IB OCR Science: Waves Key Points | IB OCR 科学：波考点精讲

📚 IB OCR Science: Waves Key Points | IB OCR 科学：波考点精讲

Waves are fundamental to understanding how energy propagates through space and matter, linking topics across physics, from sound and light to quantum mechanics. This revision guide distils the essential concepts you need to master for IB and OCR examinations, explaining each point in clear, concise language with paired English and Chinese explanations. Whether you are reviewing transverse and longitudinal waves, interference patterns, or the Doppler effect, these structured notes will reinforce your understanding and prepare you for exam-style questions.

波是理解能量如何在空间和物质中传播的基础，它将物理学的多个主题联系起来，从声音和光一直到量子力学。这份复习指南提炼了你掌握 IB 和 OCR 考试所需的核心概念，用清晰简洁的语言逐一解释，并配以中英文双语对照。无论你是在复习横波与纵波、干涉图样还是多普勒效应，这些结构化的笔记都将巩固你的理解，帮助你应对考试题型。

1. What is a Wave? | 什么是波？

A wave is a disturbance that transfers energy from one location to another without any net movement of matter. The particles of the medium oscillate about fixed positions, but the energy travels onward. Waves are generated by a vibrating source and carry information about the source’s frequency and amplitude.

波是一种扰动，它将能量从一个地方传递到另一个地方，而物质本身不发生净移动。介质的粒子围绕固定位置振动，但能量向前传播。波由振源产生，并携带着关于振源频率和振幅的信息。

In IB and OCR contexts, you must distinguish between mechanical waves, which require a medium (e.g. sound, water waves), and electromagnetic waves, which can travel through a vacuum (e.g. light, X-rays). Pulse waves are single disturbances, while continuous waves repeat at regular intervals.

在 IB 和 OCR 的语境中，你必须区分需要介质的机械波（如声波、水波）和可以在真空中传播的电磁波（如光、X 射线）。脉冲波是单一的扰动，而连续波则以固定的间隔重复。

2. Transverse and Longitudinal Waves | 横波与纵波

In a transverse wave, the particle displacement is perpendicular to the direction of energy propagation. Ripples on a string, water surface waves, and all electromagnetic waves are transverse. The wave profile shows crests (peaks) and troughs (valleys).

在横波中，粒子的振动方向与能量的传播方向垂直。弦上的涟漪、水表面波和所有电磁波都是横波。波形显示出波峰（最高点）和波谷（最低点）。

In a longitudinal wave, particle displacement is parallel to the direction of energy transfer. Sound waves in air are a key example: regions of compression (higher pressure) and rarefaction (lower pressure) move through the medium. Longitudinal waves cannot be polarised, a property unique to transverse waves.

在纵波中，粒子的振动方向与能量传递方向平行。空气中的声波就是一个典型例子：压缩区（高压）和稀疏区（低压）在介质中移动。纵波不能发生偏振，这是横波独有的特性。

3. Wave Properties: Amplitude, Wavelength, Frequency | 波的性质：振幅、波长、频率

Amplitude (A) is the maximum displacement of a particle from its equilibrium position, measured in metres for mechanical waves. It determines the energy carried by a wave: for a mechanical wave, energy is proportional to the square of the amplitude. In a sound wave, amplitude relates to loudness; in light, to brightness or intensity.

振幅（A）是粒子离开平衡位置的最大位移，对于机械波以米为单位。它决定了波携带的能量：对机械波而言，能量与振幅的平方成正比。在声波中，振幅与响度有关；在光波中，则与亮度或强度有关。

Wavelength (λ) is the distance between two consecutive points in phase, such as two adjacent crests or compressions. It is measured in metres. Frequency (f) is the number of complete oscillations passing a point per second, measured in hertz (Hz). The period (T) is the time for one full oscillation, and f = 1/T.

波长（λ）是相邻两个同相点之间的距离，比如两个相邻的波峰或压缩区。它以来计量单位。频率（f）是每秒通过某点的完整振动次数，单位为赫兹（Hz）。周期（T）是一次完整振动所需的时间，并且 f = 1/T。

Displacement-distance graphs show the waveform at an instant, enabling wavelength measurement. Displacement-time graphs track the motion of a single particle, revealing period and amplitude. Exam questions often test your ability to interpret these two representations.

