%
% Chapter 2   Wave Motion in the Atmosphere  Part I
%             2-8 GEOSTROPHIC ADJUSTMENT
%
% 2002/10/13  ���� ����   ����
% 2002/10/31  ���� ����   ����
% 2002/11/22  ���� ����   ���ߥʡ��ˤ�뽤��
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\section{�Ϲ���Ĵ��}

\begin{itembox}[l]{����}
 �礭�ʵ��ϤǸ������, �絤�Ϥۤ��Ϲ�ή�Х�󥹤��ݤäƤ���.
 �ɽ�Ū���Զѹդ���������������Ȥ����Υ��ͥ륮���򳰤����¤���
 �Ϲ�ή�Х�󥹤��ݤ����褦�ˤʤäƤ���. �����{\bfseries �Ϲ���Ĵ��}
 ({\itshape geostrophic adjustment})�Ȥ���.
 ���θ��ݤ�ץ�ߥƥ����������Ϥν���ͤ���������Ѥ���.
 
 �ޤ�, ����������������ա��ꥨ�Ѵ������������Ϥ��, ���β����ְ�¸
 �ι�Ȥ����Ǥʤ����ʬ����. �����β�Ϥ��줾���Զѹդξ��֤��Ϲ�ήĴ
 �����Ż߾��֤Τ�ΤǤ���.
 �����β���٤ϵեա��ꥨ�Ѵ���, ��®, �⤵�ν���ͤ����.
\end{itembox}

2-7��ǤϤ��줾�줬Ʊ����Ĺ�����2�Ĥν����Ȥȥ����ӡ��Ȥ���Ӥ�
��갷�ä�.
�����Ȥ�, �໤��̵���¤�ݻ�����³�����Ϲ�ή�Х�󥹤˴ؤ���
������ư�����߽Ф�.
������, �⤷������֤ˤ�����̵���ΰ���Τ���¤�줿��ʬ���Զѹդ�
ȯ�������Ȥ��Ƥ�, 
���Ū�˽����ȤϤ����Զѹդ򤽤��ΰ賰�����¤���.
���ΰ�����֤�Ĵ���ˤ��Ϲ�ή�Х�󥹤ξ��֤����.
���Υץ�������, �Դ����ʽ�����֤�Ϳ����줿����
�ץ�ߥƥ��������Ϥα�������ꤹ��.
�ºݤ�, ����Ĵ��ץ���������ǽ��³���뤪������, �絤�Ϥۤ��Ϲ�ή�Х�󥹤�
���֤��ݤäƤ���.

�Ż߾��֤���ܾ�Ȥ������������������Ϥϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 & & \DP{u}{t} - f v + g \DP{h}{x} = 0,
  \nonumber \\
 & & \DP{v}{t} + f u = 0,
  \\
 & & \DP{h}{t} + H \DP{u}{x} = 0.
  \nonumber
\end{eqnarray}
%
������(3-41)�ˤ�$U=0$�Ȥ������뤳�Ȥ��Ǥ���.
�����ˤ⼨�����褦�ˤ��ηϤ�3�Ĥ���ư������.
2�Ĥδ���������(3-44)�ȥ����ӡ���(3-50)�Ǥ���.
$H$������λ�, $\partial \overline{q} / \partial y = 0$�ʤΤ�
\footnote{
 (3-49)�����������ΤǤ���.
}
�����ӡ��Ȥ��Żߤ���
\footnote{
 (3-50)���, $\partial \overline{q} / \partial y = 0$�����$U=0$
 �ξ��, �����ӡ��Ȥΰ���®�٤ϥ����ˤʤ뤳�Ȥ�ʬ����.
}.

��������, ���֥ա��ꥨ�Ѵ����Ѥ���(3-70)���.
�ա��ꥨ�Ѵ��ϰʲ��Τ褦����������.
%
\begin{eqnarray}
 \tilde{u} (\mu , t) = \int_{- \infty}^{\infty} u (x,t) e^{- i \mu x} \ dx.
\end{eqnarray}
%
���Ѵ��ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 u (x,t) = \frac{1}{2\pi} \int_{- \infty}^{\infty} \tilde{u} (\mu , t)
  e^{i \mu x} \ d\mu.
\end{eqnarray}
%

