ˤ�����
 $\overline{p}$��$\overline{\alpha}$�β��
 �̤η��Ȥʤ�. ���η��, (3-12), (3-13)����ʸ�Τ�ΤȤ�¿���ۤʤäƤ���.
 ������, (3-14)��Ƴ�Ф���ݤˤ����ʤä�����Ǥ����ΰۤʤä���ʬ��
 ��Ȥ���, ��������絤��Ʊ��ʬ���ط���������.
 �ܤ���������(1975)�򻲾ȤΤ���.
}.
�����ν��Ͽ�ư
�ϲ��ٸ�Ψ��$\partial \overline{\theta} / \partial z > 0$�ξ��
\footnote{
 ���Τ��Ȥ�{\itshape subadiabatic}�Ȥ���.
}
�ϰ���Ǥ���.
�դ�$\partial \overline{\theta} / \partial z < 0$�ξ��
\footnote{
 ���Τ��Ȥ�{\itshape superadiabatic}�Ȥ���.
}
�Ͼ�����������
\footnote{
 $\partial \overline{\theta} / \partial z < 0$�ξ���
 (3-15)���$\nu$�ϵ����Ȥʤ�.
 ��ñ�Τ��ᤳ�ξ���$\nu$��$\nu = \nu_r i$�Ȥ����($\nu_r$�ϼ¿�),
 ��ư��λؿ���ʬ��$e^{i(\mu x + k z) - \nu_r t}$�Ȥʤ�,
 $t$�ˤĤ��ƤϿ�ư����������(�⤷���ϸ���)����.
}.

����α�ľ��������
\footnote{
 $z$�����ξ���Τ��ȤǤ���.
}
����ʿ�����������٤��礭�����, ���ʤ��$\mu^2 \gg k^2$
�Ǥ������(3-15)����
%
\begin{eqnarray}
 \nu \doteq \pm
  \left(
   \frac{g}{\overline{\theta}}
   \DP{\overline{\theta}}{z}
  \right) ^{1/2}
\end{eqnarray}
%
�Ǥ���.
����ϱ�ľ�����ο�ư�ο����ɽ��
{\bfseries �֥���-�Х����鿶ư��}({\itshape Brunt-Vaisala frequency})
\footnote{
 {\bfseries �֥���-�Х����鿶ư��}({\itshape Brunt-Vaisala frequency})
 ��{\bfseries ���Ͽ�ư��}({\itshape buoyancy frequency})
 �Ȥ�ƤФ��. ���� �絤���δĶ������ƥ��칶(2001)�򻲾ȤΤ���.
}
�ȸƤФ��.

����, ��ľ�������뤬��ʿ����������Ϥ뤫�˾��������, �ſ尵ʿ�ն����
����Ω��, (3-14)�ˤ�����$\delta_1$�ϥ����Ȥʤ�.
���ξ��, ��ư����
%
\begin{eqnarray}
 \nu^2 = \frac{\mu^2}{k^2} \frac{g}{\overline{\theta}}
  \DP{\overline{\theta}}{z}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. ���μ���$k^2 \gg \mu^2$���Ѥ���(3-15)������뤳�Ȥ�Ǥ���.
��ľ�������Ѳ���ȼ����ʿ���Ťβ�ϰʲ��Τ褦��
$+ k$��$- k$�β��­����碌�뤳�ȤǺ�뤳�Ȥ��Ǥ���
\footnote{
 �������뤳�Ȥˤ�äƱ�ľ�����ι�¤�Ͽ�ʿ�����ι�¤����֤����
 ��Ω�ˤʤ�.
 �Ĥޤ�, $x$��$t$�������Ȥ����Ť�ͤ��뤳�Ȥ��Ǥ���褦�ˤʤ�.
}:
%
\begin{eqnarray}
 e^{i (\mu x + k z - \nu t)} + e^{i (\mu x - k z - \nu t)}
  = 2 \cos kz \ e^{i (\mu x - \nu t)}.
  \nonumber
\end{eqnarray}
%
��������$\nu$��$k$���Ф�����Ω�Ǥ�����ˤ����ʤ����Ȥ��Ǥ���
\footnote{
 $k$��$\mu$��$\nu$�˰�¸���Ƥ��������ʸ�Τ褦��$+k$��$-k$��2�Ĥ�
 ξ���Ȥ�ͤ��뤳�ȤϤǤ��ʤ�.
 ($\mu$��$\nu$�ˤ�ä�$k$�����ꤷ�Ƥ��ޤ�����)
}.
���ξ��ˤ����ŤϿ�ʿ�����ˤΤߵ�����, ����®�٤�
%
\begin{eqnarray}
 c = \frac{\nu}{\mu} =
  \pm \frac{1}{k}
  \left(
   \frac{g}{\overline{\theta}} \DP{\overline{\theta}}{z}
  \right)^{1/2}
\end{eqnarray}
%
�Ȥʤ�
\footnote{
 (3-8) $c = \nu \mathcal{L} / 2 \pi$���, �����ǤϿ�ʿ�����ΰ���®�٤Τ�
 ��ͤ��Ƥ���Τ�$\mathcal{L} = 2 \pi /\mu$�ʤΤ�
 $c = \nu/\mu$�Ȥʤ�.