位移-距离图显示某一时刻的波形，可用于测量波长。位移-时间图追踪单个粒子的运动，可以读出周期和振幅。考试题经常考查你解读这两种图示的能力。

4. Wave Speed Equation | 波速方程

The relationship between wave speed (v), frequency (f) and wavelength (λ) is one of the most used equations in wave physics. It is expressed as:

波速（v）、频率（f）和波长（λ）之间的关系是波动物理学中最常用的方程之一。它表示为：

v = f λ

This equation holds for all types of waves. When a wave passes from one medium to another, its frequency remains constant because it is determined by the source. However, the speed and wavelength change, leading to refraction. In a given medium, wave speed depends on the properties of that medium, e.g. tension and mass per unit length for a string.

该方程对所有类型的波都成立。当波从一种介质进入另一种介质时，其频率保持不变，因为频率由波源决定。但波速和波长会发生变化，从而导致折射。在给定的介质中，波速取决于介质的性质，例如弦上的张力和单位长度的质量。

For electromagnetic waves in a vacuum, v = c (3.00 × 10⁸ m s⁻¹). Always remember to convert frequency to Hz and wavelength to metres when applying the formula. A typical calculation: if a wave has frequency 50 Hz and wavelength 2 m, then v = 50 × 2 = 100 m s⁻¹.

对于真空中的电磁波，v = c（3.00 × 10⁸ m s⁻¹）。应用公式时，要记得将频率换算成赫兹、波长换算成米。典型计算：若某波的频率为 50 Hz、波长为 2 m，则 v = 50 × 2 = 100 m s⁻¹。

5. Phase and Phase Difference | 相位与相位差

Phase describes the position of a point within a wave cycle, usually expressed in radians or degrees. Two points on a wave are in phase if they move in the same direction with the same displacement at the same instant. If the separation is a whole number of wavelengths, they are in phase; if it is an odd half-wavelength multiple, they are in antiphase (180° or π radians out of phase).

相位描述一个点在波周期内的位置，通常以弧度或度数表示。如果波上两点在同一瞬间运动方向相同且位移相同，则称它们同相。如果它们的间距为整数倍波长，则同相；如果为半波长的奇数倍，则为反相（相位差 180° 或 π 弧度）。

The phase difference Δφ between two points separated by Δx is given by:

相距 Δx 的两点之间的相位差 Δφ 可通过下式计算：

Δφ = 2π (Δx / λ) or Δφ = (2π / λ) × path difference

Phase difference is central to understanding interference patterns. Constructive interference occurs when waves arrive in phase (Δφ = 0, 2π, 4π…), while destructive interference requires antiphase arrival (Δφ = π, 3π…). IB and OCR exam papers often ask you to deduce phase relationships from wavefront diagrams or to calculate phase difference from a given path difference.

相位差是理解干涉图样的关键。当波同相到达时（Δφ = 0, 2π, 4π…），发生相长干涉；当波反相到达时（Δφ = π, 3π…），发生相消干涉。IB 和 OCR 的试题经常要求你根据波前图推断相位关系，或根据给定的路程差计算相位差。

6. Wave Phenomena: Reflection and Refraction | 波的现象：反射与折射

Reflection occurs when a wave strikes a boundary and bounces back. The law of reflection states that the angle of incidence equals the angle of reflection, measured from the normal. This can be demonstrated with a ripple tank. The wavelength and speed remain unchanged in the same medium.

反射发生在波遇到边界并反弹回来的时候。反射定律指出，入射角等于反射角，两者均从法线测量。这可以用水波槽来演示。在同一介质中，波长和波速保持不变。

Refraction is the change in direction of a wave as it passes from one medium into another with a different wave speed. When light enters a denser medium, such as from air into glass, it slows down and bends towards the normal. Snell’s law quantifies this behaviour:

折射是波从一种介质进入另一种波速不同的介质时方向发生改变的现象。当光进入更密的介质，例如从空气射入玻璃，它的速度减慢并向法线靠拢。斯涅尔定律定量描述了这一行为：

n₁ sinθ₁ = n₂ sinθ₂

Here n is the refractive index of each medium, and θ is the angle to the normal. The refractive index also relates wave speeds: n₁/n₂ = v₂/v₁. Remember that frequency does not change during refraction; only wavelength and speed adjust. This explains why a stick appears bent in water.

其中 n 是各介质的折射率，θ 是与法线的夹角。折射率还与波速相关：n₁/n₂ = v₂/v₁。请记住，折射过程中频率不变；只有波长和波速会发生改变。这就解释了为什么水中的棍子看起来是弯的。

7. Diffraction | 衍射

Diffraction is the spreading of a wave as it passes through a gap or around an obstacle. The effect is most significant when the gap size or obstacle is comparable to the wavelength. If the gap is much larger than the wavelength, diffraction is negligible and the wave passes through without notable spreading.

