📚 Transverse and Longitudinal Waves: Distinguishing Wave Types | 纵波与横波：波动类型辨析
A wave transfers energy and information from one place to another without transferring the material itself. In a mechanical wave, particles vibrate about their equilibrium positions as the disturbance travels through a medium. One of the most important classifications in the CIE A Level Physics syllabus is based on the direction of particle vibration relative to the direction of energy transfer: this gives rise to transverse waves and longitudinal waves.
波将能量和信息从一处传递到另一处，但并不传递物质本身。在机械波中，粒子围绕其平衡位置振动，而扰动则通过介质传播。在 CIE A Level 物理考纲中，一个非常重要的分类依据是粒子振动方向与能量传播方向之间的关系：这便引出了横波与纵波两大类型。
1. Transverse Waves | 横波
A transverse wave is a wave in which the particles of the medium oscillate perpendicular to the direction of energy propagation. The wave profile shows a series of crests and troughs as it travels along the medium.
横波是指介质中粒子的振动方向与能量传播方向相互垂直的波。波的轮廓呈现出一系列波峰和波谷，并沿介质传播。
Common examples of transverse waves include:
常见的横波实例包括：
Waves on a stretched string or rope: when one end is flicked upward, a pulse travels along the string while each segment moves up and down.
绷紧的绳子或绳索上的波：当一端向上抖动时，脉冲沿绳传播，而每一段绳只在上下方向运动。
Electromagnetic waves: light, radio waves and X-rays are transverse because the electric and magnetic fields oscillate at right angles to the direction of travel.
电磁波：光、无线电波和 X 射线都是横波，因为电场和磁场都沿与传播方向垂直的方向振动。
Water waves at the surface: these are often treated as approximately transverse, although the motion of surface water particles is actually more complicated.
水面波：它们通常被近似视为横波，但实际上表面水分子的运动更为复杂。
One key feature of transverse waves is that they can be polarised. Polarisation occurs when the vibrations of a transverse wave are restricted to a single plane. This property is impossible for purely longitudinal waves, which will be discussed later.
横波的一个关键特征是能够发生偏振。偏振是指横波的振动被限制在单一平面内的现象。这一性质对于纯粹的纵波来说是不可能实现的，这一点后文将进一步说明。
2. Longitudinal Waves | 纵波
A longitudinal wave is a wave in which the particles of the medium oscillate parallel to, or along, the direction of energy propagation.
纵波是指介质中粒子的振动方向与能量传播方向平行或相同的波。
Instead of crests and troughs, a longitudinal wave is described in terms of compressions and rarefactions. A compression is a region where particles are squeezed together, and a rarefaction is a region where particles are spread apart.
纵波不像横波那样用波峰和波谷描述，而是用疏密部描述。密部是粒子被挤压在一起的区域，疏部是粒子被分散开的区域。
The most important example of a longitudinal wave is sound in air. When a vibrating object moves forward, it compresses the air in front of it; when it moves backward, it creates a rarefaction. These compressions and rarefactions travel outward as a sound wave.
纵波最重要的例子是空气中的声波。当振动体向前运动时，它会压缩前方的空气；向后运动时，则形成疏部。这些密部和疏部向外传播，形成声波。
Seismic P-waves, which travel through the interior of the Earth, are also longitudinal waves. In a longitudinal wave, the distance between successive compressions is equal to one wavelength.
震波中的 P 波也是纵波，它可以穿过地球内部。在纵波中，相邻两个密部之间的距离等于一个波长。
It is essential to remember that in a longitudinal wave the particles do not travel with the wave; they oscillate backwards and forwards about their equilibrium positions. Energy is transferred by this local oscillation, not by the bulk movement of matter.
必须记住，在纵波中粒子并不随波前进；它们只是在平衡位置附近往复振动。能量是通过这种局部振动传递的，而不是通过物质发生整体位移。
3. Transverse or Longitudinal: Key Differences | 横波与纵波的主要区别
The table below summarises the main differences between transverse and longitudinal waves. These comparisons are frequently tested in A Level multiple-choice and structured questions.
下表总结了横波与纵波的主要区别。这些对比在 A Level 选择题和结构化题目中经常出现。
| Feature | 特征 | Transverse Wave | 横波 | Longitudinal Wave | 纵波 |
| Particle vibration | 粒子振动 | Perpendicular to propagation | 与传播方向垂直 | Parallel to propagation | 与传播方向平行 |
| Graphical description | 图形描述 | Crests and troughs | 波峰和波谷 | Compressions and rarefactions | 密部和疏部 |
| Requires medium | 是否需要介质 | Can pass through vacuum, e.g. light | 可以穿过真空，例如光 | Mechanical longitudinal waves need a medium | 机械纵波需要介质 |
| Polarisation | 偏振 | Possible | 可以发生 | Not possible | 不可以发生 |
| Examples | 实例 | Light, waves on a string | 光、绳波 | Sound in air, seismic P-waves | 空气中声波、地震 P 波 |
Remember that electromagnetic waves are always transverse, while sound waves are always longitudinal in fluids. A question may ask you to use polarisation or the ability to travel through a vacuum to distinguish between wave types.
