Waves Revision Notes | 波考点精讲

📚 Waves Revision Notes | 波考点精讲

Waves are one of the most central topics in both IB and CIE Physics, appearing in questions about oscillations, optics, sound and electromagnetism. This revision guide walks you through every key concept, equation and exam tip you need to master wave behaviour, from basic definitions to standing waves and the Doppler effect.

波是 IB 和 CIE 物理中最核心的主题之一，出现在振动、光学、声学和电磁学的考题中。这份考点精讲将带你逐一掌握波的基本定义、关键公式和解题技巧，涵盖从基础概念到驻波与多普勒效应的全部内容。

1. What is a Wave? | 什么是波？

A wave is a disturbance that transfers energy from one point to another without the net transfer of matter. In mechanical waves, particles oscillate about fixed positions, passing energy to neighbouring particles. Electromagnetic waves do not require a medium and can travel through a vacuum.

波是一种将能量从一点传递到另一点的扰动，而物质本身不发生净位移。在机械波中，粒子围绕固定位置振动，将能量传递给相邻粒子。电磁波不需要介质，可以在真空中传播。

Waves can be classified by the direction of particle oscillation relative to the direction of energy propagation: transverse waves (oscillation perpendicular to propagation, e.g. light, water ripples) and longitudinal waves (oscillation parallel to propagation, e.g. sound, seismic P-waves).

波可以根据粒子振动方向与能量传播方向的关系分类：横波（振动方向垂直于传播方向，如光、水波涟漪）和纵波（振动方向平行于传播方向，如声波、地震纵波）。

2. Transverse and Longitudinal Waves | 横波与纵波

In a transverse wave, particles move up and down while the wave travels horizontally. Crests and troughs are clearly visible. Displacement is perpendicular to propagation. This property allows polarisation – a key distinction from longitudinal waves.

在横波中，粒子上下运动，而波水平传播。波峰和波谷清晰可见，位移与传播方向垂直。这一特性使得横波可以发生偏振，这是与纵波的重要区别。

In a longitudinal wave, particles oscillate back and forth along the direction of propagation, creating compressions (high pressure) and rarefactions (low pressure). Sound waves in air are longitudinal. They cannot be polarised.

在纵波中，粒子沿传播方向前后振动，形成压缩区（高压）和稀疏区（低压）。空气中的声波是纵波，它们不能发生偏振。

Exam tip: IB and CIE often ask you to identify wave type from a diagram or to explain why sound cannot be polarised. Always link it to oscillation direction.

考试提示：IB 和 CIE 常要求根据示意图判断波的类型，或解释为什么声波不能偏振。回答时务必联系到振动方向。

3. Wave Parameters | 波的基本参数

Displacement (y): distance of a particle from its equilibrium position at a given instant. Amplitude (A): maximum displacement from equilibrium. Measured in metres.

位移 (y) ：某时刻粒子相对于平衡位置的距离。振幅 (A) ：最大位移，单位米。

Wavelength (λ): shortest distance between two points in phase, e.g. crest to crest. Frequency (f): number of complete oscillations per second, unit hertz (Hz). Period (T): time for one complete oscillation, T = 1/f.

波长 (λ) ：两个同相点之间的最短距离，如波峰到波峰。频率 (f) ：每秒完整振动的次数，单位赫兹 (Hz)。周期 (T) ：一次完整振动所需时间， T = 1/f 。

Wave speed (v): the speed at which energy is transmitted. Not the speed of the particles. For all waves, the fundamental relationship is:

波速 (v) ：能量传递的速度，不是粒子的速度。对所有波，基本关系式为：

v = f λ

4. The Wave Equation and Intensity | 波速公式与强度

The equation v = f λ links wave speed, frequency and wavelength. When a wave enters a new medium, its speed and wavelength change, but its frequency remains constant because the source determines frequency.

公式 v = f λ 将波速、频率和波长联系起来。当波进入新介质时，波速和波长会改变，但频率保持不变，因为频率由波源决定。

Intensity (I) is the power per unit area (W m⁻²). For a point source emitting spherical waves, intensity follows the inverse square law: I ∝ 1/r². Doubling the distance quarters the intensity. Amplitude is proportional to the square root of intensity, A ∝ √I.

