%
% Chapter 2   Wave Motion in the Atmosphere  Part I
%             2-3 PURE SOUND WAVES
%
% 2002/07/28  ���� ����   ����
% 2002/08/02  ���� ����   ����
% 2002/09/23  ���� ����   ����
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%         title and date setting         %%%%%%%%
\markboth{3-3 (����)����}{3-3 (����)����}
%\markright{2-3 (����)����}
%\Ddate{2002/09/23}
%\Dfile{hw-02-03.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%        page and section setting        %%%%%%%%
\setcounter{chapter}{3}
\setcounter{section}{2}
\setcounter{footnote}{0}
\setcounter{equation}{2}
%\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}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%              text start                %%%%%%%%

\section{(����)����}

\begin{itembox}[l]{����}
 ������Ǥϲ��ȤˤĤ��ƹͤ���.
 ̵���Ϥʤɤξ���Ϳ��, �������ư��Ǥ���Ȳ��ꤷ,
 ����������������Ϥ�򤯤��Ȥǲ��Ȥδط�����Ƴ��.
\end{itembox}

���Ȥ�$g=0, \ \delta_1 = \delta_2 = 1$�����,
$\overline{p}$��$\overline{\alpha}$������Ȥ��뤳�ȤǼ��Ф����Ȥ�
�Ǥ�����̩�ȤǤ���.
��, ����
\footnote{
$u'$ �� $w'$ �ʤ�.
}
�򷸿����Ĥ���$x,z,t$��Ĵ�´ؿ�
\footnote{
 {\bfseries Ĵ�´ؿ�}({\itshape harmonic function})�Ȥ�
 �ʲ��Τ褦�ʥ�ץ饹�黻�Ҥ˴ؤ����ͭ�ؿ��Τ��ȤǤ���.
 \begin{eqnarray}
  \Dlapla \phi = A \phi
   \nonumber 
 \end{eqnarray}
}
���Ȳ��ꤹ���,
%
\begin{eqnarray}
 u' = S e^{i(\mu x + k z + \nu t)}  \qquad
 w' = W e^{i(\mu x + k z + \nu t)}  \nonumber \\
 p' = P e^{i(\mu x + k z + \nu t)}  \qquad
 \alpha' = A e^{i(\mu x + k z + \nu t)} \ 
\end{eqnarray}
%
�Ǥ���.
������$x$�����$z$�������ȿ�({\itshape wave number})$\mu$, $k$��Ȥ�,
��ư��({\itshape frequency})��$\nu$��Ȥä�ʿ���ȤǤ���.
�ºݤ�ʪ���̤ϲ�ε�����̵�뤷, ��������Ф����Ȥ�������
\footnote{
 ���ʤߤ�$S$��$W$�ʤɤϼ¿��ǤϤʤ�ʣ�ǿ��Ǥ��뤳�Ȥ�����.
}.
��(3-2)�������ʤΤ�, �ɤ�������β���ȹ礻�Ǥ�򤯤��Ȥ��Ǥ���.
���Τ����Ĥ�Ĵ�´ؿ������ͤ�����ɤ�
\footnote{
 ������͡�����Ĺ, ��ư�����Ȥ�����Τ�
 $\displaystyle u' = \sum_{\mu,k,\nu} S e^{i (\mu x + kz + \nu t)}$
 �ʤɤȤ���٤�����, �������Ϥ������Ǥ��뤿��,
 �����Ĥ��Ȥ˴ؤ��Ʋ򤤤����­����碌�򤪤��ʤä�
 �򤯤��Ȥ��Ǥ���.
}.
�ؿ�(3-3)��(3-2)�����������,
�ʲ��Τ褦��, ����$S, W ,P ,A$�˴ؤ���Ʊ������������Ϥ�������.
%
\begin{eqnarray}
  (\mu U - \nu)S &+& \overline{\alpha} \mu P = 0,
 \nonumber \\
  (\mu U - \nu)W &+& \overline{\alpha} k P = 0,
 \nonumber \\
  \overline{\alpha} (\mu U - \nu) P
  &+& \overline{p} \gamma (\mu U - \nu) A = 0,
 \\
  \mu \overline{\alpha} S + \overline{\alpha} k W
  &-& (\mu U - \nu) A = 0.
 \nonumber
\end{eqnarray}
%
�����������Ϥϰʲ��Τ褦�˹�������ǽ�ɽ�����Ȥ�Ǥ���.
%
\begin{eqnarray}
 \left[
  \begin{array}{cccc}
   (\mu U - \nu ) &       0       & \overline{\alpha} \mu  &  0 \\
         0        & (\mu U - \nu) & \overline{\alpha} k    &  0 \\
         0        &       0       & \overline{\alpha} (\mu U - \nu) &
	                            \overline{p} \gamma (\mu U - \nu ) \\
   \overline{\alpha} \mu & \overline{\alpha} k & 0 & -(\mu U - \nu) 
  \end{array}
 \right]
 \left[
  \begin{array}{c}
   S \\ W \\ P \\ A
  \end{array}
 \right]
 = 0.
\end{eqnarray}
%
����$S, W, P, A$�ϥ����ˤϤʤ�ʤ��Τ�,
���μ������������Τ�Ʊ���������Ϥι��󼰤��Ǥ��ä����äƾä����
�ΤߤǤ���.
���󼰤�Ÿ������Ȱʲ��Τ褦�ʿ�ư���˴ؤ��뼰��������
\footnote{
���μ���Ƴ�Ф˴ؤ��Ƥ�{\bf 3-3-A}�򻲾�.
}.
%
\begin{eqnarray}
 \overline{\alpha}(\mu U - \nu)^2
  \left[
   - (\mu U + \nu)^2
   + \gamma \bar{p} \bar{\alpha} (k^2 + \mu^2)
  \right]
 &=& 0. 
\end{eqnarray}
%
������4�ʤΤǰʲ���4�Ĥβ�����.
%
\begin{eqnarray}
  \nu &=& \mu U  \qquad {\rm (�Ų�)}, 
 \nonumber \\
  \nu &=& \mu U \pm (k^2 + \mu^2)^{1/2} \sqrt{\gamma R \overline{T}}.
\end{eqnarray}
%
(3-7)��ñ�ʤ��ή
\footnote{
 ����®�٤�$c_x = (\nu / \mu) = U$�Ǥ��뤳�Ȥ���,
 ����ϴ��ܾ��ή��Τߤˤ�ä�ή����Ƥ���.
 ���줬�ְ�ή�פǤ���.
}
�Ǥ���.
�ޤ�, ���β�ˤϴ��ܾ��֤ξ���������
\footnote{
  $\bar{p} \bar{\alpha} = R \overline{T}$.
}
�����Ѥ���.
�ʤ�, ʿ����®��¸�ߤ�$x$�����ؤ����¤�$U$���ä���������αƶ�����
�ڤܤ��ʤ����Ȥ����դ��Ƥۤ���.

