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\Dauthor{���� �O��}
\Ddate{2010 �N 12 �� 09 ��}% ���t�̎w��
\Dtitle{GFD���[�N�m�[�g}      % ����w�b�_�[�w��
\Dfile{2010-12-ogihara.tex}
\markright{��10�� 1�����ڗ��������̐��l��@} % �E��w�b�_�̎w��

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\begin{document}
\subsection{�������[�h�ƌv�Z���[�h�ɂ���}
�������[�h��$U_1$, �v�Z���[�h��$U_2$�Ƃ���. ���ꂼ��̃��[�h�ɑ΂��鑝���W����$\lambda _1$,$\lambda_2$�Ƃ���. 
$n$�X�e�b�v�ڂ̐��l��$U_1^{(n)}, U_2^{(n)}$�͂��ꂼ��, 
\begin{align*}
	U_1^{(n)} &=\lambda_1^{n}U_1^{(0)},\\
	U_2^{(n)} &=\lambda_2^{n}U_2^{(0)}
\end{align*}
�ƂȂ�. ������, $U_1^{(0)}, U_2^{(0)}$�͏����l. $\lambda_1, \lambda_2$��(9.2.19)������, 
\begin{align*}
	\lambda_1 &= \sqrt{1 - p^2} + ip,\\
	\lambda_2 &= -\sqrt{1- p^2} + ip
\end{align*}
�ƂȂ�. $\left| p\right| < 1$�̏ꍇ���l�����$\left| \lambda_1\right| = \left| \lambda_2\right| = 1$���, 
\begin{align}
	\lambda_1 = e^{i\theta}.
\end{align}
���̂Ƃ��̈ʑ���, 
\begin{align*}
	\theta = \arctan{\frac{p}{\sqrt{1 - p^2}}}
\end{align*}
�ɂȂ�. $p>0, p<0$�̗����̏ꍇ���l�����,
\begin{align}
	\lambda_2 &= e^{i(\pm\pi - \theta)}\nonumber\\
	&=-e^{-i\theta}
\end{align}
�ƂȂ�\footnote{�ڂ����͑�9�� �Δ����������̐��l��@�̊�b2 (Mesinger and Arakawa,1976: Chapt2) ��2.3�O�i�K�X�L�[����p17-20 ���Q�Ƃ��ꂽ��. }.

%���l��$u^n_j$�𕨗����[�h$U_1^{(0)}$�ƌv�Z���[�h$U_2^{(0)}$�ł��ꂼ��\���Ă݂�. 
���l��$u^n_j$��, 
\begin{align*}
	u^n_j = \sum_k u_k^{(n\Delta t)} e^{- ik(j\Delta t)}
\end{align*}
�ƕ\����. �����������[�h�݂̂ō\�������Ƃ����, 
\begin{align}
	u^n_j 
	&= {\rm Re} \left[ \sum_k U_{1,k}^{(n)}e^{-ik(j\Delta x)} \right] \nonumber\\
	&= {\rm Re} \left[ \sum_k U_{1,k}^{(0)} \lambda_1^{(n)} e^{-ik(j\Delta x)} \right] \nonumber\\
	&= {\rm Re}\left[ \sum_k U_{1,k}^{(0)}e^{-ik(j\Delta x - \frac{n\theta}{k})}\right] \nonumber \\
	&= {\rm Re}\left[ \sum_k U_{1,k}^{(0)}e^{-ik(j\Delta x - \frac{\theta}{k\Delta t}n\Delta t)} \right] .
	\Deqlab{1.13}
\end{align}
�����v�Z���[�h�݂̂ō\�������Ƃ��Ă����l��
\begin{align}
	u^n_j
	&= {\rm Re} \left[ \sum_k U_{2,k}^{(n)}e^{-ik(j\Delta x)} \right] \nonumber\\
	&= {\rm Re} \left[ \sum_k U_{2,k}^{(0)} \lambda_2^{(n)} e^{-ik(j\Delta x)} \right] \nonumber\\
	&= {\rm Re} \left[ \sum_k (-1)^n U_{2,k}^{(0)}e^{-ik(j\Delta x + \frac{\theta}{k\Delta t} n \Delta t)} \right].
	\Deqlab{1.14}
\end{align}
\Deqref{1.13}��, \Deqref{1.14}������͉��̃t�[���G�\��
\begin{align*}
	u(x,t)={\rm Re} \left[ \sum_k U_k^{(0)}e^{-ik(x -ct)} \right]
\end{align*}
�Ɣ�ׂ��, 
�������[�h�̈ʑ����x$c_1$��, 
\begin{align*}
	c_1 = -\frac{\theta}{k\Delta t}.
\end{align*}
�v�Z���[�h�̈ʑ����x$c_2$��, 
\begin{align*}
	c_2 = \frac{\theta}{k\Delta t}
\end{align*}
�ƂȂ�. ������$\Delta t \rightarrow 0$�̋Ɍ��ł�, $\displaystyle{p = -c\frac{\Delta t}{\Delta x}\sin{k\Delta x}}$���, 
$\theta \rightarrow p$�ƂȂ�
\footnote{$\Delta t \rightarrow 0$�̋Ɍ��ł�, 
	\begin{align*}
		p &= -c\frac{\Delta t}{\Delta x}\sin{k\Delta x}\\
		&\sim 0
	\end{align*}
	�ƂȂ���, 
	\begin{align*}
		\theta &= \arctan{\left(\frac{p}{\sqrt{1 - p^2}}\right)}\\
		&\sim \arctan{\left(p -\frac{1}{2}p^3\right)}\\
		&\sim p 
	\end{align*}
	�ƂȂ�.}, �����, $\Delta x \rightarrow 0$�̋Ɍ����l�����, 
\begin{align*}
	c_1 &= -\frac{\theta}{k\Delta t}\\
	&\sim -\frac{p}{k\Delta t}\\
	&= \frac{c}{k\Delta x}\sin{k\Delta x}\\
	&\sim c
\end{align*}
�ƂȂ�. ���l�ɋ��߂��, $c_2 = -c$. 

�����, �������[�h�̈ʑ����x��$\Delta t \rightarrow 0, \Delta x \rightarrow 0$�̂Ƃ�, $c$�Ɉ�v��, �v�Z���[�h��
�ʑ����x��$\Delta t \rightarrow 0, \Delta x \rightarrow 0$�̂Ƃ�, $-c$�ƂȂ�. �X��\Deqref{1.14}����$(-1)^n$���������Ă���̂�
, �v�Z���[�h��1�X�e�b�v���Ƃɕ�����ς���. 
\end{document}