请记住：电磁波一定是横波，而流体中的声波一定是纵波。题目可能要求你利用偏振现象或能否在真空中传播来区分波的种类。
4. Wave Parameters and the Wave Equation | 波的基本参量与波速公式
Whether a wave is transverse or longitudinal, the same standard parameters describe it. The amplitude A is the maximum displacement of a particle from its equilibrium position. The wavelength λ is the distance between two adjacent points that are in phase, such as one crest to the next crest, or one compression to the next compression.
无论是横波还是纵波，都可以用相同的标准参量来描述。振幅 A 是粒子偏离平衡位置的最大位移。波长 λ 是两个相邻同相点之间的距离，例如相邻两个波峰之间，或者相邻两个密部之间的距离。
The frequency f is the number of complete oscillations made per second by a particle in the medium. The period T is the time taken for one complete oscillation, so:
频率 f 是介质中粒子每秒完成全振动的次数。周期 T 是完成一次全振动所需的时间，因此：
T = 1/f
The wave speed v is related to frequency and wavelength by the fundamental wave equation:
波速 v 与频率和波长之间的关系由基本波速公式给出：
v = fλ
source. However, its speed increases, so its wavelength must increase to keep v = fλ true.
该公式对横波和纵波都适用。例如，当声波从空气进入水中时，频率不变，因为频率由波源决定。但波速会增大，所以波长也会增大，从而满足 v = fλ。
For a sinusoidal progressive wave, the transverse displacement y at position x and time t can be described by:
对于正弦前进波，位于位置 x、时刻 t 的位移 y 可以表示为：
y = A sin(ωt − kx)
e mathematical form can also describe the displacement of particles in a longitudinal wave if y is interpreted as the displacement along the direction of propagation.
其中 ω = 2πf 是角频率，k = 2π/λ 是波数。如果将 y 理解为沿传播方向的位移，那么同一个数学形式也可以描述纵波中粒子的位移。
5. Displacement-Time and Displacement-Distance Graphs | 位移-时间图与位移-距离图
In exam questions, you must be able to interpret two different types of graphs. A displacement-distance graph is like a photograph of the wave at one instant. It shows how particle displacement varies with distance x along the wave.
在考试中，你必须能够理解两类不同的图像。位移-距离图就像某一瞬间对波拍摄的照片，它显示在某时刻粒子位移沿距离 x 的变化情况。
A displacement-time graph, on the other hand, shows how the displacement of one particular particle changes as the wave passes through it. The horizontal axis is time t, and the graph gives the period T directly from one cycle.
而位移-时间图显示的是某个特定粒子在波通过时，其位移随时间 t 的变化情况。横轴是时间 t，从图中的一个完整周期可以直接读出周期 T。
On a displacement-distance graph, the distance between adjacent crests equals one wavelength λ.
在位移-距离图中，相邻波峰之间的距离等于一个波长 λ。
On a displacement-time graph, the time between adjacent crests equals one period T.
在位移-时间图中，相邻波峰之间的时间等于一个周期 T。
A sinusoidal displacement-distance graph does not tell you alone whether the wave is transverse or longitudinal. A longitudinal wave can be plotted as particle displacement against distance, and it will also appear sinusoidal.
单靠正弦形的位移-距离图，不能判断波是横波还是纵波。纵波也可以用粒子位移对距离作图，其图像同样呈正弦形。
For longitudinal waves, pressure-distance graphs are also useful. A displacement maximum in the longitudinal direction corresponds to a velocity node or pressure extremum, depending on the convention used. Always read the question carefully to see whether the graph represents displacement or pressure.
对于纵波，压强-距离图也非常有用。在纵波传播方向上，位移的极大值可能对应压强极值，具体取决于题目所采用的约定。务必仔细阅读题目，确认图中表示的是位移还是压强。
6. Wave Speed and the Properties of the Medium | 波速与介质性质
The speed of a wave is determined by the medium or the restoring mechanism through which the wave travels, not by the frequency of the source. This is a common source of confusion.
波速由介质本身或波在传播过程中所受的恢复机制决定，而不是由波源的频率决定。这是常见的易混淆点。
For a wave on a stretched string, the speed depends on the tension T and the mass per unit length μ of the string:
对于绷紧绳上的波，波速取决于绳的张力 T 和绳的线密度 μ：
v = √T/μ
For sound in a gas, the speed increases with temperature and depends on the gas pressure and density. Sound travels faster in liquids and solids than in gases because the particles are more tightly coupled.
