%
% Chapter 4   Atmospheric Waves: Part II
%             4-5 LINEAR SHEAR
%
% 2002/12/08  ���� ����   ����
% 2002/12/09  ���� ����   ����
% 2002/12/18  ���� ����   ����ʸ���ɵ�
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   Subtitle, Date, Filename Setting     %%%%%%%%
\markboth{5-5 ��������}{5-5 ��������}
%\markright{4-5 ��������}
%\Ddate{2002/12/18}
%\Dfile{hw-04-05.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   Set Counter (chapter, section etc. ) %%%%%%%%
\setcounter{chapter}{5}
\setcounter{section}{4}
\setcounter{footnote}{0}
\setcounter{equation}{26}
%\setcounter{page}{1}
%\setcounter{figure}{0}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%        Counter Output Format           %%%%%%%%
\def\thesection{\arabic{chapter}-\arabic{section}}
\def\thesubsection{\arabic{chapter}-\arabic{section}-\arabic{subsection}}
\def\theequation{\arabic{chapter}-\arabic{equation}}
\def\thepage{\arabic{page}}
\def\thefigure{\arabic{chapter}-\arabic{figure}}
\def\thetable{\arabic{chapter}-\arabic{table}}
%\def\thefootnote{\arabic{footnote})}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%              Text Start                %%%%%%%%

\section{��������}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 �����Υ��������ή����԰������ˤĤ��Ƶ�������.

 �������������ή��ˤ�4-2��ǹͤ����褦�ʥΡ��ޥ�⡼�ɤβ������ʤ�����,
 �������������Ȥ��ư���.
 $\beta=0$, $U=U_0 + Sy$�Ȳ��ꤷ��, 4-2���(5-4)��$\psi$�˴ؤ��Ʋ�.
 ����줿��̤������ϼ�ޤäƤ������Ȥ��狼��.
\end{itembox}
 
�����ο�ʿ�̥��������ή��Ǥ��԰���ϵ���������Τ��Ȥ�����������Τ��
�Ǥ���.
��������������ͤ�����ˤ�$\beta = 0$�Ǥ���,
�������(5-6)�����������褦��, (5-5)��{\bf �Ρ��ޥ�⡼��}�β�
\footnote{$\psi = \Psi (y) \ e^{i \mu (x - ct)}$�Τ���. 4-2�Ỳ��.}��̵��.
���Τ���, ������뤿��μ�ˡ�����μ�ˡ�Ȥ������ۤʤäƤ���.

�⤦��Ĥμ�ˡ�Ȥ�, �������������Ȥ��ư�����ˡ�Ǥ���.
�������ñ�ˤ��뤿��, (5-4)�ˤ������ϵ�β�ž�θ���($\beta$�ι�)��
̵�뤷, $U=U_0 + Sy$�Ȳ��ꤹ��.
�����, (5-4)�ϰʲ��Τ褦�ʴ�ñ�ʼ��Ȥʤ�
\footnote{
 $U = U_0 + Sy$���, $\partial \overline{\zeta} / \partial y =
 - d^2 U /dy^2 = 0$�Ȥʤ�.
}.
%
\begin{eqnarray}
 \DP{\Dlapla \psi}{t} + U \DP{\Dlapla \psi}{x} = 0.
\end{eqnarray}
%
���μ��β�ϰʲ��Τ褦�ʷ��β�Ȥʤ�.
%
\begin{eqnarray}
 \Dlapla \psi = F (x - Ut).
  \nonumber
\end{eqnarray}
%
$F$��Ǥ�դδؿ��Ǥ���, $F(x)$��$t=0$�ˤ����뱲�٤ξ���Ǥ���.

����, ������֤�®�پ�ξ���ϰʲ��Τ褦�ʴ�ñ�ʷ���ɽ����ȹͤ���.
%
\begin{eqnarray}
 v = - v_0 \sin \mu x.
  \nonumber
\end{eqnarray}
%
�����,
%
\begin{eqnarray}
 \Dlapla \psi = \zeta
  = - \mu v_0 \cos \mu x,  \qquad (t = 0)
  \nonumber
\end{eqnarray}
%
�Ȥʤ�
\footnote{
 $\zeta = \partial v / \partial x - \partial u / \partial y$�ʤΤ�,
 $u$��ͤ��Ƥ��ʤ����ξ���$\zeta = \partial v / \partial x$�Ǥ���.
},
$t$�ˤ����Ƥ�
%
\begin{eqnarray}
 \Dlapla \psi = - \mu v_0 \cos \mu (x - Ut)
\end{eqnarray}
%
�Ȥʤ�.
ή���ؿ����Τ뤿��, ή���ؿ��Ⱳ�٤�Ʊ���ؿ����򤷤Ƥ���Ȳ��ꤹ��.
���ʤ��, ή���ؿ���ʲ��Τ褦�˲��ꤹ��.
%
\begin{eqnarray}
 \psi = A \cos \mu (x - Ut).
  \nonumber
\end{eqnarray}
%
�����,
%
\begin{eqnarray}
 \Dlapla \psi = - A \mu^2 \cos \mu (x - Ut)
  - A S^2 t^2 \mu^2 \cos \mu (x - Ut)
  \nonumber
\end{eqnarray}
%
�Ȥʤ�
\footnote{
 $U = U_0 + S y$�Ȳ��ꤷ�Ƥ��뤳�Ȥ�����.
}. 