 �ޤ�, �����˵�᤿
 $\nu^2 = (\mu ^2 g / k^2 \overline{\theta})
 (\partial \overline{\theta}/ \partial z)$
 �����Ҥ��������������뤳�Ȥ�(3-18)��Ƴ�ФǤ���.
}.
$k$������˴ؤ�餺, $\nu$������Υڥ��β�
$\nu = \pm (\mu / k)\Big\{( g / \overline{\theta})
(\partial \overline{\theta}/ \partial z)\Big\}^{1/2}$
����Ĥ���, (3-18)��$c$�������2�Ĥβ�����.
(3-14)�ˤ����벻�ȤȽ����Ȥβ�Ϥ��줾���ʬ���Ƽ��Ф����Ȥ��Ǥ���.
$\nu^2$���礭���Ȳ��ꤹ�뤳�Ȥǲ��Ȥ�����, $\nu^2$���������Ȳ��ꤹ�뤳�Ȥ�
�����Ȥ�������Ф����Ȥ��Ǥ���
\footnote{
 $\nu$(��ư��)���礭�������ꤹ��Ȥ������Ȥ�,
 ����Ȥ��Ȥ�������Ф��Ȥ������ȤǤ���.
 �Ĥޤ�, ����Ȥ��Ȥ�����Ȥ�����Ȳ��Ȥ����Ф�, ����Ȥ��Ȥ����
 ���Ф������������Ȥ�������Ȥ������ȤǤ���.
 �ֽ��Ϥ�������Ȳ��ȤΤߤ����Ф�, �󰵽̤ˤ��������������
 �����Ф���פȤ������ȤϤ���������Ҥ�����, ���Τ褦��
 ��ư��(���ȿ�)�ˤ����̤ˤ�äƤⲻ�Ȥ����������Ȥ�ʬΥ���뤳�Ȥ�
 �Ǥ���.
}
\footnote{
 $\nu^2$���礭���Ȳ��ꤹ���(3-14)�κ�����3�����1, ��2�����٤ƾ�����
 �Τ���Ȥ����Ȥ��Ǥ�, ��̤Ȥ��Ʋ��Ȥβ�������Ĥ�.
 ����$\nu^2$���������Ȳ��ꤹ��Ⱥ��դ���1�����2��, ��3�����٤ƾ�����
 �Τ���Ȥ����Ȥ��Ǥ�, ��̤Ȥ��ƽ����Ȥβ�Τߤ��Ĥ�.
}.

���Ȥ���Ω�äƽ��פ��ϳ�Ū����̤����褦�ʵ��ݸ��ݤ�̵��.
���Τ����������Ϥ���ϲ��Ȥ������������˾�ޤ�����礬¿��
\footnote{
 ���Ȥ�������Ф���������ͷ׻�����ñ�ˤʤ�.
 �ܤ������ȤϾ��� ¾(1996)��
 ���� �絤���δĶ������ƥ��칶(2001)�򻲾ȤΤ���.
}.
(3-14)�Ǹ�����褦��, �ſ尵���($\delta_1 = 0$)�ޤ���
�󰵽�($\delta_2 = 0$)��Ȥä����ϲ��ȤϽ��������.
���������Ȥ�Ǯ��ή�仳���ȤȤ��ä���������ξ������絤�α�ư�Ǥ�
���פ�����̤���.
Ŭ�ڤʾ��
\footnote{
 ��®�٤�������ˤʤ�褦�ʾ��Τ���.
}
��Ϳ��������������Ȥ��絤��ˤ����ƥ��ͥ륮����
�����������Ť���
\footnote{
 �������������Ȥˤ�륨�ͥ륮�������ŤˤĤ��Ƥ�
 ���� �絤���δĶ������ƥ��칶(2001)��
 ���ȤΤ���.
}.
�����ȤϤ���¾��{\bfseries �Ϲ���Ĵ��}
({\itshape geostrophic adjustment})�ȸƤФ������̤���.
����ˤĤ��Ƥϸ��2-8��ˤƵ�������.

