📚 Waves: IB & CCEA Science Exam Essentials | IB 与 CCEA 科学：波 考点精讲
Waves are a fundamental topic in both IB and CCEA science curricula, encompassing a wide range of concepts from the basic properties of waves to the behaviour of light and sound. Mastering this topic requires a clear understanding of wave types, key equations such as v = fλ, and phenomena like interference, diffraction, and the Doppler effect. This article breaks down every essential point you need to know, pairing English explanations with Chinese translations to support bilingual learners, and concludes with targeted exam tips.
波是 IB 和 CCEA 科学课程中的基础课题，涵盖从波的基本性质到光和声音行为的广泛概念。掌握这一专题需要清晰理解波形类型、关键公式（如 v = fλ）以及干涉、衍射和多普勒效应等现象。本文拆解每一个必备知识点，采用英文与中文对照讲解，帮助双语学习者巩固理解，并以针对性的考试技巧收尾。
1. Introduction to Waves | 波导论
A wave is a disturbance that transfers energy from one location to another without the net movement of matter. Waves can be classified into two main types: mechanical waves, which require a material medium to travel (e.g., sound waves, water waves, seismic waves), and electromagnetic waves, which can propagate through a vacuum (e.g., light, radio waves, X-rays).
波是一种将能量从一个位置传递到另一个位置而不引起物质净移动的扰动。波可以分为两大类：机械波（需要物质介质才能传播，例如声波、水波、地震波）和电磁波（可在真空中传播，例如光、无线电波、X射线）。
All waves exhibit characteristic behaviours, including reflection, refraction, diffraction, and interference. The energy carried by a wave depends on its amplitude and frequency, making the study of wave properties essential for understanding everything from musical instruments to modern telecommunications.
所有波都表现出反射、折射、衍射和干涉等特征行为。波所携带的能量取决于其振幅和频率，因此研究波的特性对于理解从乐器到现代电信的各种现象至关重要。
2. Transverse and Longitudinal Waves | 横波与纵波
In a transverse wave, the oscillation of particles or fields is perpendicular to the direction of energy transfer. Examples include all electromagnetic waves, waves on a string, and S-waves (secondary seismic waves). The highest points are called crests and the lowest points are called troughs.
在横波中，粒子或场的振动方向与能量传递方向垂直。例子包括所有电磁波、弦上的波以及S波（次生地震波）。最高点称为波峰，最低点称为波谷。
In a longitudinal wave, the oscillation is parallel to the direction of energy transfer. Sound waves in air and P-waves (primary seismic waves) are longitudinal. These waves consist of compressions (regions of high pressure) and rarefactions (regions of low pressure).
在纵波中，振动方向与能量传递方向平行。空气中的声波和P波（原生地震波）都是纵波。这类波由压缩区（高压区域）和稀疏区（低压区域）组成。
Some waves, such as water surface waves, exhibit a combination of transverse and longitudinal motion, but for most exam specifications, the focus is on pure transverse and longitudinal models. It is critical to remember that only transverse waves can be polarised.
有些波（例如水面波）表现出横波与纵波的复合运动，但在大多数考试大纲中，重点放在纯横波和纯纵波模型上。必须记住，只有横波才能被偏振。
3. Describing Waves: Amplitude, Wavelength, Frequency and Period | 描述波：振幅、波长、频率与周期
The displacement–distance graph of a wave shows the amplitude (A) as the maximum displacement from the equilibrium position, measured in metres. The wavelength (λ) is the distance between two consecutive points in phase, such as crest to crest or compression to compression, also measured in metres.
波的位移–距离图显示振幅（A）是离开平衡位置的最大位移，单位为米。波长（λ）是相邻两个同相点之间的距离，例如波峰到波峰或压缩区到压缩区，单位也是米。
On a displacement–time graph for a single point, the period (T) is the time taken for one complete oscillation, measured in seconds. The frequency (f) is the number of complete oscillations per second, measured in hertz (Hz). These quantities are related by f = 1 / T.
在单个质点的位移–时间图上，周期（T）是完成一次完整振动所需的时间，单位为秒。频率（f）是每秒钟完整振动的次数，单位为赫兹（Hz）。它们之间的关系为 f = 1 / T。
The wave speed (v) is the distance travelled by a wavefront per unit time and is usually given in m s⁻¹. For a periodic wave, speed, frequency and wavelength are connected by the wave equation.
波速（v）是波前在单位时间内传播的距离，常用单位是米每秒（m s⁻¹）。对于周期性波，波速、频率和波长通过波速方程联系起来。
4. The Wave Equation: v = fλ | 波速方程
The fundamental relationship for all waves is the wave equation:
所有波的基本关系式是波速方程：
v = f × λ
where v is wave speed in m s⁻¹, f is frequency in Hz, and λ is wavelength in m. This equation applies to mechanical waves and electromagnetic waves alike. For electromagnetic waves in a vacuum, the speed v is replaced by c, the speed of light in a vacuum, which is approximately 3.00 × 10⁸ m s⁻¹.