����®��
\footnote{
 ����®�٤˴ؤ��Ƥ�{\bfseries 3-3-B}����.
}
�ϰ�����($\mu x + k z =$����)����������ˡ��������ư��®�٤Ǥ���
\footnote{
 �ְ������פ˴ؤ��Ƥ�{\bf 3-3-C}����.
}.
����®��$c$�Ͽ�ư��$\nu$�Ȱʲ��Τ褦�ʴط��ˤ���. 
%
\begin{eqnarray}
 c = \nu \mathcal{L} / 2 \pi
\end{eqnarray}
%
$\mathcal{L}$�ϰ�������ˡ�������ؤ���Ĺ�Ǥ���.
��Ĺ��$\mathcal{L} = 2 \pi (k^2 + \mu^2)^{-1/2}$��ɽ����.
�����$\mathcal{L}_x = 2 \pi \mu^{-1}$, $\mathcal{L}_z = 2 \pi k^{-1}$
��Ȥ�, ����ˡ���Ѥ��뤳�Ȥ�������
\footnote{
��Ĺ$\mathcal{L}$��Ƴ�Ф˴ؤ��Ƥ�{\bf 3-3-D}����.
}.
$\mathcal{L}$�˴ؤ��뤳�δط����Ѥ����(3-8)�ϰʲ��Τ褦��
�񤯤��Ȥ��Ǥ���.
%
\begin{eqnarray}
 c = \nu (k^2 + \mu^2)^{-1/2}.
\end{eqnarray}
%
������$U=0$�Ȥ���(3-9)��(3-7)�����������,
%
\begin{eqnarray}
 c = \pm \sqrt{\gamma R \overline{T}}
\end{eqnarray}
%
�Ȥ����ɤ��Τ�줿��®�μ���������
\footnote{
���μ��˼ºݤ˿��ͤ��������Ʋ��Ȥ�®�٤���Ƥߤ��,
�����絤�Ǥ�
$c = \pm (331.1 + 0.606t)$ [m/s]�Ȥʤä�.
����Ƴ�Ф˴ؤ��Ƥ�{\bfseries 3-3-E}�򻲾�.
}.
��®�Ȥ�����Τ����Ť������ˤϰ�¸���ʤ����Ȥ��狼�������.


