📚 Wave Speed Formula and Medium Effects | 波速公式与介质影响
The speed of a wave tells us how fast a disturbance travels through a medium. For all waves, the fundamental relationship between speed v, frequency f, and wavelength λ is v = fλ. However, the actual value of v is not set by the source alone — it is determined by the properties of the medium through which the wave travels.
波速描述的是扰动在介质中传播的快慢。对于所有波，速度 v、频率 f 和波长 λ 之间的基本关系是 v = fλ。然而，速度 v 的实际值并非仅由波源决定，而是由波所传播的介质的性质所决定。
1. The Universal Wave Speed Equation | 通用波速方程
Every wave, whether mechanical or electromagnetic, obeys the equation v = fλ. Here, v is the phase speed (the speed of one point of constant phase), f is the frequency in hertz (Hz), and λ is the wavelength in metres (m). This relationship is derived from the definition of speed: in one complete oscillation, the wave advances by one wavelength in one period T, so v = λ/T = fλ.
所有波，无论是机械波还是电磁波，都遵循方程 v = fλ。其中 v 是相位速度（某一恒定相位点的速度），f 是以赫兹（Hz）为单位的频率，λ 是以米（m）为单位的波长。该关系由速度定义得出：在一个完整振动周期 T 内，波前进一个波长，因此 v = λ/T = fλ。
This equation is universal but deceptively simple. It does not tell us what determines v; it only relates v, f, and λ once the wave exists. To find the speed, we must examine the physical properties of the medium.
这个方程虽然通用，却并不简单。它并没有告诉我们什么决定 v，而只是在波存在时联系 v、f 和 λ。要找出速度，必须研究介质本身的物理性质。
2. Wave Speed on a Stretched String | 张紧弦上的波速
For a transverse wave travelling along a stretched string, the wave speed is given by:
v = √T/μ
where T is the tension in the string (in newtons) and μ (mu) is the mass per unit length of the string (in kg m⁻¹). This formula shows that increasing the tension increases the speed, while a heavier string (larger μ) slows the wave down.
对于沿张紧弦传播的横波，波速公式为：
v = √T/μ
其中 T 是弦中的张力（单位：牛顿），μ（缪）是弦的线密度，即单位长度的质量（单位：kg m⁻¹）。该公式表明，增大张力会使波速增大，而较重的弦（μ 较大）会使波速减小。
In CIE A-Level experiments, this is often tested with a vibration generator. When the tension is doubled (keeping μ constant), the speed increases by a factor of √2, not 2. Conversely, if you keep the tension constant and double the mass per unit length, the speed decreases by a factor of √2.
在 CIE A-Level 的实验中，这通常用振动发生器来测试。当张力加倍（μ 不变）时，波速变为原来的 √2 倍，而不是 2 倍。反过来，若张力不变而线密度加倍，波速则变为原来的 1/√2 倍。
3. Sound Waves in a Gas: The Role of Bulk Modulus and Density | 气体中的声波：体弹性模量与密度的作用
Sound is a longitudinal mechanical wave. In a gas, its speed is given by:
v = √B/ρ
Here B is the bulk modulus of the gas (a measure of its elasticity) and ρ is the density. A higher bulk modulus means the medium is less compressible, so disturbances are transmitted faster. A higher density means more inertia, so the wave slows down.
声波是一种纵机械波。在气体中，其速度公式为：
v = √B/ρ
其中 B 是气体的体弹性模量（衡量其弹性的物理量），ρ 是密度。体弹性模量越大，说明介质越难被压缩，扰动传播得越快；密度越大，说明惯性越大，波速则越慢。
For an ideal gas, this can be written in a more practical form:
v = √γRT/M
where γ is the adiabatic index (about 1.4 for diatomic gases like oxygen and nitrogen), R is the molar gas constant (8.31 J mol⁻¹ K⁻¹), T is the absolute temperature in kelvin, and M is the molar mass in kg mol⁻¹. This form explains why the speed of sound in air increases with temperature — about 0.6 m s⁻¹ for every 1 °C rise near room temperature.
