弦&棒的振动

弦振动

弦的振动方程

对弦上$x\rightarrow x+\mathrm{d}x$元段进行受力分析之后利用牛顿第二定律可以推出：

$$\dfrac{\partial^2\eta}{\partial x^2} = \dfrac{1}{c^2}\dfrac{\partial^2\eta}{\partial t^2}$$

其中：$\eta$是元段的垂直位移，$c^2 = \dfrac{T}{\delta}$, $\delta = \rho S$是线密度.

通解

设$\delta$和$T$均为常数，我们用行波法:

$$\begin{cases} \zeta = ct + x \\ \xi = ct - x. \end{cases}$$

弦波方程变换为：

$$\dfrac{\partial^2 \eta}{\partial x^2} - \dfrac{1}{c^2}\dfrac{\partial^2\eta}{\partial t^2} = \bigg( \dfrac{\partial}{\partial x} + \dfrac{1}{c} \dfrac{\partial}{\partial t} \bigg)\bigg( \dfrac{\partial}{\partial x} - \dfrac{1}{c} \dfrac{\partial}{\partial t} \bigg)\eta = 0 \ \ \ \rightarrow \ \ \ \dfrac{\partial^2\eta}{\partial \xi \partial \zeta} = 0$$

于是$\eta$是分别为沿$x$正反方向以速度$c$传播的行波：

$$\eta(x,t) = f(\xi)+g(\zeta)=f(ct-x)+f(ct+x)$$