波動方程式とは
音波や電磁波が従う方程式になり、双曲型偏微分方程式に分類される。
$\displaystyle\frac{\partial^2 u}{\partial t^2}=c^2\displaystyle\frac{\partial^2 u}{\partial x^2}$
$c$は光速になるが、$v$とすれば一般の波に適用できる。
解法
$\xi=x+ct,\eta=x-ct$とおき、これらの偏微分に変換する。
$\displaystyle\frac{\partial u}{\partial t}=\displaystyle\frac{\partial u}{\partial \xi}\displaystyle\frac{\partial \xi}{\partial t}+\displaystyle\frac{\partial u}{\partial \eta}\displaystyle\frac{\partial \eta}{\partial t}=c\left(\displaystyle\frac{\partial u}{\partial \xi}-\displaystyle\frac{\partial u}{\partial \eta}\right)$
$\displaystyle\frac{\partial^2 u}{\partial t^2}=c\displaystyle\frac{\partial}{\partial \xi}\left(\displaystyle\frac{\partial u}{\partial \xi}-\displaystyle\frac{\partial u}{\partial \eta}\right)\displaystyle\frac{\partial \xi}{\partial t}-c\displaystyle\frac{\partial}{\partial \eta}\left(\displaystyle\frac{\partial u}{\partial \xi}-\displaystyle\frac{\partial u}{\partial \eta}\right)\displaystyle\frac{\partial \eta}{\partial t}$
$=c^2\left(\displaystyle\frac{\partial^2 u}{\partial \xi^2}-2\displaystyle\frac{\partial^2 u}{\partial \xi\partial \eta}+\displaystyle\frac{\partial^2 u}{\partial \eta^2}\right)$
同様に
$\displaystyle\frac{\partial^2 u}{\partial x^2}=\left(\displaystyle\frac{\partial^2 u}{\partial \xi^2}+2\displaystyle\frac{\partial^2 u}{\partial \xi\partial \eta}+\displaystyle\frac{\partial^2 u}{\partial \eta^2}\right)$
これらを波動方程式に代入すると
$\displaystyle\frac{\partial^2 u}{\partial \xi\partial \eta}=0$
つまり、一般解は
$u(\xi,\eta)=f(\xi)+g(\eta)$
$u(x,t)=f(x+ct)+g(x-ct)$