对于理想气体，上式可写成更实用的形式：
v = √γRT/M
其中 γ 是绝热指数（对于氧气、氮气等双原子气体约为 1.4），R 是摩尔气体常数（8.31 J mol⁻¹ K⁻¹），T 是以开尔文为单位的绝对温度，M 是以 kg mol⁻¹ 为单位的摩尔质量。这一形式解释了为什么空气中的声速随温度升高而增大——在室温附近，每升高 1 °C，声速约增加 0.6 m s⁻¹。
4. Sound in Solids and Liquids | 固体和液体中的声速
In liquids and solids, the same general formula v = √B/ρ applies for longitudinal (compressional) waves, but the bulk modulus B is much larger than in gases. For solids, a more precise description uses Young modulus E:
v = √E/ρ
Because solids are highly elastic, their bulk modulus is enormous, and sound travels much faster in steel (about 5000 m s⁻¹) than in air (about 343 m s⁻¹). Water, being less compressible than air, also has a higher speed of sound (about 1480 m s⁻¹).
在液体和固体中，纵波（压缩波）同样适用公式 v = √B/ρ，但体弹性模量 B 远大于气体中的值。对于固体，用杨氏模量 E 描述更为准确：
v = √E/ρ
由于固体的弹性很强，其体弹性模量非常大，因此声音在钢中的传播速度（约 5000 m s⁻¹）远大于在空气中的速度（约 343 m s⁻¹）。水比空气更难压缩，所以水中的声速也更高（约 1480 m s⁻¹）。
It is important to notice that density appears in the denominator, but in solids the elasticity effect dominates. Although steel is much denser than air, its bulk modulus is millions of times larger, so the net effect is a much higher speed.
需要注意，密度出现在分母中，但在固体中弹性效应占主导。尽管钢的密度远大于空气，但其体弹性模量高出数百万倍，因此最终效果是声速远高于空气中。
5. Electromagnetic Waves in a Vacuum | 真空中的电磁波
Electromagnetic waves are unique: they do not require a medium. In a vacuum, all EM waves — radio, light, X-rays — travel at exactly the same speed:
c = 3.00 × 10⁸ m s⁻¹
This is a fundamental constant of nature. Maxwell's equations predict that c = 1/√ε₀μ₀, where ε₀ is the permittivity of free space and μ₀ is the permeability of free space. The values of these constants do not depend on the wave's frequency, amplitude, or source.
电磁波是独特的：它们不需要介质。在真空中，所有电磁波——无线电波、光、X 射线——都以完全相同速度传播：
c = 3.00 × 10⁸ m s⁻¹
这是自然界的基本常数。麦克斯韦方程组预言 c = 1/√ε₀μ₀，其中 ε₀ 是真空介电常数，μ₀ 是真空磁导率。这些常数的值与波的频率、振幅或波源无关。
In a vacuum, the wave speed does not depend on the medium because there is no medium. However, the frequency of an EM wave is determined by the source, and the wavelength adjusts according to λ = c/f. Thus, higher-frequency waves have shorter wavelengths.
在真空中，波速不依赖介质，因为没有介质。然而，电磁波的频率由波源决定，波长则根据 λ = c/f 自动调整。因此，频率越高的波波长越短。
6. Electromagnetic Waves in a Material Medium | 电磁波在物质介质中的传播
When an EM wave enters a material such as glass or water, its speed decreases. The speed v in the medium is related to the vacuum speed c by the refractive index n:
v = c/n
For glass, n lass. The refractive index arises from the interaction between the electric field of the wave and the charged particles (electrons) in the material. These charges oscillate and re-radiate, slowing down the wave.
当电磁波进入玻璃或水等物质时，其速度会减小。介质中的速度 v 与真空速度 c 的关系由折射率 n 决定：
v = c/n
对于玻璃，n 通常约为 1.5，因此光在玻璃中的传播速度约为 2.0 × 10⁸ m s⁻¹。折射率来源于波的电场与材料中带电粒子（电子）的相互作用。这些电荷会振动并重新辐射，从而减慢波的传播。
Importantly, for most transparent materials the refractive index depends on frequency. This is called dispersion. Blue light (higher frequency) travels slightly slower than red light (lower frequency) in glass, which is why a prism splits white light into a spectrum. So the medium can affect the speed of an EM wave differently for each frequency.
