Dispersion and Group Velocity | 色散与群速度

📚 Dispersion and Group Velocity | 色散与群速度

Dispersion describes the phenomenon where the phase velocity of a wave depends on its frequency (or wavelength). In many physical systems, waves of different frequencies travel at different speeds, causing a wave packet to spread out or change shape. The group velocity, defined as the velocity of the envelope of a wave packet, determines how energy and information propagate. This article explores the mathematical derivation of group velocity, its relationship with phase velocity, and its significance, using IB-level calculus and trigonometry.

色散描述了波的相速度依赖于其频率（或波长）的现象。在许多物理系统中，不同频率的波以不同速度传播，导致波包展宽或形状改变。群速度定义为波包包络的速度，决定了能量和信息的传播方式。本文用IB级别的微积分和三角函数探讨群速度的数学推导、它与相速度的关系及其重要性。

1. Wave Function and Key Parameters | 波函数与关键参数

A monochromatic traveling wave can be described by y(x,t) = A cos(kx – ωt + φ), where A is amplitude, k is the angular wavenumber (k = 2π / λ), ω is the angular frequency (ω = 2πf), and φ is the phase constant. The argument (kx – ωt) determines the phase of the wave at position x and time t.

单色行波可以用 y(x,t) = A cos(kx – ωt + φ) 描述，其中 A 为振幅，k 为角波数 (k = 2π/λ)，ω 为角频率 (ω = 2πf)，φ 为初相。辐角 (kx – ωt) 决定了波在位置 x 和时间 t 的相位。

y(x,t) = A cos(kx – ωt + φ)

2. Phase Velocity | 相速度

The phase velocity v_p is the speed at which a point of constant phase travels. Setting the phase constant: kx – ωt = constant, and differentiating with respect to t gives k (dx/dt) – ω = 0, hence v_p = dx/dt = ω / k. This velocity describes how fast individual crests or troughs move.

相速度 v_p 是恒定相位的点（例如波峰）移动的速度。设相位为常数：kx – ωt = 常数，对 t 求导可得 k (dx/dt) – ω = 0，因此 v_p = dx/dt = ω / k。该速度描述了单个波峰或波谷移动的快慢。

v_p = ω / k

3. Dispersion Relation | 色散关系

The relationship ω = ω(k) is called the dispersion relation; it emerges from the physical governing equation (e.g., the wave equation with restoring forces). If ω is directly proportional to k (ω = ck with constant c), then v_p = c is independent of frequency – there is no dispersion. If ω depends nonlinearly on k, different frequency components travel at different phase speeds, causing dispersion.

关系式 ω = ω(k) 称为色散关系，它源自物理支配方程（如带回复力的波动方程）。如果 ω 与 k 成正比（ω = ck，c 为常数），则 v_p = c 与频率无关，即无频散。如果 ω 与 k 非线性相关，不同频率分量以不同相速度传播，就会引起色散。

4. Superposition of Two Waves | 两列波的叠加

Consider two cosine waves of equal amplitude and nearly equal wavenumbers and frequencies: y₁ = A cos(k₁x – ω₁t) and y₂ = A cos(k₂x – ω₂t). Their superposition is y = y₁ + y₂. Applying the sum-to-product identity cosα + cosβ = 2 cos[(α – β)/2] cos[(α + β)/2], we obtain:

考虑振幅相等的两列余弦波，它们的波数与频率非常接近：y₁ = A cos(k₁x – ω₁t) 和 y₂ = A cos(k₂x – ω₂t)。叠加得 y = y₁ + y₂。运用和差化积公式 cosα + cosβ = 2 cos[(α – β)/2] cos[(α + β)/2]，得到：

y = 2A cos[½(k₁ – k₂)x – ½(ω₁ – ω₂)t] · cos[½(k₁ + k₂)x – ½(ω₁ + ω₂)t]

5. Envelope Speed and Group Velocity | 包络速度与群速度

The result consists of a rapidly oscillating carrier wave (the second cosine factor with average wavenumber and frequency) whose amplitude is modulated by a slowly varying envelope (the first cosine factor). The envelope travels at speed v_env = (½(ω₁ – ω₂)) / (½(k₁ – k₂)) = (ω₁ – ω₂) / (k₁ – k₂). This is the prototype of group velocity. As k₂ → k₁, the difference quotient approaches the derivative dω/dk.

合成波包含一个快速振荡的载波（第二个余弦因子，具有平均波数和频率），其振幅受到缓慢变化的包络（第一个余弦因子）的调制。包络以速度 v_env = (½(ω₁ – ω₂)) / (½(k₁ – k₂)) = (ω₁ – ω₂) / (k₁ – k₂) 传播。这正是群速度的原型。当 k₂ → k₁ 时，该差商趋近于导数 dω/dk。

v_g = lim_Δk→0 Δω / Δk = dω/dk

6. Group Velocity as a Derivative | 群速度作为导数

For a general dispersion relation ω(k), the group velocity is defined as v_g = dω/dk. It represents the speed at which the overall shape of a wave packet (and thus energy or information) moves. When d²ω/dk² ≠ 0, different frequency components travel apart and the pulse broadens – this is called group velocity dispersion.

对于一般的色散关系 ω(k)，群速度定义为 v_g = dω/dk。它表示波包的整体形状（因而能量或信息）移动的速度。当 d²ω/dk² ≠ 0 时，不同频率成分逐渐分离，脉冲展宽——这称为群速度色散。

7. Relationship Between Group and Phase Velocity | 群速度与相速度的关系

Since v_p = ω/k, we can write ω = k v_p and differentiate: v_g = d(k v_p)/dk = v_p + k (dv_p/dk). Using the wavenumber–wavelength relation k = 2π/λ, the chain rule yields an equivalent formula in terms of λ:

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）