IB Physics: Waves Key Concepts | IB 物理：波考点精讲

📚 IB Physics: Waves Key Concepts | IB 物理：波考点精讲

Waves are fundamental to understanding energy transfer without net matter transport. In IB Physics, the waves topic bridges mechanical oscillations, light, and sound, demanding both qualitative insight and quantitative skills. This revision guide distils the core ideas, from simple harmonic origins through interference and the Doppler effect, up to resolution and polarization.

波是理解能量传递而不伴随净物质输运的基础。在 IB 物理中，波这一主题连接了机械振动、光和声，要求既有定性洞察又具备定量技能。这篇复习指南凝练了核心思想，从简谐运动的起源到干涉、多普勒效应，再到分辨率与偏振。

1. Wave Properties and the Wave Equation | 波的性质与波动方程

A wave transfers energy through a medium or vacuum by means of a disturbance. Its key descriptors are displacement, amplitude, wavelength (λ), frequency (f), period (T), and wave speed (v). The period is the reciprocal of frequency: T = 1/f. The fundamental relationship linking these quantities is the wave equation.

波通过扰动在介质或真空中传递能量。它的关键描述量是位移、振幅、波长 (λ)、频率 (f)、周期 (T) 和波速 (v)。周期是频率的倒数：T = 1/f。连接这些量的基本关系就是波动方程。

v = fλ

This equation applies to all travelling waves. For a given medium, wave speed is constant, so frequency and wavelength are inversely proportional. When a wave moves from one medium to another, frequency remains unchanged while speed and wavelength adjust.

该方程适用于所有行波。在给定介质中，波速恒定，因此频率与波长成反比。当波从一种介质进入另一种介质时，频率保持不变，而波速和波长会改变。

2. Transverse and Longitudinal Waves | 横波与纵波

Waves are classified by the direction of particle oscillation relative to energy propagation. In transverse waves, particles vibrate perpendicular to the direction of energy travel; examples include light and water ripples. In longitudinal waves, particles oscillate parallel to the energy direction, as in sound waves.

波根据质点振动方向与能量传播方向的关系分类。在横波中，质点振动方向垂直于能量传播方向；例子包括光和水波涟漪。在纵波中，质点振动平行于能量传播方向，例如声波。

Longitudinal waves consist of compressions and rarefactions. The wavelength is the distance between two consecutive compressions (or rarefactions). Both transverse and longitudinal waves can be represented by displacement–distance and displacement–time graphs, where the amplitude marks the maximum displacement from equilibrium.

纵波由压缩和稀疏组成。波长是两个相邻压缩（或稀疏）之间的距离。横波和纵波都可以用位移–距离图和位移–时间图表示，其中振幅标志着离开平衡位置的最大位移。

3. Electromagnetic Spectrum | 电磁波谱

All electromagnetic (EM) waves are transverse, travel at speed c ≈ 3.00 × 10⁸ m s⁻¹ in vacuum, and do not require a medium. The spectrum, ordered by increasing frequency (and decreasing wavelength), runs: radio, microwave, infrared, visible light (ROYGBIV), ultraviolet, X-ray, gamma ray.

所有电磁波都是横波，在真空中以 c ≈ 3.00 × 10⁸ m s⁻¹ 的速度传播，且不需要介质。该谱按频率递增（波长递减）排列为：无线电波、微波、红外线、可见光（红橙黄绿蓝靛紫）、紫外线、X 射线、伽马射线。

Each region has characteristic properties and applications. For instance, microwaves are used in radar and cooking because they excite water molecules; X-rays penetrate soft tissue but are absorbed by bone, enabling medical imaging. The energy of an EM photon is E = hf, explaining why higher-frequency radiation is more ionising.

每个波段都有独特的性质和应用。例如，微波因能激发水分子而用于雷达和烹饪；X 射线能穿透软组织但被骨骼吸收，从而用于医学成像。电磁光子的能量 E = hf，这解释了为什么更高频率的辐射具有更强的电离能力。

4. Reflection, Refraction, and Snell’s Law | 反射、折射与斯涅耳定律

When a wave encounters a boundary, it can be reflected, refracted, or both. The law of reflection states that the angle of incidence equals the angle of reflection, measured from the normal.