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

%%%%%%%%           ���ֹ�Υ������饤��         %%%%%%%% 
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{A-\arabic{equation}}
\def\thefigure{A-\arabic{figure}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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


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

\subsection*{3-3-A\qquad (3-6)��Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-3-A-\arabic{equation}}
\def\thefigure{3-3-A-\arabic{figure}}


(3-5)��Ʊ���������Ϥι��󼰤������ˤʤ뤿��δط��������.
%
\begin{eqnarray}
 \left[
  \begin{array}{cccc}
   (\mu U - \nu ) &       0       & \overline{\alpha} \mu  &  0 \\
         0        & (\mu U - \nu) & \overline{\alpha} k    &  0 \\
         0        &       0       & \overline{\alpha} (\mu U - \nu) &
	                            \overline{p} \gamma (\mu U - \nu ) \\
   \overline{\alpha} \mu & \overline{\alpha} k & 0 & -(\mu U - \nu) 
  \end{array}
 \right]
 &=& 0
 \nonumber \\
 \left[
  \begin{array}{cccc}
   (\mu U - \nu ) &       0       & \overline{\alpha} \mu  &  0 \\
         0        & (\mu U - \nu) & \overline{\alpha} k    &  0 \\
         0        &       0       & \overline{\alpha} (\mu U - \nu) &
	                            \overline{p} \gamma (\mu U - \nu ) \\
         0         & \overline{\alpha} k &
   \dfrac{-(\overline{\alpha} \mu)^2}{\mu U - \nu}  & -(\mu U - \nu) 
  \end{array}
 \right]
 &=& 0
 \nonumber \\
 (\mu U - \nu )
  \left[
  \begin{array}{ccc}
   (\mu U - \nu) & \overline{\alpha} k    &  0 \\
         0       & \overline{\alpha} (\mu U - \nu) &
	                            \overline{p} \gamma (\mu U - \nu ) \\
   \overline{\alpha} k &
   \dfrac{-(\overline{\alpha} \mu)^2}{\mu U - \nu}  & -(\mu U - \nu) 
  \end{array}
 \right]
 &=& 0
 \nonumber \\
 (\mu U - \nu )
  \left[
  \begin{array}{ccc}
   (\mu U - \nu) & \overline{\alpha} k    &  0 \\
         0       & \overline{\alpha} (\mu U - \nu) &
	                            \overline{p} \gamma (\mu U - \nu ) \\
         0       &
   \dfrac{- \overline{\alpha}^2 (k^2 + \mu^2)}{\mu U - \nu}  & -(\mu U - \nu) 
  \end{array}
 \right]
 &=& 0
 \nonumber \\
 (\mu U - \nu )^2
  \left[
  \begin{array}{cc}
   \overline{\alpha} (\mu U - \nu) &
	                            \overline{p} \gamma (\mu U - \nu ) \\
   \dfrac{- \overline{\alpha}^2 (k^2 + \mu^2)}{\mu U - \nu}  & -(\mu U - \nu) 
  \end{array}
 \right]
 &=& 0
 \nonumber \\
 (\mu U - \nu)^2
  \left[
   - \overline{\alpha} (\mu U + \nu)^2
   + \gamma \bar{p} \bar{\alpha}^2 (k^2 + \mu^2)
  \right]
 &=& 0
 \nonumber \\
 \overline{\alpha}(\mu U - \nu)^2
  \left[
   - (\mu U + \nu)^2
   + \gamma \bar{p} \bar{\alpha} (k^2 + \mu^2)
  \right]
 &=& 0.
 \nonumber
\end{eqnarray}
%





