波動方程式與其應用和記法

在檢驗波函數時,我們會利用到下列微分方程式(波動方程式):
\frac{\partial^2y }{\partial t^2}=v^2\cdot \frac{\partial^2 y}{\partial x^2}

物理意義為:
只要是波,其波函數必符合此微分式
只要符合此微分式,就是波
我們可以利用 y=f(x-vt) 這類常見的波的位置和時間函數來檢驗
\frac{\partial y}{\partial t}= f'\cdot v
(連鎖率,先微外面 f 再微裡面)
也可以寫成 u=x-vt
\frac{\partial y}{\partial t}=\frac{\partial f}{\partial t} =\frac{df}{du}\cdot \frac{\partial u}{\partial t}=f'\cdot v
\frac{\partial^2 y}{\partial t^2} =\frac{\partial }{\partial t} \left(\frac{\partial y}{\partial t} \right) =\frac{\partial }{\partial t}\left(f'\cdot v\right) \\ \vspace{1 mm} \hspace{12 mm} =v\cdot \frac{\partial f'}{\partial t} =v\cdot f''\cdot v=v^2\cdot f''
若對 \frac{\partial f'}{\partial t} 有疑問的話
\frac{\partial f'}{\partial t}=\frac{df'}{du}\cdot \frac{\partial u}{\partial t} =f''\cdot v
\frac{\partial^2 y}{\partial t^2} =v^2\cdot f''
\frac{\partial y}{\partial x} =\frac{\partial f}{\partial x}=\frac{df}{du}\cdot \frac{\partial u}{\partial x} =f'\cdot 1
\frac{\partial^2 y}{\partial x^2} =\frac{\partial }{\partial x} \left(f'\right)=\frac{df'}{du} \cdot \frac{\partial u}{\partial x} =f''\cdot 1
\frac{\partial^2 y}{\partial x^2} = f''
所以
\frac{\partial^2y }{\partial t^2}=v^2\cdot \frac{\partial^2 y}{\partial x^2}
此微分方程是很重要的式子,因此我們必須熟記它
記憶的方式就是
\left(\frac{\partial x}{\partial t} \right) ^2= v^2
下一小節,我們將談論駐波是否滿足波動方程式;以及波動方程式有什麼應用

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