其中 v 是波速（m s⁻¹），f 是频率（Hz），λ 是波长（m）。这个方程既适用于机械波，也适用于电磁波。对于真空中的电磁波，波速 v 用 c 代替，即真空中的光速，约为 3.00 × 10⁸ m s⁻¹。
If any two of the three variables are known, the third can be calculated. In exam questions, always ensure units are consistent and convert wavelength to metres if given in centimetres or nanometres.
如果已知三个量中的任意两个，就可以求出第三个。在考试题中，务必保持单位一致，如果给出的波长是厘米或纳米，应转化为米。
5. Reflection and Refraction | 反射与折射
When waves encounter a boundary between two media, some of the energy is reflected and some may be transmitted. The law of reflection states that the angle of incidence (θᵢ) equals the angle of reflection (θᵣ), both measured from the normal to the surface. This is observed with light, sound and water waves.
当波遇到两种介质间的边界时，一部分能量被反射，一部分可能透射。反射定律指出入射角（θᵢ）等于反射角（θᵣ），两者都从法线量起。这适用于光波、声波和水波。
Refraction occurs when a wave passes from one medium into another, causing a change in speed and, if the incidence is oblique, a change in direction. The degree of refraction is governed by Snell’s law:
当波从一种介质进入另一种介质时会发生折射，导致速度改变，若入射是斜的，方向也会改变。折射程度由斯涅尔定律决定：
n₁ sin θ₁ = n₂ sin θ₂
where n is the refractive index of the medium, and θ is the angle between the ray and the normal. The refractive index n of a medium is defined as the ratio of the speed of light in vacuum to the speed in the medium: n = c / v.
其中 n 是介质的折射率，θ 是光线与法线的夹角。介质的折射率 n 定义为真空中光速与该介质中光速之比：n = c / v。
When light travels from a denser medium to a less dense medium, total internal reflection can occur if the angle of incidence exceeds the critical angle (θ꜀). The critical angle is given by sin θ꜀ = n₂ / n₁ (for n₁ > n₂). This principle is applied in optical fibres and prisms.
当光从光密介质射向光疏介质时，如果入射角超过临界角（θ꜀），就会发生全内反射。临界角满足 sin θ꜀ = n₂ / n₁（n₁ > n₂）。该原理应用于光纤和棱镜中。
6. Diffraction and Interference | 衍射与干涉
Diffraction is the spreading of waves when they pass through a gap or around an obstacle. The amount of diffraction increases as the gap size approaches the wavelength of the wave. This explains why sound waves with longer wavelengths diffract more noticeably around corners than light waves.
衍射是波通过狭缝或绕过障碍物时发生的扩展现象。当狭缝宽度接近波的波长时，衍射程度增加。这就解释了为什么波长较长的声波比光波更容易绕墙角传播。
Interference occurs when two or more coherent waves overlap. Coherent waves have a constant phase difference and the same frequency. The superposition results in regions of constructive interference (waves in phase, amplitude increases) and destructive interference (waves out of phase, amplitude decreases or cancels).
当两个或多个相干波重叠时会发生干涉。相干波具有恒定的相位差和相同的频率。叠加会产生相长干涉（波同相，振幅增加）和相消干涉（波反相，振幅减小或抵消）的区域。
Young’s double-slit experiment demonstrates interference of light. A fringe pattern is formed on a screen, with bright fringes where constructive interference occurs and dark fringes for destructive interference. The fringe spacing Δy is given by:
杨氏双缝实验演示了光的干涉。屏幕上形成条纹图案，亮条纹对应相长干涉，暗条纹对应相消干涉。条纹间距 Δy 由下式给出：
Δy = λ D / d
where D is the distance from the slits to the screen, and d is the separation between the slits. For a diffraction grating with many slits, the maxima occur at angles given by:
其中 D 是双缝到屏幕的距离，d 是双缝间距。对于具有许多狭缝的衍射光栅，主极大的方向满足：
d sin θ = n λ
where n is the order number (n = 0, 1, 2, …). This equation can be used to measure the wavelength of light very accurately.
其中 n 是级次（n = 0, 1, 2, …）。该方程可用于非常精确地测量光的波长。
7. Polarisation | 偏振
Polarisation is a phenomenon exclusive to transverse waves. In unpolarised light, the electric field oscillates in all possible planes perpendicular to the direction of propagation. When light is polarised, the oscillations are restricted to a single plane.
