波动

一维波动

One-Dimensional Waves

一维波动函数 (The one-dimensional wave function)

$$\psi (x,t)=f(x\mp vt)$$

一维微分波动方程 (The one-dimensional differential wave equation)

$$\frac{{\partial }^2\psi }{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^2\psi }{\partial t^2}$$

推导过程

已知一维波动函数为 $\psi (x,t)=f(x^{\prime })$，其中 $x^{\prime }=x\mp vt$。

• I

首先对 $x$ 求偏导数，$t$ 视作常数，则有

$$\begin{array}{rl}& \frac{\partial \psi }{\partial x}=\frac{\partial f}{\partial x}\\ & \frac{\partial \psi }{\partial x}=\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial x}=\frac{\partial f}{\partial x^{\prime }}\\ \text{because}& \frac{\partial x^{\prime }}{\partial x}=\frac{\partial (x\mp vt)}{\partial x}=1\end{array}$$

• II

对 $t$ 求偏导数，$x$ 视作常数，则有

$$\frac{\partial \psi }{\partial t}=\frac{\partial f}{\partial x^{\prime }}\frac{\partial x^{\prime }}{\partial t}=\frac{\partial f}{\partial x^{\prime }}(\mp v)=\mp v\frac{\partial f}{\partial x^{\prime }}$$

• III

至此，我们得到了

$$\{\begin{array}{l}\begin{array}{rlr}\frac{\partial \psi }{\partial x}& =\frac{\partial f}{\partial x^{\prime }}& \text{(1)}\\ \frac{\partial \psi }{\partial t}& =\mp v\frac{\partial f}{\partial x^{\prime }}& \text{(2)}\end{array}\end{array}$$

将上面两个式子结合，可得

$$\frac{\partial \psi }{\partial t}=\mp v\frac{\partial \psi }{\partial x}$$

• IV

继续，对 $x$ 求二阶偏导数，$t$ 依旧视作常数。根据式 (1)，有

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

• V

对 $t$ 求二阶偏导数，$x$ 依旧视作常数。根据式 (2)，有

$$\frac{{\partial }^2\psi }{\partial t^2}=\frac{\partial }{\partial t}(\mp v\frac{\partial f}{\partial x^{\prime }})=\mp v\frac{\partial }{\partial x^{\prime }}\left(\frac{\partial f}{\partial t}\right)$$

因为 ${\displaystyle \frac{\partial \psi }{\partial t}=\frac{\partial f}{\partial t}}$，所以有

$$\frac{{\partial }^2\psi }{\partial t^2}=\mp v\frac{\partial }{\partial x^{\prime }}\left(\frac{\partial \psi }{\partial t}\right)$$

对上式代入式 (2) ${\displaystyle \frac{\partial \psi }{\partial t}=\mp v\frac{\partial f}{\partial x^{\prime }}}$，可得

$$\frac{{\partial }^2\psi }{\partial t^2}=v^2\frac{{\partial }^{2}f}{\partial x^{\prime 2}}$$

• VI

整理 IV 和 V 的结果，可得

$$\{\begin{array}{l}\begin{array}{rlr}\frac{{\partial }^2\psi }{\partial x^2}& =\frac{{\partial }^{2}f}{\partial x^{\prime 2}}& \text{(3)}\\ \frac{{\partial }^2\psi }{\partial t^2}& =v^2\frac{{\partial }^{2}f}{\partial x^{\prime 2}}& \text{(4)}\end{array}\end{array}$$

合并式 (3) 和 (4)，可得

$$\boxed{{\displaystyle \frac{{\partial }^2\psi }{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^2\psi }{\partial t^2}}}$$

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谐波

Harmonic Waves

一般形式

$$\psi (x,t)=A\sin [k(x\mp vt)]=f(x\mp vt)$$

其中

• $k$ 是一个正的常数，被称作波数 (propagation number / wave number)。这里的 $k$ 是必要的，这是因为三角函数的参数必须为无量纲量。$(x\mp vt)$ 的量纲是【长度】，因此 $\sin (x\mp vt)$ 在物理量纲上是不自洽的。为此引入 $k$ 并令其量纲为【长度^-1】，这样 $[k(x\mp vt)]$ 的量纲为【无量纲】，从而 $\sin [k(x\mp vt)]$ 在物理量纲上是自洽的。
• $A$ 是振幅 (amplitude)。