(3-70)��$e^{- i \mu x}$�򤫤�, $- \infty$����$\infty$�ޤ���ʬ����.
$x$��ʬ�ι����ʬ��ʬ�����, ̵�¤ˤ����������,
�������Ϥϰʲ��Τ褦�����̤���ʬ�������ˤʤ�
\footnote{
 ���η׻��˴ؤ��Ƥ�{\bfseries 3-8-A}����.
}.
%
\begin{eqnarray}
 & & \DD{\tilde{u}}{t} = f \tilde{v} - i \mu g \tilde{h},
  \nonumber \\
 & & \DD{\tilde{v}}{t} = - f \tilde{u},
  \\
 & & \DD{\tilde{h}}{t} = - i \mu H \tilde{u}.
  \nonumber
\end{eqnarray}
%
$\tilde{v}$��$\tilde{h}$��(3-71)��Ʊ�ͤ���������.
$u$�ν���ͤΥա��ꥨ�Ѵ��ϰʲ��Τ褦��Ϳ������.
%
\begin{eqnarray}
 \tilde{u}_0 = \tilde{u} (\mu , 0) =
  \int_{- \infty}^{\infty} u (x, 0) e^{- i \mu x} dx.
\end{eqnarray}
%
$\tilde{v}_0$��$\tilde{h}_0$��Ʊ�ͤ���������.
��(3-73)��, �����������������Ϥβ�Ȥ�������줿��Τ�Ʊ��,
$e^{- i \mu c_j t}$�η��β�����.
�Ĥޤ�$U=0$�Ǥ�����ˤ�, ����®�٤�
$c_1 = 0, c_2 = \mu^{-1} \nu, c_3 = - \mu^{-1} \nu$�Ȥʤ�.
�ʤ�, �����Ǥο�ư��$\nu$��
%
\begin{eqnarray}
 \nu = \left( f^2 + \mu^2 g H \right)^{1/2}
\end{eqnarray}
%
�Ǥ���.
Ǥ�դ�$\tilde{u}_0, \tilde{v}_0, \tilde{h}_0$�˴ؤ�����̲��
(3-61)��$u$��$\tilde{u}$���֤�������ʤɤ�������,
$U=0$�Ȥ���(3-64)�������뤳�Ȥ��Ǥ���.
�������Ȥ߹�碌�뤳�Ȥǰʲ��β�������
\footnote{
 �����β��Ƴ�Ф˴ؤ��Ƥ� {\bfseries 3-8-B}����.
}.
%
\begin{eqnarray}
 \tilde{u} (\mu , t) &=& \tilde{u}_0 \cos \nu t
  + \left( \frac{f \tilde{v}_0 - i \mu g \tilde{h}_0}{\nu} \right)
  \sin \nu t,
  \nonumber \\
 \tilde{v} (\mu , t) &=&
  \frac{\mu^2 g H \tilde{v}_0 + i \mu g f \tilde{h}_0}{\nu^2}
  - \frac{f \tilde{u}_0 \sin \nu t}{\nu}
  + \left( \frac{f^2 \tilde{v}_0 - i \mu g f \tilde{h}_0}{\nu^2} \right)
  \cos \nu t,
  \\
 \tilde{h} (\mu , t) &=&
  \left( \frac{f^2 \tilde{h}_0 - i \mu H f \tilde{v}_0}{\nu^2} \right)
  - \frac{i \mu H \tilde{u}_0 \sin \nu t}{\nu}
  + \left( \frac{\mu^2 g H \tilde{h}_0 + i \mu H f \tilde{v}_0}{\nu^2} \right)
  \cos \nu t.
  \nonumber
\end{eqnarray}
%
�ɤμ��������֤���ʬ$(\quad)_S$�Ȼ����Ѳ�������ʬ$(\quad)_T$��­����碌
�ˤ�äƽ񤯤��Ȥ��Ǥ���. ���줾��򵭽Ҥ����,
%
\begin{eqnarray}
 \tilde{u}_S (\mu) &=& 0,
  \nonumber \\
 \tilde{v}_S (\mu) &=& \frac{i \mu g}{f}
  \left( \frac{f^2 \tilde{h}_0 - i \mu f H \tilde{v}_0}{\nu^2} \right),
  \\
 \tilde{h}_S (\mu) &=&
  \left( \frac{f^2 \tilde{h}_0 - i \mu f H \tilde{v}_0}{\nu^2}\right)
  \nonumber
\end{eqnarray}
%
�����,
%
\begin{eqnarray}
 \tilde{u}_T (\mu) &=& \tilde{u}_0 \cos \nu t
  + \frac{f}{\nu}
  \left( \tilde{v}_0 - i f^{-1} \mu g \tilde{h}_0 \right)
  \sin \nu t,
  \nonumber \\
 \tilde{v}_T (\mu) &=& - \frac{f}{\nu} \tilde{u}_0 \sin \nu t
  + \frac{f^2}{\nu^2}
  \left( \tilde{v}_0 - i f^{-1} \mu g \tilde{h}_0 \right)
  \cos \nu t,
  \\
 \tilde{h}_T (\mu) &=& - \frac{i \mu H}{\nu} \tilde{u}_0 \sin \nu t
  + \frac{i \mu H f}{\nu^2}
  \left( \tilde{v}_0 - i f^{-1} \mu g \tilde{h}_0 \right)
  \cos \nu t
  \nonumber
\end{eqnarray}
%
�Ǥ���.
�Ż߾��֤β�(3-77)��$\tilde{v}_S = i \mu g f^{-1} \tilde{h}_S$,
$\tilde{u}_S = 0$�ʤΤ��Ϲ����Х�󥹤ξ��֤�ɽ��.
�����(3-73)�ˤ�����$d/dt=0$�Ȥ�������Ʊ���Ǥ���.
�����Ѳ������(3-78)�Ͻ�����֤ˤ������Ϲ����Х�󥹤���Τ����ľ��
���㤷�Ƥ���. �Ϲ����Х�󥹤���Τ���Ȥ�,
$\tilde{v}_0 - i \mu g f^{-1} \tilde{h}_0$��$\tilde{u}_0$�Τ��ȤǤ���. 

