波動方程式の変換

■問題

$3$\displaystyle{y=f $ x,t $ }$ において，1次元の波動方程式

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}={c}^{2}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$

に $3$\displaystyle{{\xi}=x- ct}$ ， $3$\displaystyle{{\eta}=x+ct}$ なる変換を行うと

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{\eta}{\partial}{\xi} \hspace{2}}=0}$

となることを示せ．

■ヒント

最初に $3$\displaystyle{{\xi}=x- ct}$ ， $3$\displaystyle{{\eta}=x+ct}$ より， $3$\displaystyle{x}$ ， $3$\displaystyle{t}$ を $3$\displaystyle{{\xi}}$ ， $3$\displaystyle{{\eta}}$ で表す．

$3$\displaystyle{y}$ を $3$\displaystyle{{\xi}}$ で偏微分した後，更に $3$\displaystyle{{\eta}}$ で偏微分する．このとき，合成関数の偏導関数の公式を用いる．

■解説

$3$\displaystyle{{\xi}=x- ct}$ 　･･････(1)

$3$\displaystyle{{\eta}=x+ct}$ 　･･････(2)

とおく．

(1) を変形すると

$3$\displaystyle{x={\xi}+ct}$

これを (2) に代入して整理すると

$3$\displaystyle{{\eta}={\xi}+ct+ct}$

$3$\displaystyle{t=\frac{\hspace{2} {\eta}- {\xi} \hspace{2}}{\hspace{2} 2c \hspace{2}}}$ 　･･････(3)

となる．更に，これを (1) に代入し整理すると

$3$\displaystyle{{\xi}=x- c\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\eta}- {\xi} \hspace{2}}{\hspace{2} 2c \hspace{2}}}$

$3$\displaystyle{x=\frac{\hspace{2} {\xi}+{\eta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(4)

となる．

(4)より $3$\displaystyle{x}$ を $3$\displaystyle{{\xi}}$ で偏微分すると

$3$\displaystyle{\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}} $ \frac{\hspace{2} {\xi}+{\eta} \hspace{2}}{\hspace{2}2\hspace{2}} $ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\xi}+\frac{\hspace{2}{\eta}\hspace{2}}{\hspace{2}2\hspace{2}} $ }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(5)

次に，(3)より $3$\displaystyle{t}$ を $3$\displaystyle{{\xi}}$ で偏微分すると

$3$\displaystyle{\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}} $ \frac{\hspace{2} {\eta}- {\xi} \hspace{2}}{\hspace{2} 2c \hspace{2}} $ }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}} $ \frac{\hspace{2}{\eta}\hspace{2}}{\hspace{2} 2c \hspace{2}}- \frac{\hspace{2}{\xi}\hspace{2}}{\hspace{2} 2c \hspace{2}} $ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}}$ 　･･････(6)

同様の手順で $3$\displaystyle{x}$ ， $3$\displaystyle{t}$ を $3$\displaystyle{\text{\hspace{1} }\text{\hspace{1} }{\eta}\text{\hspace{1} }\text{\hspace{1} }}$ で偏微分をすると

$3$\displaystyle{\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2} {\xi}+{\eta} \hspace{2}}{\hspace{2}2\hspace{2}} $ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2}{\xi}\hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\eta} $ }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}}$ 　･･････(7)

$3$\displaystyle{\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2} {\eta}- {\xi} \hspace{2}}{\hspace{2} 2c \hspace{2}} $ }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2}{\eta}\hspace{2}}{\hspace{2} 2c \hspace{2}}- \frac{\hspace{2}{\xi}\hspace{2}}{\hspace{2} 2c \hspace{2}} $ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}}$ 　･･････(8)

(5)，(6)より $3$\displaystyle{y}$ を $3$\displaystyle{{\xi}}$ で偏微分すると（合成関数の偏導関数の公式を参照）

$3$\displaystyle{\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}}=\hspace{1}{f}_{x}\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}}+\hspace{1}{f}_{t}\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{x}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}+\hspace{1}{f}_{t} $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}} $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\hspace{1}{f}_{x}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\hspace{1}{f}_{t}}$ 　･･････(9)