空间周期性 (Spatial Periodicity)

谐波具有空间周期性，用数学语言描述为

$$\psi (x,t)=\psi (x\pm \lambda ,t)$$

其中

• $\lambda ={\displaystyle \frac{2\pi }{k}}$ 被称作波长 (wavelength)。

推导过程

展开分析

$$\{\begin{array}{l}\begin{array}{rl}\psi (x,t)& =A\sin k(x\mp vt)\\ & =A\sin [k(x\mp vt)\pm 2\pi ]\\ \psi (x\pm \lambda ,t)& =A\sin k[(x\pm \lambda )\mp vt]\\ & =A\sin [k(x\mp vt)\pm k\lambda ]\end{array}\end{array}$$

合并上面两个式子，可得

$$\begin{array}{rl}A\sin [k(x\mp vt)\pm 2\pi ]& =A\sin [k(x\mp vt)\pm k\lambda ]\\ \pm 2\pi & =\pm k\lambda \end{array}$$

因此

$$\lambda =\frac{2\pi }{k}$$

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时间周期性 (Temporal Periodicity)

$$\psi (x,t)=\psi (x,t\pm \tau )$$

其中

• $\tau ={\displaystyle \frac{2\pi }{kv}}$ 被称作周期 (period)。

推导过程

$$\begin{array}{rl}\psi (x,t\pm \tau )& =A\sin k[x\mp v(t\pm \tau )]\\ & \text{Apply distributive property carefully:}\\ & =A\sin k[x\mp vt\mp v(\pm \tau )]\\ & \text{Group the original phase terms:}\\ & =A\sin [\underset{\text{Original Phase}}{\underbrace{k(x\mp vt)}}\underset{\text{Shift}}{\underbrace{\mp (\pm 1)kv\tau }}]\end{array}$$

这里 $\mp (\pm 1)$ 的结果虽然看起来复杂，但本质上只是让相位增加或减少 $kv\tau$。 为了满足波的周期性 $\psi (x,t)=\psi (x,t\pm \tau )$，相位偏移量的大小必须为 $2\pi$：

$$kv\tau =2\pi$$

因此得到

$$\tau =\frac{2\pi }{kv}$$

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符号整理

符号	名称	单位 (SI)	常用关系式
$A$	振幅 (Amplitude)	m	-
$k$	角波数 (Propagation Number)	rad/m	$k={\displaystyle \frac{2\pi }{\lambda }}$
$v$	波速 (Wave Speed)	m/s	$v=\nu \lambda ={\displaystyle \frac{\omega }{k}}$
$\lambda$	波长 (Spatial Period, Wavelength)	m	$\lambda ={\displaystyle \frac{2\pi }{k}}$
$\tau$	周期 (Temporal Period, also $T$)	s	$\tau ={\displaystyle \frac{2\pi }{kv}=\frac{1}{\nu }}$
$\nu$	频率 (Temporal Frequency)	Hz	$\nu ={\displaystyle \frac{1}{\tau }}$
$\omega$	角频率 (Angular Temporal Frequency)	rad/s	$\omega ={\displaystyle \frac{2\pi }{\tau }=kv}$
$\kappa$	波数 (Spatial Frequency, Wave Number)	m⁻¹	$\kappa ={\displaystyle \frac{1}{\lambda }=\frac{k}{2\pi }}$