�絤���絬�Ͻ۴ĥ�ǥ�Ǥ��ſ尵ʿ�դ������뤿��ۤȤ�ɤβ��Ȥ�
�����Ǥ���.
������, ��ˤ��ſ尵ʿ�դ��Ѥ��Ƥ�����Ǥ��ʤ��ü�ʲ��Ȥ⤢��.
���줬{\bfseries �����}({\itshape Lamb wave})�Ǥ���.
�ʲ��ǤϤ��Υ���ȤˤĤ��ƹͤ���.
������(3-2)��(3-12)�Τ褦�ʷ��β�ʳ��ˤ������絤�ˤ�����
��ư�����äƤ���.
�����ǹͤ����Ȥϱ�ľ®�٤������, ��ʿ�����˲�®�����Ť���.
�ʲ��β��(3-2)��ľ���������뤳�Ȥ�, �����β�(3-2)��������
���Ȥ��狼��
\footnote{
 �ºݤ�(3-19)��(3-2)���������Ƥߤ���̤˴ؤ��Ƥ�
 {\bfseries 3-4-B}�򻲾ȤΤ���.
}:
%
\begin{eqnarray}
 u' &=& S e^{(\gamma - 1) z/ \gamma H} e^{i \mu (x - ct)},
  \nonumber \\
 w' &=& 0,
  \nonumber \\
 p' &=& P e^{-z/ \gamma H} e^{i \mu (x - ct)},
  \nonumber \\
 \alpha ' &=& A e^{(2 \gamma - 1) z / \gamma H} e^{i \mu (x - ct)}.
\end{eqnarray}
%
�����Ǥ�$c = \pm \sqrt{\gamma R \overline{T}}$�Ǥ���.
���β򤬥���ȤǤ���
\footnote{
 $W = 0$�Ȥ�������Ĥ�������(�Ĥޤ�ϥ���Ȥʤ櫓����)��
 ʬ���ط���Ƴ�Ф˴ؤ��Ƥ�{\bfseries 3-4-C}����.
}.
�����Ȥϼ�ưŪ��$w=0$�������ˤ���������Ω�ĤȤ�������������.
���β�ϸ����ǵ�������{\bf ����������}({\it external gravity wave\/})
�������ɤ����Ƥ���.
���Υ���Ȥ��絤��ˤ����ƥ��ͥ륮����ۤȤ��͢�����ʤ���,
����ͽ���ǥ�ǤϽ��פǤ���.
�ʤ��ʤ�, ����Ȥˤ�ä��ſ尵�μ��ο�����ʬ�ˤ���������
�����ॹ�ƥåפ˸��������¤��ä�뤫��Ǥ���
\footnote{
 ����Ȥȿ��ͷ׻��Ȥδط��ˤĤ��Ƥ�{\bfseries 3-4-D}����.
}.




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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-4-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-4-A\qquad (3-14)�κǸ�ι��Ƴ��}
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����(3-13)�����᤿�ط����κǸ�ι�
$\mu^2 g^2 (\gamma -1)$����(3-14)�κǸ�ι�
$\displaystyle \mu^2 \frac{\gamma R \overline{T} g}{\overline{\theta}}
\DP{\overline{\theta}}{z}$�ؤ��ѷ��򤪤��ʤ�.

