📚 Wave Mechanics Models and the Wave Equation | 波动力学模型与波动方程
Wave mechanics is a cornerstone of A-level physics, connecting the simple idea of a travelling disturbance with the mathematics needed to describe oscillations, sound, light, and even quantum particles. In this article, we will build the wave equation from first principles, examine key models such as the progressive wave and standing wave, and clarify common examination traps.
波动力学是 A-level 物理的核心内容，它将“扰动传播”这一简单概念与描述振动、声音、光乃至量子粒子所需的数学联系起来。在本文中，我们将从基本原理出发构建波动方程，考察行波和驻波等关键模型，并厘清常见的考试陷阱。
1. What Is a Wave? | 什么是波？
A wave is a transfer of energy and information from one point to another without any net transfer of the medium itself. The particles of the medium oscillate about their equilibrium positions, passing the disturbance along.
波是能量和信息从一处传递到另一处，而介质本身不发生净位移。介质中的粒子在其平衡位置附近振动，从而将扰动传递下去。
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Mechanical waves require a medium (e.g. sound in air, waves on a string).
机械波需要介质（例如空气中的声音、绳子上的波）。
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Electromagnetic waves can travel through a vacuum (e.g. light, radio waves).
电磁波可以在真空中传播（例如光、无线电波）。
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Transverse waves: oscillations are perpendicular to the direction of energy transfer (e.g. light, waves on a string).
横波：振动方向与能量传播方向垂直（例如光、绳子上的波）。
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Longitudinal waves: oscillations are parallel to the direction of energy transfer (e.g. sound).
纵波：振动方向与能量传播方向平行（例如声音）。
The most important mathematical description is the wave equation, which relates displacement, position, time, wavelength, frequency, and speed.
最重要的数学描述是波动方程，它将位移、位置、时间、波长、频率和速度联系起来。
2. Key Parameters of a Wave | 波的关键参数
Before writing the wave equation, we must define the variables used. Consider a sinusoidal wave travelling in the positive x-direction.
在写出波动方程之前，我们必须定义所用到的变量。考虑一个沿 x 正方向传播的正弦波。
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Amplitude A: maximum displacement from equilibrium, measured in metres (m).
振幅 A：偏离平衡位置的最大位移，单位为米（m）。
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Wavelength λ: distance between two consecutive points in phase, measured in metres (m).
波长 λ：两个相邻同相点之间的距离，单位为米（m）。
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Period T: time for one complete oscillation, measured in seconds (s).
周期 T：完成一次全振动所需的时间，单位为秒（s）。
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Frequency f: number of oscillations per second, f = 1/T, measured in hertz (Hz).
频率 f：每秒内振动的次数，f = 1/T，单位为赫兹（Hz）。
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Wave speed v: distance travelled by the wave per unit time, v = f λ.
波速 v：波在单位时间内传播的距离，v = f λ。
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Phase φ: the position of a point in its cycle, often in radians.
相位 φ：点在振动周期中所处的位置，通常用弧度表示。
The relationship v = f λ is universal for all waves and is arguably the most used formula in wave questions.
关系式 v = f λ 对所有波都成立，可以说是波动题目中最常用的公式。
3. Deriving the Progressive Wave Equation | 推导行波方程
Consider a wave source at x = 0 oscillating with simple harmonic motion. The displacement at the source is y(0, t) = A sin(ωt), where ω = 2πf is the angular frequency.
考虑一个位于 x = 0 的波源做简谐振动。波源处的位移为 y(0, t) = A sin(ωt)，其中 ω = 2πf 是角频率。
A point at position x experiences the same oscillation but delayed by time x/v, because the wave takes time to travel the distance x. Therefore:
位于位置 x 的点经历相同的振动，但延迟了时间 x/v，因为波传播距离 x 需要时间。因此：
y(x, t) = A sin[ω(t – x/v)]
Using ω = 2π/T and v = f λ = λ/T, we can rewrite this in several equivalent forms:
利用 ω = 2π/T 和 v = f λ = λ/T，我们可以将其改写成几种等价形式：
y = A sin(2πft – 2πx/λ)
or, defining the wave number k = 2π/λ, we get:
或者，定义波数 k = 2π/λ，我们得到：
y = A sin(ωt – kx)
The minus sign indicates a wave travelling in the positive x-direction. A plus sign (ωt + kx) would indicate a wave travelling in the negative x-direction.
