波动

一维波动

One-Dimensional Waves

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

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

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 $x^\prime = x \mp v t$。

• I

首先对 $x$ 求偏导数，$t$ 视作常数，则有

$$ \begin{aligned} &\frac{\partial \psi}{\partial x} = \frac{\partial f}{\partial x}\\[8pt] &\frac{\partial \psi}{\partial x} = \frac{\partial f}{\partial x^\prime} \frac{\partial x^\prime}{\partial x} = \frac{\partial f}{\partial x^\prime}\\[8pt] \text{because}\qquad &\frac{\partial x^\prime}{\partial x} = \frac{\partial (x \mp v t)}{\partial x} = 1 \end{aligned} $$

• 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

$$ \cases{\, \begin{aligned} \frac{\partial \psi}{\partial x} &= \frac{\partial f}{\partial x^\prime} &\text{(1)}\\[8pt] \frac{\partial \psi}{\partial t} &= \mp v \frac{\partial f}{\partial x^\prime} &\text{(2)} \end{aligned} } $$

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

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

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

$$ \cases{\, \begin{aligned} \frac{\partial^2 \psi}{\partial x^2} &= \frac{\partial^2 f}{\partial x^{\prime 2}} &\text{(3)} \\[8pt] \frac{\partial^2 \psi}{\partial t^2} &= v^2 \frac{\partial^2 f}{\partial x^{\prime 2}} &\text{(4)} \end{aligned} } $$

合并式 (3) 和 (4)，可得

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

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谐波

Harmonic Waves

一般形式

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

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。$(x \mp v t)$ 的量纲是【长度】，因此 $\sin(x \mp v t)$ 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 $[k(x \mp v t)]$ 的量纲为【无量纲】，从而 $\sin[k(x \mp v t)]$ 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

$$ \cases{\, \begin{aligned} \psi (x, t) &= A \sin k (x \mp v t) \\[8pt] &= A \sin [k (x \mp v t) \pm 2\pi]\\[8pt] \psi (x\pm \lambda, t) &= A \sin k \left[ (x\pm \lambda) \mp v t \right] \\[8pt] &= A \sin [k (x \mp v t) \pm k \lambda] \end{aligned} } $$

合并上面两个式子，可得

$$ \begin{aligned} A \sin [k (x \mp v t) \pm 2\pi] &= A \sin [k (x \mp v t) \pm k \lambda]\\[8pt] \pm 2\pi &= \pm k \lambda \end{aligned} $$

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

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

推导过程

$$ \begin{aligned} \psi (x, t \pm \tau) &= A \sin k [ x \mp v (t \pm \tau) ] \\[8pt] &\text{Apply distributive property carefully:}\\[8pt] &= A \sin k [ x \mp vt \mp v(\pm \tau) ] \\[8pt] &\text{Group the original phase terms:}\\[8pt] &= A \sin [ \underbrace{k (x \mp vt)}_{\text{Original Phase}} \underbrace{\mp (\pm 1) kv\tau}_{\text{Shift}} ] \end{aligned} $$

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：

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

相位和相速度

Phase

$$ \varphi (x, t) = (k x - \omega t + \varepsilon) $$

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：

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

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：

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

于是有

$$ \left( \frac{\partial x}{\partial t} \right)_\varphi = \frac{-(\partial \varphi / \partial t)_x}{(\partial \varphi / \partial x)_t} $$

$$ \left( \frac{\partial x}{\partial t} \right)_\varphi = \pm \frac{\omega}{k} = \pm v $$

(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：

$$ \frac{\partial^2 \psi_1}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi_1}{\partial t^2} \quad \text{and} \quad \frac{\partial^2 \psi_2}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi_2}{\partial t^2} $$

$$ \frac{\partial^2 \psi_1}{\partial x^2} + \frac{\partial^2 \psi_2}{\partial x^2} = \frac{1}{v^2} \left( \frac{\partial^2 \psi_1}{\partial t^2} + \frac{\partial^2 \psi_2}{\partial t^2} \right) \\[16pt] $$

$$ \frac{\partial^2}{\partial x^2} (\psi_1 + \psi_2) = \frac{1}{v^2} \frac{\partial^2}{\partial t^2} (\psi_1 + \psi_2) $$

复数表示法

The Complex Representation

$$ \begin{aligned} e^{i\theta} &= \cos \theta + i \sin \theta \\[8pt] e^{-i\theta} &= \cos \theta - i \sin \theta \end{aligned} $$

$$ \begin{aligned} \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \\[8pt] \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \end{aligned} $$

$$ \tilde{z} = r e^{i\theta} = r \cos \theta + i\, r \sin \theta $$

其中

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

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

约定

$$ \psi (x, t) = A e^{i(\omega t - k x + \varepsilon)} = A e^{i\varphi} $$

相量与波的叠加

Phasors and the Addition of Waves

$$ E^2 = E_1^2 + E_2^2 + 2 E_1 E_2 \cos \delta $$

平面波

Plane Waves

波前 (Wavefront / Equiphase Surface)

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

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

处在同一平面波的点具有相同的幅值和相位。

三维微分波动方程

The Three-Dimensional Differential Wave Equation

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

$$ \nabla^2 \psi = \frac{1}{v^2} \ddot{\psi} $$

$$ \psi(x, y, z, t) = A e^{i k (\alpha x + \beta y + \gamma z \mp v t)} $$

球面波

Spherical Waves

$$ \psi(r, t) = \left( \frac{\mathscr{A}}{r} \right) \cos [k (r \mp v t)] $$

$$ \psi(r, t) = \left( \frac{\mathscr{A}}{r} \right) e^{i k (r \mp v t)} $$

柱面波

Cylindrical Waves

$$ \psi(r, t) \approx \frac{\mathscr{A}}{\sqrt{r}} \cos [k (r \mp v t)] $$

$$ \psi(r, t) \approx \frac{\mathscr{A}}{\sqrt{r}} e^{i k (r \mp v t)} $$