これを更に $3$\displaystyle{{\eta}}$ で偏微分すると

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{\eta}{\partial}{\xi} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}{\xi} \hspace{2}} $ }$

(9)を代入する．

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\hspace{1}{f}_{x}- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}\hspace{1}{f}_{t} $ }$

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\hspace{1}{f}_{x} $ - \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}\hspace{1}{f}_{t} $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}\hspace{1}{f}_{x}- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}\hspace{1}{f}_{t}}$ 　･･････(10)

ここで， $3$\displaystyle{\hspace{1}{f}_{x}}$ ， $3$\displaystyle{\hspace{1}{f}_{t}}$ は，合成関数 $3$\displaystyle{y=f $ x,t $ }$ を $3$\displaystyle{x}$ ， $3$\displaystyle{t}$ でそれぞれ偏微分したものであるから，どちらも合成関数である．

よって， $3$\displaystyle{\text{\hspace{1} }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}\hspace{1}{f}_{x}\text{\hspace{1} },\text{\hspace{1} }\text{\hspace{1} }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}\hspace{1}{f}_{t}\text{\hspace{1} }}$ は，どちらも合成関数の偏微分となるので

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}\hspace{1}{f}_{x}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{x}\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}+\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}t \hspace{2}}\hspace{1}{f}_{x}\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{ xx }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}+\hspace{1}{f}_{ xt }\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{ xx }\text{\hspace{1} }\cdot \text{\hspace{1} } $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ +\hspace{1}{f}_{ xt }\text{\hspace{1} }\cdot \text{\hspace{1} } $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}} $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\hspace{1}{f}_{ xx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}\hspace{1}{f}_{ xt }}$ 　･･････(11)

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}\hspace{1}{f}_{t}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{t}\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}+\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}t \hspace{2}}\hspace{1}{f}_{t}\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{ tx }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}+\hspace{1}{f}_{ tt }\frac{\hspace{2} {\partial}t \hspace{2}}{\hspace{2} {\partial}{\eta} \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{ tx }\text{\hspace{1} }\cdot \text{\hspace{1} } $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ +\hspace{1}{f}_{ tt }\text{\hspace{1} }\cdot \text{\hspace{1} } $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}} $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\hspace{1}{f}_{ tx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}\hspace{1}{f}_{ tt }}$ 　･･････(12)

となる．(10)に(11)，(12)を代入する．

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{\eta}{\partial}{\xi} \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \hspace{1}{f}_{ xx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2}c\hspace{2}}\hspace{1}{f}_{ xt } $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2c \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \hspace{1}{f}_{ tx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2}c\hspace{2}}\hspace{1}{f}_{ tt } $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\hspace{1}{f}_{ xx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 4c \hspace{2}}\hspace{1}{f}_{ xt }- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4c \hspace{2}}\hspace{1}{f}_{ tx }- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4{c}^{2} \hspace{2}}\hspace{1}{f}_{ tt }}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\hspace{1}{f}_{ xx }+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 4c \hspace{2}}\hspace{1}{f}_{ xt }- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4c \hspace{2}}\hspace{1}{f}_{ xt }- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4{c}^{2} \hspace{2}}\hspace{1}{f}_{ tt }}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\hspace{1}{f}_{ xx }- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4{c}^{2} \hspace{2}}\hspace{1}{f}_{ tt }}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4{c}^{2} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}}$

これに， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}={c}^{2}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}\text{\hspace{1} }}$ を代入すると

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{\eta}{\partial}{\xi} \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4{c}^{2} \hspace{2}}\text{\hspace{1} }\cdot \text{\hspace{1} }{c}^{2}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}=0}$

となりる．

以上より

に $3$\displaystyle{{\xi}=x- ct}$ ， $3$\displaystyle{{\eta}=x+ct}$ なる変換を行うと

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{\eta}{\partial}{\xi} \hspace{2}}=0\text{\hspace{1} \hspace{1} }\text{\hspace{1} \hspace{1} }}$

となる．

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最終更新日： 2023年9月5日