减号表示波沿 x 正方向传播。加号（ωt + kx）则表示波沿 x 负方向传播。
4. The General Wave Equation | 一般波动方程
The partial differential equation that governs all classical waves is:
支配所有经典波动的偏微分方程是：
∂²y/∂x² = (1/v²) ∂²y/∂t²
This equation shows that the second spatial derivative of the displacement is proportional to the second temporal derivative, with the constant of proportionality related to the wave speed squared.
该方程表明，位移对空间的二阶导数与对时间的二阶导数成正比，比例常数与波速的平方有关。
Verification: if y = A sin(ωt – kx), then ∂y/∂x = -kA cos(ωt – kx) and ∂²y/∂x² = -k²y. Similarly, ∂²y/∂t² = -ω²y. Since ω = vk, we have k² = ω²/v², hence ∂²y/∂x² = (1/v²) ∂²y/∂t².
验证：若 y = A sin(ωt – kx)，则 ∂y/∂x = -kA cos(ωt – kx)，∂²y/∂x² = -k²y。类似地，∂²y/∂t² = -ω²y。由于 ω = vk，我们有 k² = ω²/v²，因此 ∂²y/∂x² = (1/v²) ∂²y/∂t²。
Candidates should be able to substitute a given waveform into this equation to verify it is a solution, and to determine the wave speed from a graph or data.
考生应能将给定的波形代入该方程以验证其是否为解，并能从图像或数据中确定波速。
5. Phase Difference and Path Difference | 相位差与路程差
Phase difference is a common exam topic. Two points on a wave are separated by a path difference Δx, which corresponds to a phase difference Δφ.
相位差是常见考点。波上两个点的路程差为 Δx，对应相位差 Δφ。
Δφ = 2π Δx / λ
For a path difference of one wavelength, the phase difference is 2π radians (360°), meaning the points are in phase. For half a wavelength, the phase difference is π radians (180°), meaning the points are in antiphase.
路程差为一个波长时，相位差为 2π 弧度（360°），即两点同相。路程差为半个波长时，相位差为 π 弧度（180°），即两点反相。
Example: A wave of frequency 500 Hz travels at 340 m/s. What is the phase difference between two points 0.17 m apart?
示例：频率为 500 Hz 的波以 340 m/s 传播。两个相距 0.17 m 的点之间的相位差是多少？
First, λ = v/f = 340/500 = 0.68 m. Then Δφ = 2π × 0.17 / 0.68 = π/2 radians (90°).
首先，λ = v/f = 340/500 = 0.68 m。然后 Δφ = 2π × 0.17 / 0.68 = π/2 弧度（90°）。
6. The Principle of Superposition | 叠加原理
When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This is the principle of superposition.
当两个或多个波在同一点相遇时，合位移等于各分位移的矢量和。这就是叠加原理。
Mathematically, if y₁ and y₂ are two waves, then:
数学上，若 y₁ 和 y₂ 是两个波，则：
y = y₁ + y₂
Superposition explains interference patterns, beats, and standing waves. It applies to both transverse and longitudinal waves, as long as the amplitudes are not too large (linear regime).
叠加原理解释了干涉图样、拍频和驻波。它适用于横波和纵波，只要振幅不太大（线性区域）。
In A-level questions, you may be asked to sketch the resultant of two waves, or to calculate the amplitude when two waves of the same frequency are superposed with a known phase difference.
在 A-level 题目中，你可能需要画出两个波的合成图，或者计算两个同频波在已知相位差下叠加后的振幅。
7. Standing Waves | 驻波
A standing wave (or stationary wave) is formed when two waves of the same frequency and amplitude travel in opposite directions. For example, a wave on a string and its reflection from a fixed end.
驻波（或定波）由两个频率和振幅相同的波沿相反方向传播叠加而成。例如，绳子上的波与其在固定端的反射波。
Using the trigonometric identity for the sum of two sine waves, we obtain:
利用两个正弦波之和的三角恒等式，我们得到：
y = 2A sin(kx) cos(ωt)
In this expression, the position-dependent part sin(kx) and the time-dependent part cos(ωt) are separated. This means all particles vibrate with the same frequency, but their amplitude depends on their position.
在这个表达式中，与位置相关的部分 sin(kx) 和与时间相关的部分 cos(ωt) 是分离的。这意味着所有粒子都以相同的频率振动，但它们的振幅取决于其位置。
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Nodes: positions where sin(kx) = 0, so the amplitude is always zero. They occur at x = 0, λ/2, λ, 3λ/2, …
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