当波遇到界面时，它可能被反射、折射或两者兼有。反射定律指出，入射角等于反射角，均从法线量起。

Refraction occurs when a wave passes from one medium into another with a different speed. The change in speed causes the wave to change direction unless it is incident along the normal. Snell’s law quantifies this:

当波从一种介质进入另一种波速不同的介质时会发生折射。速度的变化导致波改变方向，除非是沿法线入射。斯涅耳定律对此进行了量化：

n₁ sin θ₁ = n₂ sin θ₂

Here n is the refractive index, a ratio of the speed of light in vacuum to that in the medium: n = c/v. Optically denser media have higher n; light bends toward the normal when entering a denser medium. Total internal reflection happens when the angle of incidence exceeds the critical angle θc, given by sin θc = n₂/n₁ (where n₁ > n₂).

这里 n 是折射率，即真空中光速与介质中光速之比：n = c/v。光密介质的 n 值更高；光进入光密介质时会向法线偏折。当入射角超过临界角 θc（sin θc = n₂/n₁，其中 n₁ > n₂）时发生全内反射。

5. Diffraction and Interference | 衍射与干涉

Diffraction is the spreading of a wave around obstacles or through apertures. It is most noticeable when the aperture size is comparable to the wavelength. Light passing a narrow slit creates a diffraction pattern of bright and dark fringes.

衍射是波绕过障碍物或穿过孔径时发生的扩展现象。当孔径尺寸与波长相近时衍射最为显著。光通过狭缝会产生明暗条纹的衍射图样。

Interference occurs when two or more coherent waves overlap. Constructive interference yields a resultant amplitude equal to the sum of individual amplitudes (path difference = nλ). Destructive interference occurs when waves meet exactly out of phase (path difference = (n + ½)λ). Coherence requires a constant phase relationship and the same frequency.

当两列或更多相干波叠加时发生干涉。相长干涉产生的合振幅等于各振幅之和（波程差 = nλ）。当波恰好反相相遇时发生相消干涉（波程差 = (n + ½)λ）。相干性要求相位关系恒定且频率相同。

6. Young’s Double-Slit Experiment | 杨氏双缝实验

Young’s double-slit experiment provided the first definitive evidence for the wave nature of light. Monochromatic light passing through two narrow, closely spaced slits produces an interference pattern of equally spaced bright and dark fringes on a screen.

杨氏双缝实验为光的波动性提供了首个确凿证据。单色光通过两条狭窄、间距很小的缝，在屏幕上产生等间距的明暗干涉条纹。

The fringe spacing Δy is given by:

条纹间距 Δy 由下式给出：

Δy = λD / d

where λ is the wavelength, D the distance from slits to screen, and d the slit separation. This formula assumes D ≫ d and small angles. White light produces a central white fringe flanked by spectra because each colour component has a different fringe width.

其中 λ 为波长，D 为缝到屏幕的距离，d 为缝间距。此公式假设 D ≫ d 且角度很小。白光产生中央白色条纹，两侧伴有光谱，因为不同颜色的条纹宽度不同。

7. Standing Waves and Harmonics | 驻波与谐波

A standing (stationary) wave is formed by the superposition of two identical travelling waves moving in opposite directions. Unlike progressive waves, standing waves do not transfer net energy, and they exhibit nodes (zero displacement) and antinodes (maximum displacement).

驻波是由两列相同且反向传播的行波叠加形成的。与行波不同，驻波不传递净能量，并且表现出波节（位移为零）和波腹（位移最大）。

Standing waves arise in bounded systems such as strings and pipes. For a string fixed at both ends, the allowed wavelengths are λₙ = 2L/n where n = 1,2,3…, corresponding to the fundamental frequency f₁ = v/(2L) and its harmonics fₙ = n f₁. In a pipe open at both ends, the same harmonic series holds. For a pipe closed at one end, only odd harmonics exist: λₙ = 4L/n with n = 1,3,5… and fₙ = n v/(4L).

驻波出现在弦和管等有限系统中。对于两端固定的弦，允许的波长为 λₙ = 2L/n，其中 n = 1,2,3…，对应基频 f₁ = v/(2L) 及其谐波 fₙ = n f₁。在两端开口的管中，谐波序列相同。一端封闭的管则只存在奇次谐波：λₙ = 4L/n，n = 1,3,5…，且 fₙ = n v/(4L)。

8. Doppler Effect | 多普勒效应

The Doppler effect is the observed change in frequency (and wavelength) when a wave source and an observer move relative to each other. The general formula for sound, with a stationary medium, is:

多普勒效应是当波源与观察者相对运动时观测到的频率（和波长）变化。对于在静止介质中的声波，一般公式为：

f’ = f (v ± v₀) / (v ∓ vₛ)

where v is the wave speed, v₀ the observer’s speed (positive if moving toward source), and vₛ the source’s speed (positive if moving toward observer). The signs are chosen so that approaching motion raises the observed frequency (higher pitch) and receding lowers it.