�Ż߾��֤β�ϸ�Ǥ��ܺ٤�Ĵ�٤뤳�Ȥˤ���,
�����ǤϤޤ������Ѳ�����򤬻��֤����ä���ˤĤ�ƾä��뤳�Ȥ򼨤�.
���Ѵ�(3-72)��Ȥ��Ȼ����Ѳ������ϰʲ��Τ褦�˽񤯤��Ȥ��Ǥ���.
%
\begin{eqnarray}
 u_T (x,t) &=& \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \tilde{u}_0 \cos \nu t \ e^{i \mu x} \ d\mu
  + \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \frac{f}{\nu} \tilde{d} (\mu , 0) \sin \nu t \ e^{i \mu x} \ d\mu,
  \nonumber \\
 v_T (x,t) &=& - \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \frac{f \tilde{u}_0}{\nu} \sin \nu t \ e^{i \mu x} \ d\mu
  + \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \frac{f^2}{\nu^2} \tilde{d} (\mu , 0) \cos \nu t \ e^{i \mu x} \ d\mu,
  \nonumber \\
 h_T (x,t) &=& - \frac{i}{2\pi} \int_{- \infty}^{\infty}
  \frac{\mu H}{\nu} \tilde{u}_0 \sin \nu t \ e^{i \mu x} \ d\mu
  + \frac{i}{2\pi} \int_{- \infty}^{\infty}
  \frac{\mu H f}{\nu^2} \tilde{d} (\mu , 0) \cos \nu t \ e^{i \mu x} \ d\mu.
  \nonumber \\
\end{eqnarray}
%
�ʤ�,
%
\begin{eqnarray}
 d(x,t) = \nu (x,t) - g f^{-1} \DP{h}{x} (x,t)
  \nonumber
\end{eqnarray}
%
�Ǥ���, ������Ϲ����Х�󥹤���Τ�����̣����
\footnote{
 ���μ���$e^{- i \mu x}$�򤫤�, $- \infty$����$\infty$�ޤ���ʬ�����,
 %
 \begin{eqnarray}
  \int_{-\infty}^{\infty} d (x,t) e^{- i \mu x} dx
   &=& \int_{-\infty}^{\infty} v (x,t) e^{- i \mu x} dx
   + (g/f)\int_{-\infty}^{\infty} (\partial h / \partial x) e^{- i \mu x} dx
   \nonumber \\
  \tilde{d} (\mu, t) &=& \tilde{v} (\mu ,t)
   + (g/f)
   \left\{
    \left[h e^{- i \mu x} \right]_{-\infty}^{\infty}
    + i \mu \int_{-\infty}^{\infty} h e^{-i \mu x} dx
   \right\}
   \nonumber \\
  &=& \tilde{v} (\mu ,t)
   + (i \mu g/f) \tilde{h} (\mu,t)
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. �������$t=0$��ͤ����,
 \begin{eqnarray}
  \tilde{d} (\mu ,0 ) = \tilde{v}_0 + (i \mu g / f) \tilde{h}_0
   \nonumber
 \end{eqnarray}
 %
 �Ǥ���.
}.
Schoenstadt(1977)
��{\bfseries ����ˡ}({\itshape saddle point method}), �ޤ���
{\bfseries ��α��ˡ}({\itshape stationary phase method})�ȸƤФ��
��ˡ�򤳤���������(3-79)���Ѥ���.
$x$����ꤷ, $|x| \ll L_R f t = (gH)^{1/2} t$�Ǥ����, ��������
�ʲ��Τ褦�ˤʤ�
\footnote{
 ��α��ˡ�˴ؤ��Ƥ�ͭ�� ¾(1991), ����(1995),
 ��ƣ ¾(1979), ��(1990)�ʤɤ򻲾ȤΤ���.
}.
%
\begin{eqnarray}
 u_T &\sim& \frac{1}{(2\pi \alpha)^{1/2}}
  \left[
   |\tilde{u} (x/\alpha , 0)| \cos (ft + \phi)
   + |\tilde{d} (x/ \alpha , 0)| \sin (ft + \psi)
  \right],
  \nonumber \\
 v_T &\sim& \frac{1}{(2\pi \alpha)^{1/2}}
  \left[
   - |\tilde{u} (x/\alpha , 0)| \sin (ft + \phi)
   + |\tilde{d} (x/ \alpha , 0)| \cos (ft + \psi)
  \right],
  \\
 h_T &\sim& \frac{1}{(2\pi \alpha)^{1/2}} \frac{(-Hx)}{f \alpha}
  \left[
   |\tilde{u} (x/\alpha , 0)| \sin (ft + \phi)
   - |\tilde{d} (x/ \alpha , 0)| \cos (ft + \psi)
  \right].
  \nonumber
\end{eqnarray}
%
������$\psi$��$\phi$�Ϥ�ä�����Ѳ����������٤Ǥ���.
$L_R=(gH)^{1/2}/f$��{\bfseries �����ӡ����ѷ�Ⱦ��}
({\itshape Rossby radius of deformation})
\footnote{
 $\sqrt{gH}$�Ͽ�����(������)������®�٤Ǥ���$L_R$�ϲ�ž�������٤λ���
 $f$�δ֤ǿ����Ȥ���ã�Ǥ����Υ���б����Ƥ���.
}
�ȸƤФ���̤Ǥ���.
�ʤ�, $\alpha = L_R^{2} f t$�Ǥ���.

�˸¤Ǥβ�(3-80)��, $x$����ꤷ�����,
�����Ѳ�����®��$u_T, v_T$��$t^{-1/2}$�����㤷�ƾä�,
$h_T$��$t^{-3/2}$�����㤷�ƾä��Ƥ������Ȥ򼨤��Ƥ���
\footnote{
 $\alpha = L_R^2 f t$�Ǥ��뤳�Ȥ�����.
}.
������Զѹդϴ��������Ȥ���������, ���δ��������Ȥϥ��ͥ륮���򤽤ν��
���Զѹդ��ΰ褫��󤯤˱��֤��Ȥ��餳�ο����������.
�ºݤˤ�, ���ͥ륮����{\bfseries ��®��}({\itshape group velocity})
�����¤���. ��®�٤ϰʲ��μ���Ϳ������.
%
\begin{eqnarray}
 G = \DP{\nu}{\mu} = \DP{}{\mu} (f^2 + \mu^2 g H)^{1/2}
  = \mu g H (f^2 + \mu^2 g H)^{-1/2}.
\end{eqnarray}
%
���μ���, û�ȥ��ͥ륮��
\footnote{
 �����ӡ����ѷ�Ⱦ��$L_R$����٤�``û��''�ȤΤ��ȤǤ���.
}
�������Ȥ�®�٤ǰ�ư���Ƥ������Ȥ򼨤��Ƥ���
\footnote{
 �����Ϥ����ȥ��ͥ륮���μ�����Ƥ���Ǥʤ���,
 $G = \partial \mu / \partial \mu$�Ȥϸ����ʤ���
 �����ǤϤ��μ���Ƴ�ФϹԤ�ʤ�.
}.
����, ����礭�ʥ�������Ǥϥ��ͥ륮���Ϥ���ä����ư���뤳�Ȥ�ʬ����
\footnote{
 $f$�ϥ��ꥪ��ѥ�᡼���ʤΤǵ��Ϥ����������ˤϸ��̤��夯
 $\mu g H (f^2 + \mu^2 g H)^{-1/2} \simeq \sqrt{gH}$�ȶ���Ǥ���.
 �դ��絬�Ϥˤʤä����ϥ��ꥪ��ѥ�᡼���������Ƥ���Τ�
 $f$�ϻĤ�, ��®��$G$�Ͼ������ʤ�.
}.
�����Ѳ������Ȥ�������Զѹ��ΰ褫�����¤��Ƥ��ä���ˤϰ��ꤷ�����֤�����
�Ĥ�. �����(3-80)�ˤ����ƿ���Ū�˼�����Ƥ���.
���μ��Ǥ�, $x$����ꤷ�����֤Ǥϲ���ʡ��������ʤäƤ���.
(���ʤ��, �����Ѳ������Ȥ�$x$�Ȥϴط��ʤ�ư��).