衍射是指波通过狭缝或绕过障碍物时发生扩散的现象。当狭缝或障碍物的尺寸与波长相当时，衍射效应最为显著。如果缝宽远大于波长，衍射可以忽略不计，波通过时不会有明显的扩散。

For a single slit, the diffraction pattern consists of a bright central maximum and progressively dimmer, narrower fringes on either side. The angular width of the central maximum is related to wavelength and slit width. Diffraction limits the resolving power of optical instruments, a key topic in IB HL and OCR options. Diffraction also enables waves to reach regions behind obstacles, which is why you can hear sounds around corners even though the source is not directly in view.

对于单缝衍射，图样由一个明亮的中央极大和两侧渐暗、渐窄的条纹组成。中央极大的角宽度与波长和缝宽有关。衍射限制了光学仪器的分辨能力，这是 IB HL 和 OCR 选修中的关键主题。衍射也使波能够到达障碍物后面的区域，这就是为什么你能听到拐角处的声音，尽管声源不在视线内。

8. Interference and Superposition | 干涉与叠加

The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This applies to all wave types and leads to interference patterns when coherent sources are used.

叠加原理指出，当两个或多个波在某点相遇时，合位移等于各个位移的矢量和。这适用于所有类型的波，并在使用相干源时产生干涉图样。

Coherent sources emit waves with a constant phase relationship and the same frequency. For constructive interference, the path difference must be a whole number of wavelengths (nλ). For destructive interference, the path difference is (n + ½)λ. These conditions produce alternating bright and dark fringes in a double-slit experiment.

相干源发出的波具有恒定的相位关系和相同的频率。要产生相长干涉，路程差必须是波长的整数倍（nλ）。要产生相消干涉，路程差为 (n + ½)λ。这些条件在双缝实验中产生明暗相间的条纹。

Young’s double-slit formula gives the fringe spacing Δy on a screen:

杨氏双缝公式给出了屏幕上条纹间距 Δy：

Δy = λD / d

where D is the slit-to-screen distance and d is the slit separation. This experiment provides a method to measure the wavelength of light. In sound, interference leads to variations in loudness; in microwaves, it creates standing wave patterns detectable with a probe.

其中 D 是双缝到屏幕的距离，d 是双缝间距。该实验提供了一种测量光波长的方法。在声音中，干涉导致响度变化；在微波中，干涉产生可用探针检测的驻波图样。

9. Standing Waves | 驻波

A standing wave is formed when two identical waves travelling in opposite directions superpose. This often occurs through reflection at a boundary. The resulting pattern has points of zero displacement called nodes, and points of maximum displacement called antinodes. Energy is stored, not transferred, in a standing wave system.

驻波由两列相同但传播方向相反的波叠加而成。这通常通过边界反射产生。形成的图样具有位移为零的点（称为波节）和位移最大的点（称为波腹）。在驻波系统中，能量被储存而非传播。

In a string fixed at both ends, the allowed wavelengths for standing waves are λₙ = 2L/n, where n = 1, 2, 3… The corresponding frequencies are fₙ = nv/(2L). The fundamental frequency corresponds to n = 1. These are called harmonics. For a pipe open at both ends, the harmonics follow the same pattern, while for a pipe closed at one end, only odd harmonics exist (n = 1, 3, 5…).

在两端固定的弦上，允许的驻波波长为 λₙ = 2L/n，其中 n = 1, 2, 3…。对应的频率为 fₙ = nv/(2L)。基频对应 n = 1。这些称为谐波。对于两端开放的管，谐波遵循相同的模式；而对于一端封闭的管，仅存在奇次谐波（n = 1, 3, 5…）。

Measuring the distance between adjacent nodes (λ/2) allows you to determine the wavelength of the wave. Standing waves explain the resonant frequencies of musical instruments and are crucial in microwave and laser technologies.

测量相邻波节之间的距离（λ/2）可以确定波的波长。驻波解释了乐器的共振频率，并且在微波和激光技术中至关重要。

10. The Doppler Effect | 多普勒效应

The Doppler effect is the change in observed frequency when there is relative motion between a wave source and an observer. When the source and observer move towards each other, the observed frequency is higher (blueshift for light, higher pitch for sound). When they move apart, the frequency is lower (redshift, lower pitch).