对于气体中的声波，波速随温度升高而增大，并且取决于气体的压强和密度。声波在液体和固体中比在气体中传播更快，因为粒子之间的耦合更强。
nsparent material, the speed is reduced, and the refractive index n is defined by n = c/v.
对于电磁波，其在真空中的速度是 c = 3.00 × 10⁸ m s⁻¹。在透明材料中波速会减小，折射率 n 定义为 n = c/v。
per second. Since v changes, the wavelength must also change. This is expressed by v = fλ.
当波从一种介质进入另一种介质时，其频率保持不变，因为每秒产生多少次完整振动由波源决定。由于波速 v 改变，波长 λ 也必然改变。这一点由 v = fλ 表示。
7. Polarisation: A Distinguishing Feature | 偏振：区分波型的重要特征
Polarisation is one of the strongest pieces of evidence that a wave is transverse. An unpolarised transverse wave vibrates in many different planes perpendicular to the direction of propagation. A polarising filter allows only the components vibrating in a particular plane to pass through.
偏振是证明波为横波的最有力证据之一。未偏振的横波在垂直于传播方向的许多不同平面内振动。偏振片只允许在某一特定平面内振动的分量通过。
If a beam of light is passed through a polarising filter, the transmitted intensity is reduced. When a second polarising filter is placed after the first with its axis at 90°, no light is transmitted. This occurs only because light is a transverse wave.
当一束光通过偏振片时，透射强度会减小。当第二个偏振片放在第一个偏振片之后，且其透振方向与第一个成 90° 时，没有光能透射过去。这种现象之所以会发生，正是因为光是横波。
The same experiment cannot be performed with a sound wave. Because particles in a sound wave vibrate along the direction of propagation, there is no plane of vibration to restrict. Therefore, longitudinal waves cannot be polarised.
同样的实验无法用声波完成。因为声波中的粒子沿传播方向振动，没有可供限制的振动平面。因此，纵波不能发生偏振。
In exam questions, polarisation is often used to demonstrate that electromagnetic waves are transverse. It also explains why glare from reflective surfaces can be reduced using polarised sunglasses.
在考试中，偏振常被用来证明电磁波是横波。它也可以解释为什么使用偏振太阳镜可以减少反射面产生的眩光。
8. Worked Examples | 典型例题分析
Example 1 | 例题 1: A sound wave in air has a frequency of 2.00 kHz and a wavelength of 0.170 m. Calculate the speed of sound and the period of the wave.
例题 1：空气中的声波频率为 2.00 kHz，波长为 0.170 m。计算声速和波的周期。
Using v = fλ:
利用 v = fλ：
v = fλ = 2.00 × 10³ × 0.170 = 340 m s⁻¹
The period is calculated from T = 1/f:
周期通过 T = 1/f 计算：
T = 1 / (2.00 × 10³) = 5.00 × 10⁻⁴ s
This classic result for the speed of sound in air under typical conditions is worth remembering.
这个结果是在典型条件下空气中的声速，值得记住。
Example 2 | 例题 2: A wave on a string has frequency 25 Hz and speed 60 m s⁻¹. Determine the wavelength.
例题 2：绳波的频率为 25 Hz，波速为 60 m s⁻¹。求波长。
Rearranging v = fλ gives:
将 v = fλ 变形可得：
λ = v/f = 60 / 25 = 2.4 m
Notice that the wave type does not affect the calculation: the same formula works for transverse and longitudinal waves.
注意，波的类型不影响该计算：同一个公式对横波和纵波都成立。
Example 3 | 例题 3: On a displacement-distance graph, adjacent crests are separated by 1.5 m. A particle takes 0.50 s to complete one oscillation. Calculate the wave speed.
例题 3：在位移-距离图中，相邻波峰之间的距离为 1.5 m。某粒子完成一次全振动需要 0.50 s。求波速。
The wavelength is λ = 1.5 m and the period is T = 0.50 s. Since v = λ/T:
波长为 λ = 1.5 m，周期为 T = 0.50 s。因为 v = λ/T：
v = 1.5 / 0.50 = 3.0 m s⁻¹
ernatively, you could calculate f = 1/T = 2.0 Hz, and then v = fλ = 2.0 × 1.5 = 3.0 m s⁻¹.
也可以先计算 f = 1/T = 2.0 Hz，再由 v = fλ = 2.0 × 1.5 = 3.0 m s⁻¹ 得到相同结果。
9. Common Misconceptions and Exam Tips | 常见误区与应试要点
The following misconceptions appear frequently in A Level wave questions. Avoid them to improve your accuracy and speed.
以下误区在 A Level 波类题目中经常出现。避免这些错误可以更好地提高准确率和做题速度。
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