偏振是横波独有的现象。在非偏振光中，电场在所有垂直于传播方向的平面上振动。当光被偏振后，振动被限制在一个平面上。
Polarisation can be achieved by passing unpolarised light through a polarising filter. According to Malus’s law, if completely polarised light of intensity I₀ passes through a second polariser (analyser) at an angle θ to the plane of polarisation, the transmitted intensity is I = I₀ cos²θ. This provides evidence for the transverse nature of light and is widely applied in sunglasses, LCD screens and photography.
偏振可以通过让非偏振光通过偏振片来实现。根据马吕斯定律，如果完全偏振光强度为 I₀，通过第二个偏振片（检偏器），其偏振化方向与入射偏振光的透振方向成 θ 角，则透射光强为 I = I₀ cos²θ。这为光的横波本质提供了证据，并广泛应用于太阳镜、液晶显示屏和摄影中。
Sound waves cannot be polarised because they are longitudinal. This distinction is a frequent exam question.
声波不能被偏振，因为它们是纵波。这一区别是常见的考试题。
8. Electromagnetic Waves | 电磁波
Electromagnetic (EM) waves consist of oscillating electric and magnetic fields at right angles to each other and to the direction of propagation. They form a continuous spectrum ordered by wavelength or frequency. In order of decreasing wavelength (increasing frequency), the main regions are: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays.
电磁波由相互垂直且与传播方向垂直的振荡电场和磁场组成。它们构成一个按波长或频率排序的连续谱。按照波长递减（频率递增）的顺序，主要区域是：无线电波、微波、红外线、可见光、紫外线、X射线和伽马射线。
All EM waves travel at the speed of light c in a vacuum (c = 3.00 × 10⁸ m s⁻¹) and satisfy the wave equation c = f λ. Different regions have distinct properties and uses: radio waves for communication, microwaves for cooking and radar, infrared for thermal imaging, visible light for sight, ultraviolet for sterilisation and fluorescence, X-rays for medical imaging, and gamma rays for cancer treatment and sterilisation.
所有电磁波在真空中都以光速 c（c = 3.00 × 10⁸ m s⁻¹）传播，并满足波速方程 c = f λ。不同区域具有不同的性质和用途：无线电波用于通信，微波用于烹饪和雷达，红外线用于热成像，可见光用于视觉，紫外线用于杀菌和荧光，X射线用于医学成像，伽马射线用于癌症治疗和消毒。
It is important to memorise that the energy of an EM photon is directly proportional to its frequency: E = h f, where h is Planck’s constant. Thus, gamma rays have the highest energy per photon, and radio waves the lowest.
重要的是要记住，电磁波光子的能量与其频率成正比：E = h f，其中 h 是普朗克常数。因此，伽马射线具有最高的单光子能量，无线电波具有最低的。
9. Sound Waves | 声波
Sound waves are longitudinal mechanical waves that require a medium to travel. The speed of sound depends on the medium and its temperature. In air at room temperature, the speed of sound is approximately 340 m s⁻¹. The relationship between speed, frequency and wavelength still holds: v = f λ.
声波是需要介质传播的纵波机械波。声速取决于介质及其温度。在室温下的空气中，声速约为 340 m s⁻¹。速度、频率和波长之间的关系依然成立：v = f λ。
The human ear can typically detect frequencies between 20 Hz and 20 kHz. Frequencies above this range are called ultrasound, which have many applications such as medical scanning (sonography), sonar, and industrial cleaning.
人耳通常能听到 20 Hz 到 20 kHz 的频率。高于此范围的频率称为超声波，有许多应用，如医学扫描（超声成像）、声纳和工业清洗。
Sound waves undergo reflection (echoes), refraction (due to temperature gradients), diffraction (around obstacles) and interference. Standing waves in air columns produce resonant frequencies that form the basis of musical instruments. For a pipe open at both ends, the harmonic frequencies are fₙ = n v / (2L); for a pipe closed at one end, fₙ = n v / (4L) where n is an odd integer.
声波会发生反射（回声）、折射（由于温度梯度）、衍射（绕过障碍物）和干涉。空气柱中的驻波产生共振频率，构成了乐器发声的基础。对于两端开口的管，谐频为 fₙ = n v / (2L)；对于一端封闭的管，fₙ = n v / (4L)，其中 n 为奇数。
10. Standing Waves | 驻波
A standing wave is formed when two identical waves travelling in opposite directions superpose. This occurs when a wave is reflected back along its path, as in a string fixed at both ends or in an air column. The pattern consists of nodes (points of zero displacement) and antinodes (points of maximum displacement).