其中 v 为波速，v₀ 为观察者速度（朝向波源运动为正），vₛ 为波源速度（朝向观察者运动为正）。符号选择应使接近运动提高观测频率（音调升高），远离运动降低频率。

For electromagnetic waves (light), relativistic Doppler shift applies, but the IB syllabus focuses on sound. The Doppler effect also underpins redshift of galaxies, supporting the expansion of the universe.

对于电磁波（光），需使用相对论性多普勒频移，但 IB 大纲侧重于声音。多普勒效应也是星系红移的基础，支撑了宇宙膨胀的证据。

9. Polarization | 偏振

Polarization is a phenomenon exclusive to transverse waves. It refers to the restriction of oscillations to a single plane. Unpolarized light has electric field vectors vibrating in all directions perpendicular to the propagation axis. A polarizer transmits only the component parallel to its transmission axis.

偏振是横波独有的现象。它指的是将振动限制在单一平面内。非偏振光的电场矢量在垂直于传播轴的所有方向上振动。偏振片只透过与其透振轴平行的分量。

According to Malus’s law, when completely plane-polarized light passes through a second polarizer (analyser) at angle θ, the transmitted intensity is:

根据马吕斯定律，当完全平面偏振光通过与其透振轴成 θ 角的第二个偏振片（检偏器）时，透射光强为：

I = I₀ cos² θ

Polarization provides evidence that light is a transverse wave. It is used in LCD screens, polarised sunglasses, and stress analysis in materials.

偏振证明了光是横波。它被用于液晶显示器、偏振太阳镜以及材料应力分析中。

10. Single-Slit Diffraction and Resolution | 单缝衍射与分辨率

When monochromatic light passes through a single slit of width a, it produces a central maximum flanked by dimmer secondary maxima. The angular position of the first minimum is given by:

当单色光通过宽度为 a 的单缝时，产生中央亮纹，两侧是较暗的次级亮纹。第一极小值的角位置由下式给出：

a sin θ = λ

For a circular aperture, diffraction limits the ability to resolve two point sources. The Rayleigh criterion states that two sources are just resolved when the central maximum of one coincides with the first minimum of the other. For a circular aperture of diameter D, the minimum resolvable angular separation is:

对于圆孔，衍射限制了分辨两个点光源的能力。瑞利判据指出，当一个像的中央极大与另一个像的第一极小重合时，恰好可分辨。对于直径为 D 的圆孔，最小可分辨角间距为：

θ ≈ 1.22 λ / D

This principle governs the resolution of telescopes, microscopes, and the human eye.

这一原理决定了望远镜、显微镜和人眼的分辨本领。

11. Intensity and Amplitude | 强度与振幅

Wave intensity I is the power transmitted per unit area perpendicular to the propagation direction. For a wave spreading uniformly from a point source, intensity obeys the inverse-square law: I ∝ 1/r², where r is distance from the source.

波的强度 I 是垂直于传播方向单位面积传递的功率。对于从点源均匀扩散的波，强度遵循平方反比定律：I ∝ 1/r²，其中 r 是到波源的距离。

Intensity is proportional to the square of the amplitude (I ∝ A²). This relationship links the energy carried by a wave to its displacement. When combining waves, understanding intensity is essential: constructive interference of two equal-amplitude waves can produce four times the intensity of a single wave.

强度与振幅的平方成正比 (I ∝ A²)。这一关系将波携带的能量与其位移联系起来。在波的合成中，理解强度至关重要：两列等幅波的相长干涉可产生四倍于单列波的强度。

12. Wavefronts and Rays | 波前与射线

A wavefront is a surface connecting adjacent points in phase, such as all the crests of a wave. Wavefronts propagate at the wave speed and are perpendicular to the direction of energy flow. Rays are lines drawn perpendicular to wavefronts, indicating the direction of wave travel.

波前是连接相邻同相位点的表面，例如波的所有波峰。波前以波速传播，并与能量流动方向垂直。射线是垂直于波前绘制的线，表示波传播的方向。

Huygens’ principle treats every point on a wavefront as a source of secondary spherical wavelets. The envelope of these wavelets constructs the new wavefront, explaining reflection, refraction, and diffraction. This model unifies many wave phenomena in a single geometric framework.

惠更斯原理将波前上的每一点都视为次级球面子波的波源。这些子波的包络面构成了新的波前，从而解释了反射、折射和衍射。该模型在一个统一的几何框架下解释了众多波动现象。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Physics: Waves Key Concepts | IB 物理：波 考点精讲