�ǽ�Ū�ʲ����$L_R^2/L^2$�˰�¸����. ��Ĺ�Υ��������$L=1/\mu$�Ǥ���.
(3-77)��񤭴������,
%
\begin{eqnarray}
 \tilde{v}_S &=& \frac{1}{1 + (L_R^2/L^2)}
  \left(
   \frac{L_R^2}{L^2} \tilde{v}_0 + i \mu f^{-1} g \tilde{h}_0
  \right),
  \nonumber \\
 \tilde{h}_S &=& \frac{1}{1 + (L_R^2/L^2)}
  \left(
   \frac{L_R^2}{L^2} \frac{f \tilde{v}_0}{i \mu g} + \tilde{h}_0
  \right)
\end{eqnarray}
%
�Ȥʤ�.
������֤Ǥϥ��ꥪ���ϤȰ��ϸ��ۤ�Ʊ�������������Ȳ��ꤹ��.
���Τ��Ȥ�$\tilde{v}_0$��$i \mu f^{-1} g \tilde{h}_0$��Ʊ������������
���ĤȤ������ȤǤ���
\footnote{
 $\tilde{v}_0$�����ꥪ���Ϥ�, $i \mu f^{-1} g \tilde{h}_0$�����ϸ��ۤ�
 ɽ��. �����ι���̤��(3-73)��1���ܤμ��α��դ���1��$f \tilde{v}$,
 ��2��$i \mu g \tilde{h}$�Ǥ���, �����̤��(3-70)��1���ܤμ��κ��դ�
 ��2��$fv$, ��3��$g \partial h / \partial x$�Ǥ���.
 ����$g \partial h / \partial x$��2-5��ˤ�
 $g \partial h / \partial x = (1/\overline{\rho}) \partial p /\partial x$
 �Ȥ����ط����ˤ�äư��ϸ��ۤ�񤭴�������Ǥ���.
 ���Τ��Ȥ���$\tilde{v}_0$�����ꥪ���Ϥ�, $i \mu f^{-1} g \tilde{h}_0$��
 ���ϸ��ۤ�ɽ�����Ȥ�ʬ����.
}.
�ޤ��ǽ��$L^2 \ll L_R^2$�Ǥ������ͤ���. ���ξ���(3-82)��
�ʲ��Τ褦�˶�������.
%
\begin{eqnarray}
 \tilde{v}_S = \tilde{v}_0,
  \quad
  \tilde{h}_S = \frac{f \tilde{v}_0}{i \nu g}.
\end{eqnarray}
%
�ǽ�Ū����®�Ͻ�����֤���®��������, �ǽ�Ū�ʹ��٤Ͻ�����֤���®���
�Ϲ�ήŪ�ʴط��ˤ��뤳�Ȥ�ʬ����.

$L^2 \gg L_R^2$�Ȥ�������(3-82)���Ѥ���Ȱʲ��Τ褦�˶�������.
%
\begin{eqnarray}
 \tilde{v}_S = i \mu f^{-1} g \tilde{h}_0 ,
  \quad
  \tilde{h}_S = \tilde{h}_0.
\end{eqnarray}
%
���ξ��, �ǽ�Ū�ʾ��֤����ƽ�����֤ι⤵�ˤ�äƷ��ꤵ���.
���ο����񤤤ϼ��Τ褦�����󤹤뤳�Ȥ��Ǥ���.
�������뤬$L_R$��꾮�������, �ǽ�Ū�ʾ��֤Ͻ�����֤���®��
��äƷ��ꤵ��, �������뤬$L_R$����礭�����, �ǽ�Ū�ʾ��֤Ͻ�����֤�
���̾�ˤ�äƷ��ꤵ���.
����������줿ñ��ʥ����ƥ�Ǥη�̤�, ���ʣ���ʥ����ƥ�Ǥ�
����Ω��.
���η�̤�2�����簵�絤
\footnote{
 {\bfseries �簵�絤}({\itshape barotropic atmosphere})�Ȥ�
 ��ʿ�����˲��٤����ͤǤ���, ̩�٤����ϤΤߤδؿ��Ǥ���褦���絤�Τ���
 �Ǥ���. ���Τ褦���絤��Ǥ����Ϲ⤵�����˰��ͤǤ���.
}
�ˤ��Ѥ�����.
$L_R = c_g / f$�ξ���Ϣ³���ؤ��絤
\footnote{
 ���ܾ��̩�٤��⤵�ˤ�ä��Ѳ�����褦��¿�ع�¤�����絤�Τ���.
 ������ǤΤ����ޤǤε�����, ���٤ư��ؤ��絤�Ǥ��äǤ��ä�.
}
�ˤ��η�̤���Ѥ��뤳��
���Ǥ���. $c_g$�ϴ��������Ȥΰ���®�٤Ǥ���
\footnote{
 2-6���(3-44)���$c_g = \pm \sqrt{gH + f^2/\mu^2}$�Ǥ���,
 $L_R=(gH)^{1/2}/f$�Ǥ��뤫��, �Ĥޤꥳ�ꥪ���Ϥ������ʤ�����
 ���α��Ѥϲ�ǽ�ˤʤ�. 
}.
���η�̤�, ��ľ��¤���ʿ��¤��ʬΥ���ƹͤ�,
��ͭ�ؿ����������Ƶ��뤳�Ȥ��Ǥ���.

����٤ˤ�����簵�絤��$H \approx 10$km�ξ��,
$L_R = 3000$km�Ǥ���, $2 \pi L_R = 18,000$km�Ǥ���
\footnote{
 $H \approx 10^{4}$ [m], $g \simeq 10$ [m/s$^2$]�Ǥ���,
 $\varphi = 60^{\circ}$�Ȥ����,
 $f = 2 \Omega \sin \varphi = 7.27 \times 10^{-5}$ [1/s]�Ǥ���.
 �ʤ�, $\Omega = 2\pi / 86400 = 7.27 \times 10^{-5}$ [1/s]��Ȥä�.
 �����$L_R = (gH)^{1/2} / f \simeq (316 / 7.27) \times 10^{5} \simeq
 4.35 \times 10^{6}$ [m]$=4350$ [km]�Ȥʤ�.
 ��ʸ�ȤϾ����ۤʤ뤬, ��������Ʊ�����餤���ͤȤʤ�.
}.
���ξ��, �����褽5000km����Ĺ��������ѵ��ϤǤ���ư�����餫�ˡ�û����
�Ǥ���
\footnote{
 $L=1/\mu$�Ǥ��뤫��, ��Ĺ��$2\pi L$�Ǥ���.
}.
������, �����絤�ξ��ϴ����Ȥ�®�٤�$c_g \sim 100$ m sec$^{-1}$�Ȥʤ�
����$L_R \sim 1000$ km �Ȥʤ�, ���ѵ��ϤǤ���ư�Υ��������ɤŨ����.
���Τ褦��, ������֤Ǥμ���, ��®�줬�������Ȥν���ͤ�����ˤϽ��פǤ���.
�ʤ�, Ǯ�ӤǤ�$f$������������$L_R$���礭���ʤ�, �ۤȤ�ɤ��Ȥϡ�û����.
�Ĥޤ�Ǯ�ӤǤϽ�����֤���®�줬�ǽ�Ū�ʾ��֤���ꤹ��.