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

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

{\bf ����®��}({\it phase velocity\/})�Ȥ���������($\theta =$ const)
�����־���ư����®�٤Ǥ���.
����Τ褦��$x$-$z$ʿ�̤ǹͤ���Ȱ����
%
\begin{eqnarray*}
 \theta = \mu x + k z - \nu t
\end{eqnarray*}
%
�Ǥ���. ������$x$�����ؤΰ���®�٤�ͤ���($z=$const).
��������$\delta t$���вᤷ����, $x$��$\delta x$�Ѳ�����.
���κݤ�$\theta =$const �ʤΤ�,
%
\begin{eqnarray*}
 \mu x + kz - \nu t = \mu(x + \delta x) + kz - \nu (t + \delta t)
\end{eqnarray*}
%
�Ǥ���. ���μ����ѷ������,
%
\begin{eqnarray*}
 \frac{\delta x}{\delta t} = \frac{\nu}{\mu}
\end{eqnarray*}
%
�Ȥʤ�. �Ĥޤ�, $x$�����ؤΰ���®��$c_x$��
%
\begin{eqnarray*}
 c_x \equiv \frac{\delta x}{\delta t} = \frac{\nu}{\mu}
\end{eqnarray*}
%
�Ȥʤ�.
Ʊ�ͤˤ���, $z$�����ؤΰ���®�٤�
$c_z = (\delta z / \delta t) = \nu / k$�Ȥʤ�.




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

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

��ʸ�Ρְ������פȤϤɤ�������Τ��ͤ���.
�ޤ�, $x$������ư���Ⱦ�Τ�������ͤ�,
�����Ȥ򲣤��鸫���,
\Dfigref{hw-02-03-�����鸫����}
�Τ褦�˸����������.
����,
\Dfigref{hw-02-03-�����鸫����}
��夫�鸫�Ƥߤ��
\Dfigref{hw-02-03-�夫�鸫����}
�Τ褦�ˤʤ�,
��ۤɤ���������ɽ�������.
����������ʸ�Ǹ�����Ȥ����Ρذ������٤Ǥ���.
\Dfigref{hw-02-03-�夫�鸫����}���Ȥ�
$x$�����ؤΤ����Ť����Ȥ���, $x$-$z$ʿ�̾�����Ť����Ȥ�
�ͤ����,
\Dfigref{hw-02-03-xzʿ����}
�Τ褦�ˤʤ�.
�����ȤϿޤ�����������������Ť���. ����®�٤Ϥ�������������ؤ�
®�٤Ǥ���, ������������ʸ�Ǹ����ذ�������ˡ�������٤Ǥ��뤳�Ȥ�
�ޤ���狼�������.
�ʤ�, ���β���ˤ���¼(1983)�򻲾Ȥ���.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/phase_point.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �����鸫����}
  }
  \Dfiglab{hw-02-03-�����鸫����}
 \end{quote}
\end{figure}
%
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/phase_line1.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �夫�鸫����}
  }
  \Dfiglab{hw-02-03-�夫�鸫����}
 \end{quote}
\end{figure}
%
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/phase_line2.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize $x$-$z$ʿ�̾�����Ť�����}
  }
  \Dfiglab{hw-02-03-xzʿ����}
 \end{quote}
\end{figure}
%






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

\subsection*{3-3-D\qquad ��������ˡ�������ؤ���Ĺ��Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-3-D-\arabic{equation}}
\def\thefigure{3-3-D-\arabic{figure}}