���̤δط���
\footnote{
 $\overline{\theta}, \overline{T}, \overline{p}$�Ϥ��줾��
 �������ؾ��֤Ǥβ���, ����, �������̣����.
}
$\overline{\theta} = \overline{T}(p_s/\overline{p})^{\kappa}$
��$z$����ʬ�����,
%
\begin{eqnarray}
 \DP{\overline{\theta}}{z} &=&
  \overline{T} p_s^{\kappa} \DP{\overline{p}^{-\kappa}}{z}
  \nonumber \\
 &=& - \overline{T} \left( \frac{p_s}{\overline{p}} \right)^{\kappa}
  \frac{\kappa}{\overline{p}} \DP{\overline{p}}{z}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. �������ſ尵�δط���
$\partial \overline{p} / \partial z = - g / \overline{\alpha}$
�Ȥ����ۤɻ��Ѥ������̤δط������Ѥ����,
%
\begin{eqnarray}
 \DP{\overline{\theta}}{z} &=&
  \overline{\theta} \frac{\kappa g}{\bar{p} \bar{\alpha}}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. ������ſ尵ʿ�դμ�
$\bar{p} \bar{\alpha} = R \overline{T}$��Ȥ����Ȥ�,
%
\begin{eqnarray}
 \DP{\overline{\theta}}{z} &=&
  \overline{\theta} \frac{\kappa g}{R \overline{T}}
  \nonumber \\
 \frac{\gamma R \overline{T} g}{\overline{\theta}}
  \DP{\overline{\theta}}{z}
 &=&
  g^2 \gamma \left( \frac{R}{c_p} \right)
  \nonumber \\
 &=&
  g^2 \gamma \left( \frac{c_p - c_v}{c_p} \right)
  \nonumber \\
 &=&
  g^2 \gamma \left( 1- \frac{c_v}{c_p} \right)
  \nonumber \\
 &=&
  g^2 \gamma \left( 1- \frac{1}{\gamma} \right)
  \nonumber \\
 &=&
  g^2 \left( \gamma -1 \right)
  \nonumber \\
 \mu^2 \frac{\gamma R \overline{T} g}{\overline{\theta}}
  \DP{\overline{\theta}}{z}
 &=&
  \mu^2 g^2 \left( \gamma -1 \right)
\end{eqnarray}
%
�Ȥʤ뤳�Ȥ��狼��.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-4-B      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-4-B\qquad ����Ȥβ�}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-4-B-\arabic{equation}}
\def\thefigure{3-4-B-\arabic{figure}}

����Ȥβ�(3-19)��������(3-2)�����������Ȥ򼨤�.

�ʲ���(3-2)�μ�:
%
\begin{eqnarray}
  \DP{u'}{t} + U \DP{u'}{x} + \overline{\alpha} \DP{p'}{x}
 &=& 0,
 \nonumber \\
  \delta_1
   \left(
    \DP{w'}{t} + U \DP{w'}{x}
   \right)
  + \overline{\alpha} \DP{p'}{z}
  - \frac{g \alpha ' }{\overline{\alpha}}
 &=& 0,
 \nonumber \\
 \Deqlab{hw-02-04-��ư����������}
  \overline{\alpha}
  \left(
   \DP{p'}{t} + U \DP{p'}{x}
  \right)
  - g w'
  + \overline{p} \gamma
   \left(
    \DP{\alpha '}{t} + U \DP{\alpha'}{x} + w' \DP{\overline{\alpha}}{z}
   \right)
 &=& 0,
  \\
 \left(
  \DP{u'}{x} + \DP{w'}{z}
 \right)
 \overline{\alpha}
 - \delta_2
  \left(
   \DP{\alpha '}{t} + U \DP{\alpha '}{x}
  \right)
 - w' \DP{\overline{\alpha}}{z}
 &=& 0
 \nonumber
\end{eqnarray}
%
�˰ʲ��Υ���Ȥβ�(3-19)����������.
%
\begin{eqnarray}
 u' &=& S e^{(\gamma - 1) z/ \gamma H} e^{i \mu (x - ct)},
  \nonumber \\
 w' &=& 0,
  \nonumber \\
 p' &=& P e^{-z/ \gamma H} e^{i \mu (x - ct)},
  \nonumber \\
 \Deqlab{hw-02-04-����Ȥβ�}
  \alpha ' &=& A e^{(2 \gamma - 1) z / \gamma H} e^{i \mu (x - ct)}.
\end{eqnarray}
%
�������ä�, \Deqref{hw-02-04-��ư����������}�Τ��줾��μ���
%
\begin{eqnarray}
 c S e^{z/H} - \overline{\alpha} P &=& 0,
  \nonumber \\
 (\overline{\alpha} / \gamma H) P
  + (g / \overline{\alpha}) A e^{2 z / H} &=& 0,
  \nonumber \\
 \overline{\alpha} c P + \overline{p} \gamma c A e^{2z / H} &=& 0,
  \nonumber \\
 \overline{\alpha} S e^{z/H} + c A e^{2z / H} &=& 0.
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. �ʤ�, $U = 0$�Ȥ��Ƥ���. ��������ǽ񤯤�,
%
\begin{eqnarray}
 \left[
  \begin{array}{ccc}
   c e^{z/H} & - \overline{\alpha} & 0 \\
   0 & \overline{\alpha} / \gamma H & (g / \overline{\alpha}) e^{2z/H} \\
   0 & \overline{\alpha} c & \overline{p} \gamma c e^{2z/H}\\
   \overline{\alpha} e^{z/H} & 0 & c e^{2z/H} \\
  \end{array}
 \right]
 \left[
  \begin{array}{c}
   S \\ P \\ A
  \end{array}
 \right]
 = 0
\end{eqnarray}
%
�Ȥʤ�. ���ι��󼰤�Ÿ�������, 
%
\begin{eqnarray}
 (\overline{\alpha} / \gamma H) \overline{p} \gamma c e^{2z/H}
  - \overline{\alpha} c (g/\overline{\alpha}) e^{2z/H}
 &=& c e^{2z/H}
  \left\{
   \bar{\alpha} \bar{p}/H - g
  \right\}
  \nonumber \\
 &=& c e^{2z/H}
  \left\{
   g (\bar{\alpha} \bar{p}/ R \overline{T}) - g
  \right\}
  \nonumber \\
 &=& 0
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. ����Ǿ���������$\bar{p} \bar{\alpha} = R \overline{T}$��
��������ϥ���$H = R \overline{T} / g$�����Ѥ���.