������֤Ǥ��Ϲ���ʿ�դ���Τ������ʬ�ˤ�äƺǽ�Ū�ʹ��پ�򵭽Ҥ��뤳
�Ȥ�ͭ�ѤǤ���.
(3-77)��3���ܤμ����Ѥ���$h_S$�ε��Ѵ���(3-72)���Ѥ��ưʲ��Τ褦�˵���
���뤳�Ȥ��Ǥ���
\footnote{
 (3-77)���,
 %
 \begin{eqnarray}
  \tilde{h}_S (\mu) &=& \frac{f^2}{\nu^2} - \frac{i \mu f H}{\nu^2} \tilde{v}_0
   \nonumber \\
  &=& \tilde{h}_0 + \frac{f^2 - \nu^2}{\nu^2} \tilde{h}_0
   - \frac{i \mu f H}{\nu^2} \tilde{v}_0
   \nonumber \\
  &=& \tilde{h}_0 + \frac{\mu^2 gH}{\nu^2} \tilde{h}_0
   - \frac{i \mu f H}{\nu^2} \tilde{v}_0
   \nonumber \\
  &=& \tilde{h}_0 - \frac{H f i \mu}{\nu^2}
   \left(
    \tilde{v}_0 + \frac{i \mu g}{f} \tilde{h}_0
   \right)
   \nonumber \\
  &=& \tilde{h}_0 - \frac{H f i \mu}{\nu^2} \tilde{d}_0
   \nonumber 
 \end{eqnarray}
 %
 �Ǥ���. �ʤ�, (3-79)���о줷���ݤˤ����Ƥ�Ҥ٤���, 
 $\tilde{d} (\mu ,0 ) = \tilde{v}_0 + (i \mu g / f) \tilde{h}_0$�Ǥ���.
 ���μ���$2\pi e^{i \mu x}$�򤫤���$- \infty$����$\infty$�ޤ�
 ��ʬ�����(3-72)���(3-85)��������.
 %
}.
%
\begin{eqnarray}
 h_S (x) = h(x,0) - \frac{1}{2 \pi}
  \int_{- \infty}^{\infty} \frac{Hfi \mu}{\nu^2}
  \tilde{d}_0 e^{i\mu x} \ d\mu.
\end{eqnarray}
%
$d_0$�Ͻ�����֤Ǥ��Ϲ���ʿ�դ���Τ���Ǥ���.
�����Ǥ�{\bfseries �����߹��ߤ�����}({\itshape convolution theorem})
���Ѥ���������Ǥ���.
����ϰʲ��Τ褦��ɽ�����Ȥ��Ǥ���.
%
\begin{eqnarray}
 \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \tilde{F}(\mu) \tilde{G} (\mu) e^{i \mu x} \ d\mu
  = \int_{- \infty}^{\infty} F(x') G (x -x') \ dx'.
\end{eqnarray}
%
$\tilde{F}$��$\tilde{G}$��$F$��$G$���Ѵ�������ΤǤ���.
�ա��ꥨ�Ѵ��ǰʲ��Τ褦�˵��ҤǤ��뼰��ͤ���.
%
\begin{eqnarray}
 \int_{- \infty}^{\infty} e^{- |x| / L_R} e^{- i \mu x} \ dx
  = \frac{2 L_R}{1 + \mu^2 L_R^2}
  = \frac{2 L_R f^2}{\nu^2}.
  \nonumber
\end{eqnarray}
%
$G(x)= e^{-|x|/L_R}$, $F(x)=\partial d(x,0)/\partial x$�Ȥ���(3-86)��
�Ȥ���(3-85)�ϰʲ��Τ褦�ˤʤ�
\footnote{
 ���μ���Ƴ�Ф˴ؤ��Ƥ�{\bfseries 3-8-C}�򻲾ȤΤ���.
}.
%
\begin{eqnarray}
 h_S(x) = h(x,0) - \frac{H}{2 L_R f} \int_{- \infty}^{\infty}
  e^{|x - x'|/L_R} \DP{d(x',0)}{x'} \ dx'.
  \nonumber
\end{eqnarray}
%
���β����ʬ��ʬ����Ȱʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 h_S(x) = h(x,0) + \frac{H}{2 L_R^2 f} \int_{- \infty}^{\infty}
  \frac{x-x'}{|x-x'|} e^{|x - x'|/L_R} d(x',0) \ dx' .
\end{eqnarray}
%
���η�̤�, ������������ˤ����������֤��Ϲ���ʿ�դ���Τ���αƶ���
��Υ$L_R$���礭���ʤ�ˤĤ��̵���ʤäƤ������Ȥ򼨤��Ƥ���.
�Ż߾��֤Ǥ�®�پ��(3-87)�Ȱʲ����Ϲ���Ū�ط����������뤳�Ȥ��Ǥ���.
%
\begin{eqnarray}
 v_S (x) = f g^{-1} \partial h_S / \partial x,
  \quad
  u_S = 0.
\end{eqnarray}
%