��������ˡ������(�Ĥޤ����οʹ�����)�ؤ���Ĺ��,
$x$, $z$�����Τ��줾�����Ĺ�����ñ�˵��뤳�Ȥ��Ǥ���.
\Dfigref{hw-02-03-��ʬ���Ȥ���Ĺ�δط�}
�Ǥ�, $x$, $z$�����ؤ���������줾��������ؤ���Ĺ,
$L$�Ƚ񤤤Ƥ��������������οʹ������ؤ���Ĺ�Ǥ���.
���Ȥϴ�ñ�ʴ����ؤǤ���.
\Dfigref{hw-02-03-���ѷ�}
�Ǥϻ��ѷ������Ѥ���
$(1/2)ab = (1/2)cd$�δط���������Ω��.
��ä�,
%
\begin{eqnarray*}
 \mathcal{L}_x \mathcal{L}_z
  &=& \mathcal{L} \sqrt{\mathcal{L}_x^2 + \mathcal{L}_z^2}
  \\
 \mathcal{L} 
  &=& \frac{\mathcal{L}_x \mathcal{L}_z}{\sqrt{\mathcal{L}_x^2 + \mathcal{L}_z^2}}
  \\
  &=& \frac{4 \pi^2 (\mu k)^{-1}}
      {\sqrt{4 \pi^2 \mu^{-2} + 4 \pi^2 k^{-2}}}
  \\
  &=& \frac{2 \pi}
      {\sqrt{\mu^{2} + k^{2}}}
\end{eqnarray*}
%
�Ȥʤ�.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/phase_line3.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize ��������ˡ�������ؤ���Ĺ�ȳ���ʬ����Ĺ�Ȥδط�}
  }
  \Dfiglab{hw-02-03-��ʬ���Ȥ���Ĺ�δط�}
 \end{quote}
\end{figure}
%
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/trigono.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize ���ѷ�}
  }
  \Dfiglab{hw-02-03-���ѷ�}
 \end{quote}
\end{figure}
%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-3-E      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-3-E\qquad �����絤�Ǥβ�®}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{3-3-E-\arabic{equation}}
\def\thefigure{3-3-E-\arabic{figure}}

��(3-10)
\footnote{
\begin{eqnarray*}
 c = \pm \sqrt{\gamma R \overline{T}}.
\end{eqnarray*}
}
���Ѥ�, �����絤�Ǥμºݤβ�®����Ƥߤ�.
�������갵��Ǯ, ������Ǯ�����줾��
$c_p = 1004$ [J K$^{-1}$ kg$^{-1}$],
$c_v = 717$ [J K$^{-1}$ kg$^{-1}$]�ʤΤ�,
$\gamma = c_p / c_v = 1.4$
�Ȥʤ�.
���絤�Τ�ʬ���̤�$W_d = 28.97 \times 10^{-3}$ [kg mol$^{-1}$]
�ʤΤ�, �����絤�ε������$R_d$��
$R_d = R^{*} / W_d = 287$ [J K$^{-1}$ kg$^{-1}$]�Ȥʤ�.
���ʤߤ�$R^{*} = 8.3143 $ [J K$^{-1}$ mol$^{-1}$]�ϰ��̵�������Ǥ���.
��äƲ�®��,
%
\begin{eqnarray*}
 c &=&
  \pm \sqrt{1.4 \times 287 T}
 \\
   &=&
   \pm 20.04 \sqrt{T}
\end{eqnarray*}
%
�Ȥʤ�. �������ݻ�$t = T - 273$ [��]�����������,
%
\begin{eqnarray}
 \Deqlab{hw-02-03-��®�μ�}
 c = \pm 20.04 \sqrt{273 + t}
\end{eqnarray}
%
�Ȥʤ�. ������, $f(t) = \sqrt{273 + t}$ �Ȥ�����$f(t)$��$t=0$�Τޤ��
�ǥƥ��顼Ÿ�����Ƥߤ��,
%
\begin{eqnarray*}
 f(t) &=& \sum_{n=0}^{\infty} \frac{f^n (0)}{n!} x^n
  \\
      &=& \sqrt{273} + \frac{t}{2 \sqrt{273}} - \cdots
  \\
      &\simeq& \sqrt{273} + \frac{t}{2 \sqrt{273}}
\end{eqnarray*}
%
�Ȥʤ�. �����
\Deqref{hw-02-03-��®�μ�}�����������,
%
\begin{eqnarray*}
 c = \pm (331.1 + 0.606t)
\end{eqnarray*}
%
�Ȥʤä�.

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

%\subsection*{Appendix \qquad }

%\subsubsection*{}

%%%%%%%%          ���ֹ�Υ������Х饤��        %%%%%%%% 
\setcounter{equation}{10}
\setcounter{figure}{1}
\def\theequation{\arabic{chapter}-\arabic{equation}}
\def\thefigure{\arabic{chapter}-\arabic{figure}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%              text end                  %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