强度 (I) 指单位面积的功率（ W m⁻²）。对于点源发出的球面波，强度遵循平方反比定律： I ∝ 1/r² 。距离加倍，强度变为四分之一。振幅与强度的平方根成正比， A ∝ √I 。

CIE particularly tests the relationship I ∝ A². In IB, this often appears in data-based questions about sound or light.

CIE 特别考查 I ∝ A² 的关系。在 IB 中，这常出现在关于声音或光的实验数据题。

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

Phase describes the stage an oscillation has reached within a complete cycle, often expressed in radians or degrees. Two points on a wave are in phase if they have the same displacement and velocity; they are separated by a whole number of wavelengths, Δx = nλ.

相位描述振动在一个完整周期中所处的阶段，通常用弧度或角度表示。波上两点如果位移和速度相同则为同相，它们相距波长的整数倍， Δx = nλ 。

Points are in antiphase (exactly out of phase) when one displacement is opposite the other, separated by (n + ½)λ. Phase difference in radians is given by:

当两点的位移正好相反时，它们反相，相距 (n + ½)λ 。相位差（弧度）为：

Δφ = 2π Δx / λ

This concept is vital for understanding interference and standing waves. IB questions often ask you to state phase difference between two marked points on a drawn wave.

这一概念对理解干涉和驻波至关重要。IB 常常要求你在绘制的波形图上标出两点之间的相位差。

6. Reflection, Refraction and Snell’s Law | 反射、折射与斯涅尔定律

When a wave hits a boundary between two media, part of it is reflected and part transmitted. The law of reflection states that the angle of incidence equals the angle of reflection, measured from the normal.

波遇到两种介质的界面时，一部分被反射，一部分透射。反射定律指出入射角等于反射角，均从法线测量。

Refraction occurs because the wave speed changes. If it slows down, the wave bends towards the normal. Snell’s Law quantifies this:

折射发生的原因是波速改变。若波速减小，波向法线方向偏折。斯涅尔定律给出定量关系：

n₁ sin θ₁ = n₂ sin θ₂

where n = c/v is the absolute refractive index. Critical angle and total internal reflection occur when light travels from a denser to a less dense medium at angles greater than the critical angle: sin θ_c = n₂ / n₁ (n₁ > n₂).

其中 n = c/v 为绝对折射率。当光从光密介质射向光疏介质且入射角大于临界角时，发生全内反射： sin θ_c = n₂ / n₁ （ n₁ > n₂ ）。

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 pronounced when the size of the gap is comparable to the wavelength. If the gap is much larger than λ, spreading is minimal.

衍射是波穿过狭缝或绕过障碍物时发生的展宽现象。当狭缝尺寸与波长相近时，衍射效果最明显。如果狭缝远大于波长 λ ，展宽程度很小。

For a single slit, the central maximum is twice as wide as the subsidiary maxima. The condition for minima in single-slit diffraction is a sin θ = nλ, where a is the slit width. This formula is required in both IB and CIE.

对于单缝衍射，中央明纹的宽度是次级明纹的两倍。单缝衍射极小值的条件是 a sin θ = nλ ，其中 a 为缝宽。IB 和 CIE 都要求掌握这一公式。

Diffraction is essential for understanding resolution limits and the operation of diffraction gratings. A typical exam question might ask why you can hear sound around a corner but not see light – because sound wavelengths are much larger.

衍射对理解分辨率极限和衍射光栅的工作原理至关重要。常见考题会问为什么声音可以绕过墙角而光不能——因为声波的波长要长得多。

8. Interference and Superposition | 干涉与叠加

The principle of superposition states that when two waves meet, the resultant displacement at a point is the vector sum of the individual displacements. Constructive interference occurs when waves are in phase (path difference = nλ), giving a larger amplitude. Destructive interference occurs when they are in antiphase (path difference = (n + ½)λ), giving a smaller or zero amplitude.

叠加原理指出，两列波相遇时，某点的合位移是各波位移的矢量和。同相时发生相长干涉（路程差 = nλ ），振幅增大；反相时发生相消干涉（路程差 = (n + ½)λ ），振幅减小或为零。

For two coherent sources producing an interference pattern on a screen, the fringe spacing is given by:

对于两个相干光源在屏幕上产生的干涉图样，条纹间距为：

Δy = λ D / d

where D is the distance from slits to screen, d is slit separation. This is a classic Young’s double-slit experiment equation. Coherent sources have constant phase difference and the same frequency.