���μ��α��դ�(5-28)����$A$���ͤ���ꤹ�뤳�Ȥ������.
���, $\psi$�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \psi = \frac{v_0 \cos \mu (x - Ut)}{\mu (1 + S^2 t^2)}.
\end{eqnarray}
%
��®����ʬ�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 u = U - \DP{\psi}{y}
  &=& U - \frac{v_0 \mu S t \sin \mu (x - Ut)}{\mu (1 + S^2 t^2)},
  \nonumber \\
 v &=& \DP{\psi}{x}
  = - \frac{v_0 \sin \mu (x - Ut)}{1 + S^2 t^2}.
  \nonumber
\end{eqnarray}
%
�弰�������餫�ʤ褦��, $\psi$��$v$��$t$��2���ȿ���㤷,
$u$��$t/(1+t^2)$���Ѳ�����.
�����®��$U=U_0 + Sy$�ȤȤ�˰�ư����Τ�, �Ȥ�(���֤ȤȤ�˼�ޤäƤ�
���Τ�Ʊ�ͤ�)�����������˷����Ƥ���.
���ä����Ȥ����Υ����������˷����Ϥ����Ⱦ��𤬼�ޤäƤ������Ȥ�
���ͥ륮���μ�(5-20)�ˤ�äƼ�����Ƥ���, �����ǤϤ����Ʊ�����Ȥ�
�����äƤ���Τ�4-3��μ�(5-18), (5-20), �������(5-29)����狼��
\footnote{
 4-3���(5-18)
 %
 \begin{eqnarray}
  \psi (x, y, t) = \Psi (y,t) \cos \left[ \mu x - \theta (y,t) \right]
   \nonumber
 \end{eqnarray}
 %
 �Ȥ������(5-29)����Ӥ����,
 %
 \begin{eqnarray}
  \Psi (y,t) = v_0 / \mu (1 + S^2 t^2), \qquad
   \theta (y,t) = \mu t (U_0 + Sy)
   \nonumber
 \end{eqnarray}
 %
 �Ǥ���. ���Τ��Ȥ���, $(\partial \theta / \partial y)(dU/dy) > 0$
 �Ǥ���Ȥ�����.
}.
���β�ϥΡ��ޥ�⡼�ɤ�Ϣ³���ڥ��ȥ�
$c(y') = U(y')$�ι�ǽ񤯤��Ȥ������.
�����$y=y'$�ˤ�����ͣ�챲�٤����, ����¾�Ǥϱ��٤ϥ����Ǥ���.
���β�Ϥ����δؿ��Ƚ�����֤α��٤���ʬ��ɽ�����.

�԰�������δ��ܾ�ΥΡ��ޥ�⡼�ɤβ�Ϥ��Ф��д����Ϥβ��
�����Ǥ��ʤ�.
���Τ���, �Ρ��ޥ�⡼�ɤβ�����Ƥ������ʽ�����֤��Ф�����̲��
��������Τ����Ѥ��뤳�Ȥ�����ʤ��ä�.
�ʤ�, �����ӡ��Ȥβ�(5-8)�Ϥ�������Ѥ��뤳�Ȥ����褿.
Case(1960)��, ����ͤμ�ˡ���Ѥ���,
(5-5)��(5-6)��������줿Υ���Ρ��ޥ�⡼�ɤ�,
�嵭�ˤƵ��Ҥ��줿��ΤȻ���, ���̲��ɬ�פȤ����Ϣ³�⡼�ɤ�ȯ������.
����Ϣ³�⡼�ɤϤ������Ǥ�ʿ����®��Ʊ������®�٤����, ��餫�ʽ�����
�����Ѥ��줿���ˤϾ嵭��������Τ�Ʊ���褦�˸��ꤷ�Ƥ���.
���Τ褦��, ������֤ηв��ˤ�Ϣ³���ڥ��ȥ�⡼�ɤ�, �����ʤ�
Υ���⡼�ɤ�������Ǥ��Ƥ�����ͽ�ۤ����.



%\begin{center}
% {\bf 
% --------------------------------------
% ��ʸ�����ޤ�
% --------------------------------------
% } 
%\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             �ʲ���Appendix             %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%   Appendix �Ѥ� Counter, OutputFormat ������   %%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   Set Counter (chapter, section etc. ) %%%%%%%%
%\setcounter{chapter}{2}
%\setcounter{section}{0}
%\setcounter{footnote}{0}
\setcounter{equation}{0}
%\setcounter{page}{1}
\setcounter{figure}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%        Counter Output Format           %%%%%%%%
\def\theequation{A-\arabic{equation}}
%\def\thesection{\arabic{chapter}-\arabic{section}}
%\def\thesubsection{\arabic{chapter}-\arabic{section}-\arabic{subsection}}
\def\thefigure{A-\arabic{figure}}
\def\thetable{A-\arabic{table}}



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

%\subsection*{Appendix \qquad }

%\subsubsection*{}


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