重要的是，对于大多数透明材料，折射率取决于频率。这被称为色散。蓝光（频率较高）在玻璃中比红光（频率较低）传播得稍慢，这就是为什么棱镜能将白光分解为光谱。因此，介质对电磁波速度的影响因频率而异。
7. What Determines Wave Speed? | 什么决定波速？
In general, wave speed is controlled by two competing properties of the medium: elasticity (the restoring force) and inertia (the mass that resists motion). A stiffer medium returns to its original shape faster, so the wave travels faster. A denser medium has more inertia, so the wave travels slower (for the same elasticity).
一般来说，波速由介质的两种竞争性质决定：弹性（恢复力）和 惯性（阻碍运动的惰性质量）。越刚硬的介质恢复原状越快，因此波速越快。同一弹性下，密度越大、惯性越大，波速则越慢。
This principle unifies the formulas for a string (v = √T/μ) and for sound (v = √B/ρ). In both cases, tension or bulk modulus represents elasticity, while μ or ρ represents inertia.
这一原理统一了弦上的公式（v = √T/μ）和声音的公式（v = √B/ρ）。在这两个公式中，张力或体弹性模量代表弹性，而 μ 或 ρ 代表惯性。
For mechanical waves, the medium is essential and its state (temperature, pressure, composition) directly affects the speed. For electromagnetic waves, the medium determines the phase speed through the refractive index, but the wave does not require a medium to exist.
对于机械波，介质必不可少，其状态（温度、压力、成分）直接影响波速。对于电磁波，介质通过折射率决定相位速度，但波并不需要介质才能存在。
8. Factors That Do NOT Change Wave Speed | 不影响波速的因素
Many students confuse the relationship v = fλ with causality. A change in frequency does not change the speed of a wave in a given medium (unless the medium is dispersive for that type of wave). For example, increasing the frequency of a sound wave in air does not make it travel faster; instead, the wavelength decreases so that v stays constant.
许多学生会混淆 v = fλ 中的因果关系。在给定介质中，频率的改变并不会改变波速（除非该介质对这类波有色散）。例如，提高空气中声波的频率并不会使其传播得更快；相反，波长会减小，从而保持 v 不变。
- Amplitude: Louder sounds or brighter light do not travel faster. Energy increases, but speed remains the same.
- Frequency: In non-dispersive media, speed is independent of frequency. In dispersive media (e.g. glass for light), speed changes with frequency, but this is a medium property, not a source property.
- Source motion: Moving the source changes the frequency observed (Doppler effect), but the wave speed relative to the medium is unchanged.
- 振幅：更响的声音或更亮的光并不会传播得更快。能量增加，但波速不变。
- 频率：在非色散介质中，波速与频率无关；在色散介质中（例如玻璃对光），波速随频率变化，但这是介质性质，而非波源性质。
- 波源运动：波源运动会改变观察到的频率（多普勒效应），但波相对于介质的速度不变。
9. Worked Example: String under Tension | 例题：张紧的弦
stretched with a tension of 80 N. Calculate the speed of a transverse pulse on the string.
一根弦的线密度为 2.0 × 10⁻³ kg m⁻¹，受到 80 N 的张力拉伸。求横脉冲沿弦传播的速度。
Solution: Using v = √T/μ,
v = √80 / 2.0 × 10⁻³ = √4.0 × 10⁴ = 200 m s⁻¹
The speed is 200 m s⁻¹. Note that the speed does not depend on how the pulse was created — it is purely a property of the string's tension and linear density.
解答：利用 v = √T/μ，
v = √80 / 2.0 × 10⁻³ = √4.0 × 10⁴ = 200 m s⁻¹
速度为 200 m s⁻¹。注意速度与脉冲产生的具体方式无关——它只取决于弦的张力和线密度。
10. Worked Example: Sound in Helium vs Air | 例题：氦气与空气中的声速
ave γ = 1.4 and T = 300 K. Find the speed of sound in each gas using R = 8.31 J mol⁻¹ K⁻¹.