其中 D 是双缝到屏幕的距离， d 是缝间距。这是经典的杨氏双缝干涉公式。相干光源具有恒定的相位差和相同的频率。

9. Standing Waves | 驻波

Standing waves form when two identical waves travelling in opposite directions superpose. They are characterised by nodes (points of zero displacement) and antinodes (points of maximum amplitude). Energy is stored, not transferred along the medium.

驻波由两列相同但反向传播的波叠加形成。其特点是存在波节（位移为零的点）和波腹（振幅最大的点）。能量被储存在介质中，而不是沿线传播。

For a string fixed at both ends, the standing wave can only exist if the length L satisfies L = n λ/2. The nth harmonic has n antinodes. Frequencies are given by f_n = n v / (2L). For a pipe open at both ends, the same condition applies; for a pipe closed at one end, L = (2n-1) λ/4, and only odd harmonics are present.

对于两端固定的弦，驻波存在的条件是弦长 L 满足 L = n λ/2 。第 n 次谐波有 n 个波腹。频率为 f_n = n v / (2L) 。对于两端开口的管，条件相同；一端闭口的管则满足 L = (2n-1) λ/4 ，且只出现奇次谐波。

IB and CIE both require you to sketch standing wave patterns and label nodes/antinodes, and to calculate frequencies for strings and air columns.

IB 和 CIE 都要求你绘制驻波图形并标出波节和波腹，以及计算弦和气柱的频率。

10. Doppler Effect | 多普勒效应

The Doppler effect is the change in observed frequency when a wave source moves relative to an observer. If the source and observer approach each other, the observed frequency is higher (blue shift for light); if they recede, it is lower (red shift).

多普勒效应是波源与观察者有相对运动时，观测频率发生变化的现象。若两者相互靠近，观测频率升高（光的蓝移）；相互远离，则频率降低（红移）。

For sound, the general formula is: f’ = f (v ± vₒ) / (v ∓ v_s), where v is the speed of sound, vₒ is observer speed and v_s is source speed. Use the upper signs when they approach, lower signs when receding. In IB, only the move-source/stationary-observer or stationary-source/move-observer cases are examined; CIE may use the full formula.

对声波，通用公式为： f’ = f (v ± vₒ) / (v ∓ v_s) ，其中 v 是声速， vₒ 是观察者速度， v_s 是波源速度。相互靠近时用上面的符号，远离时用下面的符号。IB 只考查波源运动观察者静止或波源静止观察者运动的情况；CIE 可能使用完整公式。

For electromagnetic waves, the relativistic Doppler formula is used. A key application: red shift of galaxies gives evidence for the expanding Universe.

对于电磁波，使用相对论性多普勒公式。一个重要应用是：星系的红移为宇宙膨胀提供了证据。

11. Polarisation | 偏振

Polarisation is the restriction of wave oscillations to a single plane. Only transverse waves can be polarised. Unpolarised light has oscillations in all planes perpendicular to propagation; after passing through a polariser, vibrations occur in one plane only.

偏振是将波的振动限制在一个平面内。只有横波才可以偏振。非偏振光在与传播方向垂直的所有平面上都有振动；通过偏振片后，振动只在一个平面内发生。

Malus’s law states that when completely linearly polarised light passes through a second polariser (analyser), the transmitted intensity is I = I₀ cos² θ, where θ is the angle between the transmission axes. This is examined in both IB and CIE.

马吕斯定律指出，当完全线偏振光通过第二个偏振片（检偏器）时，透射强度为 I = I₀ cos² θ ，其中 θ 是两个透射轴之间的夹角。IB 和 CIE 都会考查。

Applications include polaroid sunglasses, stress analysis, and LCD displays. In the lab, microwaves can be polarised by a metal grille; this is a classic demonstration that microwaves are transverse.

应用包括偏光太阳镜、应力分析和液晶显示器。在实验中，微波可以通过金属栅格偏振，这是证明微波是横波的经典演示实验。

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Waves Revision Notes | 波 考点精讲