2�Ϥ�7,8��Ǥ����Ϲ���Ū������֤����2�Ĥΰۤʤä��������갷�ä�.
2-6��ǤϽ�����֤ϰ�ĤΥ���������������Ȥη����򤷤Ƥ���,
����Ϲ���Ū���֤ˤ�����(����Ū�ˤ����Ū�ˤ�)�ݸ¤ʤ���ư����.
������ǤϽ�����֤��Զѹդϸ��ꤵ�줿�ΰ�
�˶ɺߤ�, �����Ȥ��Զѹդ��ΰ�γ������¤���ˤĤ����Ż߾��֤˶��դ�.
���Υץ�������2�����ΰ�ǤϤ����®�����Ѥ���.
�����, Ʊ��������˽����Ȥ����ͥ륮���򤵤�˹����ΰ�˹����뤫��Ǥ���.
����Ū��, ������֤��ԶѹդΥ������뤬�ΰ�Υ����������٤ƾ��������,
Ĵ����Ѥ����˸�ΨŪ�ˤ����ʤ���.
�������ߤ��Υ������뤬Ʊ�����餤�ξ��, ���ξ��Ʊ�����֤��вᤷ�Ƥ��Ƥ�
�ޤ����ꤻ���˿�ư���Ƥ���.
�ǽ�Ū�ʾ���, �Ĥޤ��Ĵ�᤬�����ʤ�줿��ξ���, �ϴ�����$(L_R/L)^{2}$
�˰�¸���Ƥ���. $L_R= c_g/f$�Ǥ���, $c_g$�Ϥ��ηϤˤ�����Ŭ���ʽ����Ȥ�
����®�٤Ǥ���.
������֤�$L$�Υ������뤬$L_R$����٤ƾ��������, �ǽ�Ū�ʾ��֤Ͻ������
����®�β�ž����ʬ�Ƿ��ꤵ���. ����, $L$��$L_R$����٤��礭�����, �ǽ�
Ū�ʾ��֤Ͻ�����֤μ��̾�ˤ�äƷ��ꤵ���. �ץ�ߥƥ����������Ϥˤ���
�ƽ���ͤ�����򤪤��ʤ��ݤˤ���Ĵ��ץ��������θ����Τ����˽��פǤ�
��.












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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             �ʲ���Appendix             %%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-8-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-8-A\qquad ���������������Υա��ꥨ�Ѵ�}
\setcounter{equation}{0}
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\def\theequation{3-8-A-\arabic{equation}}
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����������������(3-70)��ա��ꥨ�Ѵ�����(�Ĥޤ��(3-71)�����Ѥ���),
(3-73)��Ƴ��.

(3-70)��1���ܤμ���$e^{- i \mu x}$�򤫤�,
$- \infty$����$\infty$�ޤ���ʬ�����,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�����������������ѷ�1}
  \int_{- \infty}^{\infty} \DP{u}{t} e^{- i \mu x} \ dx
  - \int_{- \infty}^{\infty} f v e^{- i \mu x} \ dx
  + \int_{- \infty}^{\infty} g \DP{h}{x} e^{- i \mu x} \ dx
  = 0
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-08-�����������������ѷ�1}����1���ͤ���.
���ι�Ǥ�����ʬ�ؿ���$t$�˰�¸����Τ�$u$�ΤߤʤΤ�
\footnote{
 ��ʬ$\int u e^{- i \mu x} dx$��$t$�Τߤδؿ��ˤʤ�Τ�,
 ���դˤ�����$d/dt$��$\partial / \partial t$��������.
},
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�����������������ѷ�2}
  \int_{- \infty}^{\infty} \DP{u}{t} e^{- i \mu x} \ dx
  = \DD{}{t} \int_{- \infty}^{\infty} u e^{- i \mu x} \ dx
\end{eqnarray}
%
�Ǥ���. ������2���ͤ���. $f$��$x$�δؿ��ǤϤʤ��ΤǴ�ñ������ʬ�ؿ�
���鳰�����Ȥ��Ǥ���. �Ǹ����3���ͤ���.
$g$������Ǥ��뤫������ʬ�ؿ�����Ϥ������Ȥ��Ǥ��뤬,
$h$�����$e^{- i \mu x}$��$x$�δؿ��Ǥ���.
���ι�ϵեա��ꥨ�Ѵ�(3-72)���Ѥ���. �����,
%
\begin{eqnarray}
 g \int_{- \infty}^{\infty} \DP{h}{x} e^{- i \mu x} \ dx
  &=& g \int_{- \infty}^{\infty} \DP{}{x}
  \left\{
   \frac{1}{2 \pi} \int_{- \infty}^{\infty} \tilde{h} (\mu') \ e^{i \mu' x}
   \ d\mu'
  \right\}
  e^{- i \mu x} \ dx
  \nonumber \\
 &=& \frac{i \mu g}{2 \pi} \int_{- \infty}^{\infty}
  \int_{- \infty}^{\infty} \tilde{h} (\mu') \ e^{i (\mu' - \mu) x}
  \ d\mu' \ dx
  \nonumber \\
 &=& \frac{i \mu g}{2 \pi} \int_{- \infty}^{\infty} \tilde{h} (\mu') 
  \int_{- \infty}^{\infty} e^{i (\mu' - \mu) x}
  \ dx \ d\mu' 
  \nonumber \\
 &=& i \mu g \int_{- \infty}^{\infty} \tilde{h} (\mu') \
  \delta (\mu' - \mu) \ d\mu' 
  \nonumber
\end{eqnarray}
%
�Ȥʤ�
\footnote{
 �����Ǥϰʲ��Υǥ륿�ؿ���������Ѥ���.
 %
 \begin{eqnarray}
  \frac{1}{2\pi} \int_{- \infty}^{\infty} e^{i \omega (x - x')} \ d \omega
   = \delta (x - x').
   \nonumber
 \end{eqnarray}
}.
�����, �ǥ륿�ؿ�������
\footnote{
 �ǥ륿�ؿ��ϰʲ��δط�����������.
 %
 \begin{eqnarray}
  \int_{- \infty}^{\infty} f(x') \ \delta(x - x') \ dx' = f(x).
   \nonumber
 \end{eqnarray}
}
���,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�����������������ѷ�3}
 g \int_{- \infty}^{\infty} \DP{h}{x} e^{- i \mu x} \ dx
  =  i \mu g \tilde{h}(\mu)
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-08-�����������������ѷ�1}��
\Deqref{hw-02-08-�����������������ѷ�2}��
\Deqref{hw-02-08-�����������������ѷ�3}
��������, �ա��ꥨ�Ѵ��μ�(3-71)���Ѥ��뤳�Ȥ�
�ʲ��μ���������.
%
\begin{eqnarray}
 \DP{\tilde{u}}{t} = f \tilde{v} - i \mu g \tilde{h}.
  \nonumber 
\end{eqnarray}
%
$U=0$, 
(3-73)�ΰʲ���1���ܤμ���������.
2���ܤμ���3���ܤμ���Ʊ�ͤˤ���������.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-8-B      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-8-B\qquad �ա��ꥨ�Ѵ����줿��®�β�}
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�����Ǥϥա��ꥨ�Ѵ����줿��®�β�(3-76)�����.
����Ƴ�Фˤ�(3-60), (3-61), (3-63), (3-64)�μ����Ѥ���.
�ޤ�, �����Ǥ�$\tilde{u}$��Ƴ�ФΤߤ����ʤ�.
$\tilde{v}, \tilde{h}$��Ƴ�Ф�Ʊ�ͤη׻���ˡ�Ǥ����ʤ����Ȥ��Ǥ���.