多普勒效应是指当波源和观察者之间存在相对运动时，观测到的频率发生改变的现象。当波源和观察者相互靠近时，观测频率变高（光波蓝移，声波音调变高）；当相互远离时，频率变低（红移，音调变低）。

For sound waves, the observed frequency f’ is given by:

对于声波，观测频率 f’ 由下式给出：

f’ = f (v ± vₒ) / (v ∓ vₛ)

where v is the speed of sound in the medium, vₒ is the observer’s velocity, and vₛ is the source’s velocity, with signs chosen according to direction. This formula must be applied carefully: both numerator and denominator affect the shift. In IB and OCR, you may need to explain the Doppler effect in contexts like redshift of galaxies (evidence for expanding universe) or radar speed guns.

其中 v 是声波在介质中的速度，vₒ 是观察者的速度，vₛ 是波源的速度，符号根据运动方向选取。这个公式应用时必须谨慎：分子和分母都会影响频移。在 IB 和 OCR 考试中，你可能需要解释多普勒效应在星系红移（宇宙膨胀的证据）或雷达测速枪等情境中的应用。

11. Polarisation | 偏振

Polarisation is a property exclusive to transverse waves: it restricts the oscillations of the wave to a single plane. An unpolarised wave has vibrations in many planes perpendicular to the direction of propagation. A polarising filter transmits only the component of the wave parallel to its transmission axis.

偏振是横波独有的特性：它将波的振动限制在单一平面内。非偏振波在与传播方向垂直的各个平面内振动。偏振滤光片只允许平行于其透振轴的分量通过。

If two polarisers are placed with their axes at an angle θ, the intensity of transmitted light follows Malus’s Law: I = I₀ cos²θ. Polarisation provides evidence that light is a transverse wave. Longitudinal waves cannot be polarised, which is why sound does not exhibit this effect. Applications include Polaroid sunglasses (reduce glare from reflected light), LCD screens, and stress analysis in materials.

如果两个偏振片的透振轴成 θ 角放置，透射光强遵循马吕斯定律：I = I₀ cos²θ。偏振为光是横波提供了证据。纵波不能发生偏振，因此声音没有这种效应。应用包括偏振太阳镜（减少反射眩光）、LCD 屏幕和材料的应力分析。

12. Electromagnetic Waves | 电磁波

Electromagnetic (EM) waves are transverse oscillations of electric and magnetic fields that propagate through a vacuum at the speed of light c = 3.00 × 10⁸ m s⁻¹. They are generated by accelerating charges and do not require a medium. The EM spectrum, in order of increasing frequency and decreasing wavelength, is: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

电磁波是电场和磁场的横向振荡，在真空中以光速 c = 3.00 × 10⁸ m s⁻¹ 传播。它们由加速电荷产生，不需要介质。电磁波谱按频率递增、波长递减的顺序为：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 share the same fundamental nature but differ in how they interact with matter due to their varying photon energies (E = hf). The visible spectrum ranges from red (≈700 nm) to violet (≈400 nm). In IB and OCR, you might be asked to recall properties, typical wavelengths, uses, and hazards of different regions. For instance, microwaves cause heating through molecular resonance, ultraviolet can ionise atoms and damage DNA, and X-rays have high penetrating power for medical imaging.

所有电磁波具有相同的本质，但由于光子能量不同（E = hf），它们与物质的相互作用也不同。可见光谱从红色（约 700 nm）到紫色（约 400 nm）。在 IB 和 OCR 中，你可能会被要求回忆不同波段的性质、典型波长、用途和危害。例如，微波通过分子共振产生热效应，紫外线能使原子电离并损伤 DNA，X 射线具有强穿透力，用于医学成像。

Region	Wavelength Range	Key Application
Radio	> 10 cm	Communications
Microwaves	1 mm – 10 cm	Radar, cooking
Infrared	700 nm – 1 mm	Thermal imaging
Visible	400 – 700 nm	Vision, lasers
Ultraviolet	10 – 400 nm	Sterilisation
X-rays	0.01 – 10 nm	Medical imaging
Gamma	< 0.01 nm	Cancer therapy

Using the wave equation c = f λ, you can calculate frequency or wavelength for any EM wave. Remember that higher frequency means higher photon energy, which determines ionising capability. These ideas integrate wave properties with quantum phenomena, bridging your understanding for topics like the photoelectric effect.

利用波动方程 c = f λ，你可以计算任何电磁波的频率或波长。记住，频率越高，光子能量越大，这决定了电离能力。这些概念将波动性质与量子现象结合起来，为你理解光电效应等主题架起了桥梁。

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IB OCR Science: Waves Key Points | IB OCR 科学：波 考点精讲