����ˤ�ä�, ����Ȥβ�(3-19)��(3-2)�����������Ȥ������줿.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-4-C      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-4-C\qquad ����Ȥ�ʬ���ط���Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-4-C-\arabic{equation}}
\def\thefigure{3-4-C-\arabic{figure}}

$W = 0$���ȤǤ������Ȥ�ʬ���ط���Ƴ�Ф���.
$W = 0$�Ȥ��Ʊ�ľ�����α�ư��������ͤ��ʤ����, (3-13)��
�ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \left[
  \begin{array}{ccc}
   - \nu & \mu & 0 \\
   0 & \nu & \gamma R \overline{T} \nu \\
   \mu & 0 & \delta_2 \nu \\
  \end{array}
 \right]
 \left[
  \begin{array}{c}
   S \\ P \\ A
  \end{array}
 \right]
 = 0.
 \nonumber
\end{eqnarray}
%
���ι��󼰤��������������Ȥ����, �ʲ��δط�����Ƴ�����.
%
\begin{eqnarray}
 \nu \left\{ \delta_2 \nu^2 - \mu^2 \gamma R \overline{T} \right\} = 0.
  \nonumber
\end{eqnarray}
%
�󰵽�($\delta_2 = 1$)�Ǥ�����ϰʲ���ʬ���ط�����Ƴ�����.
%
\begin{eqnarray}
 \nu = \pm \mu \sqrt{\gamma R \overline{T}}.
  \nonumber
\end{eqnarray}
%
(3-7)
\footnote{
 $c = \nu / \sqrt{k^2 + \mu^2}.$
}
�����Ѥ��뤳�Ȥǥ���Ȥΰ���®�٤�������:
%
\begin{eqnarray}
 c = \pm \sqrt{\gamma R \overline{T}}.
  \nonumber
\end{eqnarray}
%
���β���������Ȥ�Ʊ���Ǥ���. (����Ȥⲻ�ȤʤΤ���������).
������, ����Ȥ����ȤȰ㤦����ſ尵ʿ�դζ���ǤϾ��Ǥ��ʤ����Ǥ���.
���Τ�������, ���ͷ׻��ˤ����Ƥϥ���Ȥ��θ����ɬ�פ�������ΤǤ���.