当两列完全相同的波沿相反方向传播并叠加时，会形成驻波。这发生在波沿原路径反射回来时，例如在两端固定的弦中或在空气柱内。驻波图案由波节（位移为零的点）和波腹（位移最大的点）组成。
The distance between two adjacent nodes or two adjacent antinodes is half a wavelength (λ/2). For a string fixed at both ends, the fundamental frequency (first harmonic) has a node at each end and an antinode in the centre: L = λ/2. The nth harmonic has wavelength λₙ = 2L / n and frequency fₙ = n f₁.
相邻两波节或相邻两波腹之间的距离是半个波长（λ/2）。对于两端固定的弦，基频（第一谐波）两端为波节，中心为波腹：L = λ/2。第 n 次谐波波长为 λₙ = 2L / n，频率为 fₙ = n f₁。
In a closed pipe (one end closed), only odd harmonics are present because a displacement node must exist at the closed end and an antinode at the open end. In an open pipe, both ends are antinodes, and all integer harmonics are possible. These principles are often tested with calculations of fundamental frequency and the effect of changing length or tension.
在一端封闭的管（闭管）中，由于封闭端必须是位移波节，开口端是波腹，因此只能产生奇次谐波。在一端开口的管（开管）中，两端都是波腹，所有整数次谐波都可能出现。这些原理常伴随计算基频以及改变长度或张力的影响而进行考查。
11. The Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency due to relative motion between a wave source and an observer. When the source moves towards the observer, the wavefronts are compressed, leading to a higher observed frequency (blueshift for light, higher pitch for sound). When the source moves away, the observed frequency is lower (redshift, lower pitch).
多普勒效应是由于波源与观察者之间的相对运动而引起的观测频率变化。当波源朝向观察者运动时，波前被压缩，导致观察到的频率升高（光表现为蓝移，声音为音调升高）。当波源远离时，观测频率降低（红移，音调降低）。
For sound, when the source moves at speed vₛ and the observer is stationary, the observed frequency f’ is:
对于声波，当波源以速度 vₛ 运动而观察者静止时，观测频率 f’ 为：
f’ = f × v / (v ± vₛ)
Use the minus sign when the source moves towards the observer and the plus sign when it moves away. If the observer moves relative to a stationary source, the formula becomes f’ = f × (v ± vₒ) / v, with appropriate signs. In the general case, both motions can be combined.
当波源朝向观察者运动时用减号，远离时用加号。如果观察者相对于静止波源运动，公式变为 f’ = f × (v ± vₒ) / v，符号取法相应而定。一般情况可以合并两种运动。
The Doppler effect provides crucial evidence for the expansion of the universe (cosmological redshift), and is used in speed cameras, weather radar, and echocardiography. For electromagnetic waves, the relativistic formula is required at high speeds, but for most exam contexts the approximate low-speed shift Δf / f ≈ v/c is sufficient.
多普勒效应为宇宙膨胀（宇宙学红移）提供了关键证据，并应用于测速摄像头、气象雷达和超声心动图。对于电磁波，在高速情况下需要使用相对论公式，但在大多数考试情境中，低速近似公式 Δf / f ≈ v/c 就足够了。
12. Exam Tips and Common Pitfalls | 考试技巧与常见误区
When drawing wave diagrams, always label the amplitude and wavelength clearly, and indicate whether you are plotting displacement–distance or displacement–time. A common mistake is confusing these two graphs, especially when determining period and wavelength.
在画波图时，始终清楚地标出振幅和波长，并注明你画的是位移–距离图还是位移–时间图。常见的错误是混淆这两种图，尤其是在确定周期和波长时。
For calculations, always convert units to metres, seconds and hertz unless specified otherwise. Remember that 1 kHz = 10³ Hz and 1 nm = 10⁻⁹ m. Use the wave equation v = fλ as your starting point for numerical problems; rearranging correctly is a core skill.
在计算时，除非另有说明，始终将单位转换为米、秒和赫兹。记住 1 kHz = 10³ Hz，1 nm = 10⁻⁹ m。以波速方程 v = fλ 作为数值题的起点；正确变换公式是核心技能。
Be prepared to explain why only transverse waves can be polarised, and to describe experimental evidence for the wave nature of light and sound. Standard demonstrations such as the ripple tank for water waves, the double-slit for light, and the polarisation of microwaves are frequently examined.
要准备好解释为什么只有横波能被偏振，并描述证明光和声音具有波动性的实验证据。标准演示实验，如水波的波纹槽、光的双缝实验以及微波的偏振实验，常被考查。
Finally, when dealing with standing waves, correctly identify nodes and antinodes. Do not assume that the number of nodes equals the harmonic number; for a string fixed at both ends, the fundamental has two nodes (at ends) and one antinode. Practice sketching the first three harmonics for both strings and air columns.
最后，在处理驻波时，正确识别波节和波腹。不要认为波节数等于谐波次数；对于两端固定的弦，基频有两个波节（在两端）和一个波腹。练习绘制弦和空气柱的前三个谐振模式。
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