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