\subsubsection*{3-4-C-a \quad ��ľ�����ؤ�����}

��ʸ��Ǥϥ���ȤϿ�ʿ�����ˤΤ����¤���ȽҤ٤Ƥ���.
����ϥ���Ȥ�$W=0$�Ǥ���Ȥ�����������Τ뤳�Ȥ��Ǥ���.
���Ȥ�Ʊ��������(3-4)��2���ܤμ�
\footnote{
 $(\mu U - \nu) W + \overline{\alpha} k P = 0$.
}
��$W=0$���Ѥ����$\overline{\alpha} k P = 0$�Ȥʤ�.
$\overline{\alpha}$��$P$�Ͼ�˥����Ȥʤ�櫓�ǤϤʤ��Τ�
$k = 0$�Ȥʤ�. �Ĥޤ��ľ�ȿ��������ˤʤ�Ȥ������ȤʤΤ�
��ľ�����ˤ��Ȥ����Ť��ʤ����Ȥ��狼��.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-4-D      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-4-D\qquad ����Ȥȿ��ͷ׻��Ȥδط�}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-4-D-\arabic{equation}}
\def\thefigure{3-4-D-\arabic{figure}}

��ʸ�Ǥ�Ҥ٤Ƥ��뤬, ���ͷ׻��򤹤�ݤˤϤǤ���������ȤϽ���������
�ɤ�. ���̤β��Ȥ��絬�ϥ�ǥ�ǻ��Ѥ����ſ尵ʿ�ն���ˤ�ä�
���������Τ���, ���Υ���Ȥ����Ϥ��ζ���ˤ�äƼ��������Ȥ�
�Ǥ��ʤ�
\footnote{
 �����Ƥ����ͷ׻��Ǥϥץ�ߥƥ�����������($p$���ľ��ɸ�Ȥ�����������
 �Τ���. 1-9�Ỳ��)����Ѥ���. �����������ϤǤϰ츫�����Ȥ���̩�٤ι��
 �������Ϥ���ä��Ƥ��뤬, �褷���󰵽̤��ꤷ����ΤǤϤʤ��Τ�
 ����ȤϾ��Ǥ��ʤ�.
}.
�ʲ��ǤϤʤ����Ȥ�������Ȥ�ɬ�פȤ���, ����Ȥ��ĤäƤ��ޤ����Ȥ������
�ʤ�Τ���Ҥ٤�.

���ͷ׻��Ǥ�{\bfseries CFL���}
({\itshape Courant-Friedrichs-Lewy condition})�Ȥ�����Τ�
¸�ߤ���. ����ϰʲ��Τ褦�ʾ��Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-02-04-CFL���}
  \Delta t < \Delta x / c. \hspace{50mm}
  \\
 c: {\rm ®��}, \quad \Delta t : {\rm ������ʬ�Υ����ॹ�ƥå�} , \quad
  \Delta x: {\rm ���֥��ƥå�}
  \nonumber
\end{eqnarray}
%
���ξ�郎��������ʤ����, ���ͷ׻����԰���ˤʤäƤ��ޤ�
\footnote{
 CFL���˴ؤ��Ƥϴ��(1997)�򻲾ȤΤ���.
}.
���Ȥ����������Ȥ���٤�®��$c$�������礭������,
���Ȥ��θ�������ϥ����ॹ�ƥå�$\Delta t$�򾮤������ʤ����
�ʤ�ʤ�.
�դ˸�����, ����Ȥ�̵�뤷�ƥ����ॹ�ƥåפ��礭����äƤ��ޤä����ˤ�
���ͷ׻��Ϥɤ�ɤ��԰���ˤʤäƤ��ޤ��櫓�Ǥ���.
���줬��ʸ�Ρ֥���Ȥˤ�äƺ���Υ����ॹ�ƥåפ˸��������¤��ä���
���Ȥΰ�̣�Ǥ���.

�ʤ�, ������ǤϹ����(����), �����(����������)���ü�ʲ��ȤǤ�������
���о줷���櫓����, 
\Dfigref{hw-02-04-����������}��
�������Ȥ�ʬ���ط���ɽ����������դǤ���.
�ܤ����Ͼ���(1978)�򻲾Ȥ��Ƥ�餤����.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/gravity_wave.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �����, �����, ����Ȥ�ʬ���ط�
  (���ҵ���, 1978:
   �����ϳ�����.
   �����ؽ��ǲ�, 249pp. ���) .}
  }
  \Dfiglab{hw-02-04-����������}
 \end{quote}
\end{figure}
%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      Appendix        %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection*{Appendix \qquad }

%\subsubsection*{}


%%%%%%%%              Text End                  %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
