波動方程式與其應用和記法

在檢驗是否為波時,我們會利用到下列微分方程式(波動方程式)來確認位置與時間的函數:

$$\frac{\partial^2 y}{\partial t^2} = v^2 \cdot \frac{\partial^2 y}{\partial x^2}$$


物理意義為:只要是波,其波函數必符合此微分方程式。只要符合此微分方程式,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 $u = x – vt$ 則

$$
\begin{aligned}
\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} (\frac{\partial y}{\partial t}) = \frac{\partial}{\partial t} (f’ \cdot v) \\ &= v \cdot \frac{\partial f’}{\partial t} = v \cdot f^{\prime\prime} \cdot v = v^2 \cdot f^{\prime\prime}
\end{aligned}
$$

若對 $\frac{\partial f’}{\partial t}$ 有疑問的話

$$\frac{\partial f’}{\partial t} = \frac{df’}{du} \cdot \frac{\partial u}{\partial t} = f^{\prime\prime} \cdot v$$

$$故 \frac{\partial^2 y}{\partial t^2} = v^2 \cdot f^{\prime\prime}$$

$$\frac{\partial y}{\partial x} = \frac{\partial f}{\partial x} = \frac{df}{du} \cdot \frac{\partial u}{\partial x} = f’ \cdot 1$$

$$\frac{\partial^2 y}{\partial x^2} = \frac{\partial}{\partial x} (f’) = \frac{df’}{du} \cdot \frac{\partial u}{\partial x} = f^{\prime\prime} \cdot 1$$

$$故 \frac{\partial^2 y}{\partial x^2} = f^{\prime\prime}$$

所以

$$\frac{\partial^2 y}{\partial t^2} = v^2 \cdot \frac{\partial^2 y}{\partial x^2}$$

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

$$(\frac{\partial x}{\partial t})^2 = v^2$$

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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