在相同温度和压强下，将氦气（摩尔质量 4.0 × 10⁻³ kg mol⁻¹）与空气（摩尔质量 29 × 10⁻³ kg mol⁻¹）进行比较。两者 γ = 1.4，T = 300 K。已知 R = 8.31 J mol⁻¹ K⁻¹，求各气体中的声速。
For helium:
v = √1.4 × 8.31 × 300 / 4.0 × 10⁻³ = √873 × 10³ ≈ 934 m s⁻¹
For air:
v = √1.4 × 8.31 × 300 / 29 × 10⁻³ = √120 × 10³ ≈ 347 m s⁻¹
Light gases give faster sound. This is why a voice in helium sounds higher-pitched — the resonant frequencies of the vocal tract increase because the wave speed is higher.
对于氦气：
v = √1.4 × 8.31 × 300 / 4.0 × 10⁻³ = √873 × 10³ ≈ 934 m s⁻¹
对于空气：
v = √1.4 × 8.31 × 300 / 29 × 10⁻³ = √120 × 10³ ≈ 347 m s⁻¹
轻气体中的声速更快。这就是吸入氦气后声音变得尖细的原因——声道中的共振频率因波速增大而升高。
11. Exam Tips: Common Pitfalls | 考试提示：常见陷阱
In CIE A-Level papers, wave speed questions often ask you to explain the effect of changing a variable. A frequent error is to say that increasing frequency increases speed because v = fλ. Remember: frequency and wavelength are not independent; they are linked through the medium's speed. The speed is set by the medium, not by the source.
在 CIE A-Level 考试中，波速题常要求解释改变某个变量的影响。一个常见错误是认为增加频率会增加速度，因为 v = fλ。请记住：频率和波长不是独立的；它们通过介质的速度相互联系。速度由介质决定，而不是由波源决定。
| Wave type | Speed formula | Key medium factors |
|---|---|---|
| Transverse wave on a string | v = √T/μ | Tension; mass per unit length |
| Sound in gas | v = √γRT/M | Temperature; molar mass; γ |
| Light in a transparent medium | v = c/n | Refractive index (frequency-dependent) |
v = √T/μ / v = √γRT/M / v = c/n
| 波的种类 | 速度公式 | 关键介质因素 |
|---|---|---|
| 弦上的横波 | v = √T/μ | 张力；线密度 |
| 气体中的声波 | v = √γRT/M | 温度；摩尔质量；γ |
| 透明介质中的光 | v = c/n | 折射率（与频率有关） |
12. Summary | 总结
The wave speed equation v = fλ applies to every wave, but the speed itself is a property of the medium. For mechanical waves on a string, speed depends on tension and linear density. For sound waves, speed depends on elasticity and density — in a gas, this is equivalent to temperature and molar mass. For electromagnetic waves, the speed in a vacuum is constant c, but in a material it is reduced to v = c/n. Frequency is fixed by the source; wavelength and speed are determined by the medium.
波速方程 v = fλ 适用于所有波，但速度本身是介质的性质。对于弦上的机械波，速度取决于张力和线密度；对于声波，速度取决于弹性和密度——在气体中，这相当于温度和摩尔质量；对于电磁波，真空中的速度为常数 c，但在物质中降为 v = c/n。频率由波源决定；波长和速度由介质决定。
Understanding how the medium affects wave speed is essential for explaining real-world phenomena: why sound is faster in warm air, why steel transmits sound faster than air, and why a prism splits white light. These ideas form a core part of the CIE A-Level Physics syllabus and appear frequently in multiple-choice and structured questions.
理解介质如何影响波速，对于解释现实世界中的现象至关重要：为什么声音在温暖的空气中传播更快，为什么钢传播声音比空气快，以及为什么棱镜能分解白光。这些概念是 CIE A-Level 物理课程的核心部分，在选择题和结构化题目中经常出现。
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