其它

最常使用的两种谐波表达式：

• $\psi =A\sin [k(x\mp vt)]$
• $\psi =A\sin (kx\mp \omega t)$

相位和相速度

Phase

$$\phi (x,t)=(kx-\omega t+\epsilon )$$

固定位置 $x$ 不变，对时间 $t$ 求偏导数，可得 the rate-of-change of phase with time：

$$\left|{\left(\frac{\partial \phi }{\partial t}\right)}_x\right|=\omega$$

类似地，固定时间 $t$ 不变，对位置 $x$ 求偏导数，可得 the rate-of-change of phase with distance：

$$\left|{\left(\frac{\partial \phi }{\partial x}\right)}_t\right|=k$$

于是有

$${\left(\frac{\partial x}{\partial t}\right)}_{\phi }=\frac{-(\partial \phi /\partial t{)}_x}{(\partial \phi /\partial x{)}_t}$$$${\left(\frac{\partial x}{\partial t}\right)}_{\phi }=\pm \frac{\omega }{k}=\pm v$$

(Problem 2.34)

$$\pm v=\frac{-(\partial \psi /\partial t{)}_x}{(\partial \psi /\partial x{)}_t}$$

叠加原理

The Superposition Principle

对于波速 (wave speed) 相同的两种波，叠加后仍然满足波动方程：

$$\frac{{\partial }^2{\psi }_1}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^2{\psi }_1}{\partial t^2}\text{and}\frac{{\partial }^2{\psi }_2}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^2{\psi }_2}{\partial t^2}$$$$\frac{{\partial }^2{\psi }_1}{\partial x^2}+\frac{{\partial }^2{\psi }_2}{\partial x^2}=\frac{1}{v^2}(\frac{{\partial }^2{\psi }_1}{\partial t^2}+\frac{{\partial }^2{\psi }_2}{\partial t^2})$$$$\frac{{\partial }^2}{\partial x^2}({\psi }_1+{\psi }_2)=\frac{1}{v^2}\frac{{\partial }^2}{\partial t^2}({\psi }_1+{\psi }_2)$$

复数表示法

The Complex Representation

$$\begin{array}{rl}{e}^{i\theta }& =\cos \theta +i\sin \theta \\ e^{-i\theta }& =\cos \theta -i\sin \theta \end{array}$$$$\begin{array}{r}\cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2}\\ \sin \theta =\frac{e^{i\theta }-e^{-i\theta }}{2i}\end{array}$$$$\tilde{z}=r{e}^{i\theta }=r\cos \theta +ir\sin \theta$$

其中

• $r$ is the magnitude of $\tilde{z}$
• $\theta$ is the phase of $\tilde{z}$

$$\psi (x,t)=\text{Re}\left[A{e}^{i(\omega t-kx+\epsilon )}\right]$$

约定

$$\psi (x,t)=A{e}^{i(\omega t-kx+\epsilon )}=A{e}^{i\phi }$$

相量与波的叠加

Phasors and the Addition of Waves

$$E^2=E_1^2+E_2^2+2E_1{E}_2\cos \delta$$

平面波

Plane Waves

波前 (Wavefront / Equiphase Surface)

$$\psi (\mathbf{r},t)=A\cos (\mathbf{k}\cdot \mathbf{r}\mp \omega t)$$$$\psi (\mathbf{r},t)=A{e}^{i(\mathbf{k}\cdot \mathbf{r}\mp \omega t)}$$

三维微分波动方程

The Three-Dimensional Differential Wave Equation

$$\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}+\frac{{\partial }^2\psi }{\partial z^2}=\frac{1}{v^2}\frac{{\partial }^2\psi }{\partial t^2}$$$${\mathrm{\nabla }}^2\psi =\frac{1}{v^2}\ddot{\psi }$$$$\psi (x,y,z,t)=A{e}^{ik(\alpha x+\beta y+\gamma z\mp vt)}$$

球面波

Spherical Waves

$$\psi (r,t)=\left(\frac{\mathcal{A}}{r}\right)\cos [k(r\mp vt)]$$$$\psi (r,t)=\left(\frac{\mathcal{A}}{r}\right)e^{ik(r\mp vt)}$$

柱面波

Cylindrical Waves

$$\psi (r,t)\approx \frac{\mathcal{A}}{\sqrt{r}}\cos [k(r\mp vt)]$$$$\psi (r,t)\approx \frac{\mathcal{A}}{\sqrt{r}}{e}^{ik(r\mp vt)}$$