(3-61)��$u$��ʬ�μ���Ÿ�����ƽ񤯤�,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��1}
  u = a_1 u_1 e^{i \mu (x - c_1 t)}
  + a_2 u_2 e^{i \mu (x - c_2 t)}
  + a_3 u_3 e^{i \mu (x - c_3 t)}
\end{eqnarray}
%
�Ȥʤ�.
�����Ǥ�$U=0$�Ȥ��Ƥ��뤿��,
$\nu = \left( f^2 + \mu^2 g H \right)^{1/2}$�Ȥ����,
(3-63)���
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��2}
  c_1 = 0, \quad c_2 = \nu / \mu , \quad c_3 = - \nu / \mu
\end{eqnarray}
%
�Ȥʤ�. (���Τ��Ȥ���ʸ�Ǥ�Ҥ٤��Ƥ��뤬).
�ޤ�, \Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��2}
���(3-60)�ˤ�����$u_j$��
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��3}
  u_1 = 0, \quad u_2 = - \nu / \mu , \quad u_3 = \nu / \mu
\end{eqnarray}
%
�Ȥʤ�.
�ޤ�$U=0$���, (3-64)��
%
\begin{eqnarray}
 a_1 &=& - \frac{i \mu f^{-1} h H v_0 - h_0}{\alpha},
  \nonumber \\
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��4}
  a_{2} &=& - u_0 / 2 \alpha^{1/2}
  - \frac{if \mu^{-1} v_0 + g h_0}{2 \alpha},
  \\
 a_{3} &=&   u_0 / 2 \alpha^{1/2}
  - \frac{if \mu^{-1} v_0 + g h_0}{2 \alpha}.
  \nonumber
\end{eqnarray}
%
�Ȥʤ�.
$\alpha \equiv gH + f^2 \mu^{-2}$�Ǥ���,
$\nu = \left( f^2 + \mu^2 g H \right)^{1/2}$�ʤΤ�
$\alpha = \nu^2 / \mu^2$�Ǥ���. �����
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��4}
�����������,
%
\begin{eqnarray}
 a_1 &=& - \frac{i \mu f^{-1} h H v_0 - h_0}{\nu^2 / \mu^2},
  \nonumber \\
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��5}
  a_{2} &=& - \frac{u_0}{2 \nu / \mu} 
  - \frac{if \mu^{-1} v_0 + g h_0}{2 \nu^2 / \mu^2},
  \\
 a_{3} &=& \frac{u_0}{2 \nu / \mu}
  - \frac{if \mu^{-1} v_0 + g h_0}{2 \nu^2 / \mu^2}.
  \nonumber
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��2},
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��3},
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��5}��
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��1}�����������,
%
\begin{eqnarray}
 u &=& \Bigg[
  \left( - \frac{u_0}{2 \nu / \mu} 
   - \frac{if \mu^{-1} v_0 + g h_0}{2 \nu^2 / \mu^2}
  \right)
  \left( - \nu / \mu \right) e^{- i \nu t}
  \nonumber \\
  & & \hspace{20mm} + \left( \frac{u_0}{2 \nu / \mu} 
     - \frac{if \mu^{-1} v_0 + g h_0}{2 \nu^2 / \mu^2}
    \right)
  \left( \nu / \mu \right) e^{i \nu t} \Bigg]
  e^{i \mu x}
  \nonumber \\
 &=& \Bigg[
  \left( \frac{u_0}{2} 
   + \frac{if \mu^{-1} v_0 + g h_0}{2 \nu / \mu}
  \right)
  \cos \nu t
  \nonumber \\
 & & \hspace{10mm}
 \left( \frac{u_0}{2} 
  + \frac{if \mu^{-1} v_0 + g h_0}{2 \nu / \mu}
 \right)
 \left( - i \sin \mu t \right)
 \nonumber \\
 & & \hspace{20mm}
   + \left( \frac{u_0}{2} 
      - \frac{if \mu^{-1} v_0 + g h_0}{2 \nu / \mu}
     \right) \cos \nu t
   \nonumber \\
 & &
  \hspace{30mm}
  + \left( \frac{u_0}{2} 
     - \frac{if \mu^{-1} v_0 + g h_0}{2 \nu / \mu}
    \right)
  \left( - i \sin \mu t \right)
  \Bigg]
  e^{i \mu x}
  \nonumber \\
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��6}
  &=& \Bigg[
  u_0 \cos \nu t +
  \left(
   \frac{f v_0 - i \mu g h_0}{\nu}
  \right)
  \sin \nu t
  \Bigg] e^{i \mu x}
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��6}
��ξ�դ�$e^{- i \mu x}$�򤫤�, $x$��$- \infty$����$\infty$�ޤ���ʬ�����,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��7}
  \int_{- \infty}^{\infty} u (x, t) e^{- i \mu x} dx
  &=& \int_{- \infty}^{\infty}
  \Bigg[
  u_0 \cos \nu t +
  \left(
   \frac{f v_0 - i \mu g h_0}{\nu}
  \right)
  \sin \nu t
  \Bigg] dx
\end{eqnarray}
%
�Ȥʤ�.
$u(x,0) = u_0 e^{i \mu x}$�ʤΤ�
\footnote{
 2-7 ��ˤƤ����������Ƥ���.
},
(3-74)���
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��8}
  \int_{- \infty}^{\infty} u (x,0) e^{- i \mu x} \ dx
  = \int_{- \infty}^{\infty} u_0 \ dx
  = \tilde{u}_0
\end{eqnarray}
%
�Ǥ���.
$v$��$h$�˴ؤ��Ƥ�Ʊ�ͤǤ���.
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��8}��
\Deqref{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��7}�����������,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ա��ꥨ�Ѵ����줿��®�β��Ƴ��9}
 \tilde{u} (\mu , t) &=& \tilde{u}_0 \cos \nu t
  + \left( \frac{f \tilde{v}_0 - i \mu g \tilde{h}_0}{\nu} \right)
  \sin \nu t 
\end{eqnarray}
%
�Ȥʤ�.

$v$��$h$�˴ؤ��Ƥ�Ʊ�ͤ���ˡ�ǵ��뤳�Ȥ��Ǥ���.





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%%%%      3-8-C      %%%%
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\subsection*{3-8-C\qquad �ǽ�Ū�ʹ⤵$h_S(x)$�μ���Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-8-C-\arabic{equation}}
\def\thefigure{3-8-C-\arabic{figure}}

�ǽ�Ū�ʹ⤵$h_S(x)$�˴ؤ��뼰(3-87)��(3-85)����Ƴ�Ф���.
��ʸ�ǻ��Ѥ��Ƥ��뤿���߹��ߤ������ʤɤ�Ƴ�Ф����Ѥ���.

(3-85)�ϰʲ��μ��Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��1}
  h_S (x) = h(x,0) - \frac{1}{2 \pi}
  \int_{- \infty}^{\infty} \frac{Hfi \mu}{\nu^2}
  \tilde{d}_0 e^{i\mu x} \ d\mu.
\end{eqnarray}
%
�����߹��ߤ�����(3-86)�ϰʲ��μ��Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��2}
  \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \tilde{F}(\mu) \tilde{G} (\mu) e^{i \mu x} \ d\mu
  = \int_{- \infty}^{\infty} F(x') G (x -x') \ dx'.
\end{eqnarray}
%
��ʸ�̤�, $G(x)= e^{-|x|/L_R}$, $F(x)=\partial d(x,0)/\partial x$
�Ȥ���. �ա��ꥨ�Ѵ����Ѥ����$\tilde{G}$��
%
\begin{eqnarray}
 \tilde{G}(\mu) &=& \int_{- \infty}^{\infty} e^{-|x|/L_R} e^{- i \mu x} dx
  \nonumber \\
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��3}
  &=& \frac{2 L_R f^2}{\nu^2}
\end{eqnarray}
%
�Ȥʤ�. �������ʸ�ε��Ҥ򤽤Τޤް��Ѥ�����ΤǤ���.
����$\tilde{F}$��,
%
\begin{eqnarray}
 \tilde{F} (\mu) &=& \int_{- \infty}^{\infty} \DP{d_0 (x)}{x} e^{- i \mu x} dx
  \nonumber \\
 &=& \left[ d_0 (x) e^{- i \mu x} \right]_{- \infty}^{\infty}
  - (- i \mu) \int_{- \infty}^{\infty} d_0 (x) e^{- i \mu x} dx
  \nonumber \\
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��4}
  &=& i \mu \tilde{d}_0
\end{eqnarray}
%
\Deqref{hw-02-08-�ǽ����֤ι⤵��Ƴ��3}��
\Deqref{hw-02-08-�ǽ����֤ι⤵��Ƴ��4}���
%
\begin{eqnarray}
 \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \tilde{F}(\mu) \tilde{G} (\mu) e^{i \mu x} \ d\mu
  &=& \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \frac{2 i L_R f^2 \mu}{\nu^2} \tilde{d}_0 e^{i \mu x} d\mu
  \nonumber 
\end{eqnarray}
%
�Ǥ���, $L_R = (gH)^{1/2}/ f$���,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��5}
  \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \tilde{F}(\mu) \tilde{G} (\mu) e^{i \mu x} \ d\mu
  &=& \frac{2 L_R f}{H} \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \frac{H f i \mu}{\nu^2} \tilde{d}_0 e^{i \mu x} d\mu
\end{eqnarray}
%
�Ǥ���.
\Deqref{hw-02-08-�ǽ����֤ι⤵��Ƴ��2}��
\Deqref{hw-02-08-�ǽ����֤ι⤵��Ƴ��5}���,
%
\begin{eqnarray}
 \frac{1}{2\pi} \int_{- \infty}^{\infty}
  \frac{H f i \mu}{\nu^2} \tilde{d}_0 e^{i \mu x} d\mu
  &=&
  \frac{H}{2 L_R f} \int_{- \infty}^{\infty} F(x') G (x -x') \ dx'
  \nonumber \\
 &=&
  \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��6}
  \frac{H}{2 L_R f} \int_{- \infty}^{\infty}
  e^{|x - x'|/L_R} \DP{d(x',0)}{x'} \ dx'
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-08-�ǽ����֤ι⤵��Ƴ��1}��
\Deqref{hw-02-08-�ǽ����֤ι⤵��Ƴ��6}���,
%
\begin{eqnarray}
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��7}
  h_S (x) = h(x,0)
  - \frac{H}{2 L_R f} \int_{- \infty}^{\infty}
  e^{|x - x'|/L_R} \DP{d(x',0)}{x'} \ dx'
\end{eqnarray}
%
�Ȥʤ�.
���μ��α�����2�����ʬ��ʬ�����,
%
\begin{eqnarray}
  h_S (x) &=& h(x,0)
  - \frac{H}{2 L_R f}
  \left\{
   \left[ e^{|x - x'|/L_R} d(x',0) \right]_{- \infty}^{\infty}
   - \int_{- \infty}^{\infty}
   \frac{x-x'}{|x-x'|} \frac{1}{L_R} e^{|x - x'|/L_R} d(x',0) \ dx' 
  \right\}
  \nonumber \\
 \Deqlab{hw-02-08-�ǽ����֤ι⤵��Ƴ��8}
  &=& h(x,0)
  + \frac{H}{2 L_R^2 f}
  \int_{- \infty}^{\infty}
  \frac{x-x'}{|x-x'|} e^{|x - x'|/L_R} d(x',0) \ dx' 
\end{eqnarray}
%
�Ȥʤ�. �������ʸ��(3-87)�Ǥ���.



%\subsubsection*{}


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%\subsection*{Appendix \qquad }

%\subsubsection*{}


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