📚 IB Physics: Waves – Key Concepts and Exam Tips | IB 物理：波 考点精讲
Waves form a core part of the IB Physics syllabus, bridging fundamental concepts with more advanced applications such as interference, standing waves and the Doppler effect. Whether you are tackling Standard Level or Higher Level, mastering wave behaviour is essential for top marks on both Paper 1 and Paper 2. This guide unpacks the key ideas, essential equations and common pitfalls to help you revise efficiently.
波是 IB 物理教学大纲的核心内容，它将基本概念与干涉、驻波和多普勒效应等更深入的应用联系起来。无论你参加的是标准级还是高级考试，掌握波的行为对于在卷一和卷二中取得高分都至关重要。本指南梳理了核心概念、必备方程和常见易错点，帮助你高效复习。
1. Wave Properties | 波的基本性质
A wave is a propagating disturbance that transfers energy without permanently displacing the medium. Key descriptors include amplitude (A), the maximum displacement from equilibrium, and wavelength (λ), the distance between two successive points in phase. The time for one complete oscillation is the period (T), and the frequency (f) counts oscillations per second.
波是一种传播的扰动，它传递能量而不引起介质的永久位移。关键描述量包括振幅 (A)，即离开平衡位置的最大位移；波长 (λ)，即两个相继同相点之间的距离。完成一次完整振动的时间为周期 (T)，频率 (f) 则指每秒振动的次数。
The relationship between period and frequency is reciprocal, and wave speed is linked to wavelength and frequency. Graphs of displacement against distance (snapshot) or against time (history) for a point allow you to extract A, λ and T directly. Remember that the speed of a mechanical wave depends only on the medium, not on frequency or amplitude.
周期与频率互为倒数，波速则与波长、频率相联系。位移–距离图（快照）或某一点的位移–时间图（历史）可以让你直接提取 A、λ 和 T。记住，机械波的速度仅取决于介质，与频率或振幅无关。
f = 1 / T v = f λ
2. Transverse and Longitudinal Waves | 横波与纵波
In a transverse wave, particles oscillate perpendicular to the direction of energy transfer; electromagnetic waves and waves on strings are typical examples. In a longitudinal wave, particles oscillate parallel to the propagation direction, as in sound waves. Only transverse waves can be polarised, a property exploited in IB question scenarios to distinguish wave types.
在横波中，质点的振动方向与能量传递方向垂直；电磁波和绳上的波是典型的例子。在纵波中，质点沿传播方向平行振动，例如声波。只有横波才能被偏振，这一性质在 IB 考题中常被用来区分波的类型。
For any wave, displacement against position graphs help visualise compressions and rarefactions in longitudinal waves by mapping pressure or density variations. The wavelength remains the distance between successive compressions or crests.
对于任何波，位移–位置图可通过映射压强或密度变化来直观显示纵波中的疏部和密部。波长仍为相继密部或波峰之间的距离。
3. The Wave Equation and Wave Speed | 波速方程与波速
The equation v = fλ is central to wave calculations. When a wave passes from one medium to another, its speed and wavelength change, but the frequency stays fixed because it is set by the source. For a string under tension, the speed depends on tension and linear density; for sound in air, on temperature.
方程 v = fλ 是波计算的核心。当波从一种介质进入另一种介质时，波速和波长发生变化，但频率保持不变，因为它由波源决定。对于张紧的弦，波速取决于张力和线密度；对于空气中的声波，取决于温度。
v = f λ and v = √(T/μ) for a stretched string
An IB problem might ask you to compute wavelength given speed and frequency, or deduce how the wavelength alters when a water wave enters shallower water. Always identify which quantity remains constant before applying the formula.
IB 考题可能会要求你根据波速和频率计算波长，或推断水波进入浅水区时波长如何变化。应用公式前，请先确定哪个量保持不变。
4. Wavefronts, Rays and Huygens’ Principle | 波前、射线与惠更斯原理
A wavefront joins points in phase, such as crests, and rays show the direction of energy flow, perpendicular to wavefronts. Huygens’ principle explains propagation, reflection and refraction by treating every point on a wavefront as a source of secondary spherical wavelets. The new wavefront is the envelope of these wavelets.
波前连接同相位的点，如波峰，射线则表示能量流动方向，与波前垂直。惠更斯原理把波前上的每一点都视为次级球面子波的波源，从而解释传播、反射和折射。新的波前是这些子波的包络面。
Understanding wavefronts is particularly useful when drawing refraction diagrams: the change in spacing of wavefronts indicates the change in speed and wavelength. These sketches often appear in IB examinations as part of qualitative questions.
理解波前在绘制折射图时特别有用：波前间距的改变表明波速和波长的变化。这类示意图经常出现在 IB 考试中，作为定性分析题的一部分。
5. Reflection, Refraction and Diffraction | 反射、折射与衍射
Waves obey the law of reflection: angle of incidence equals angle of reflection. Refraction occurs when waves cross a boundary into a medium of different wave speed, changing direction unless incidence is normal. Snell’s law quantifies this.
波遵守反射定律：入射角等于反射角。当波穿过边界进入波速不同的介质时发生折射，除非垂直入射，否则传播方向会改变。斯涅尔定律给出定量描述。
n₁ sinθ₁ = n₂ sinθ₂ and sinθc = n₂ / n₁ (n₁ > n₂)
Diffraction is the spreading of a wave when it passes through a gap or around an obstacle. It is most pronounced when the aperture size is comparable to the wavelength. This fundamental behaviour underpins single-slit patterns and the resolution of optical instruments.
衍射是波通过缝隙或绕过障碍物时发生的展宽现象。当缝隙尺寸与波长相近时，衍射最为显著。这一基本行为是单缝图样和光学仪器分辨率的基础。
6. Superposition and Interference | 叠加与干涉
When two or more waves meet, the net displacement is the vector sum of individual displacements – the principle of superposition. Constructive interference occurs when waves are in phase, giving maximum amplitude; destructive interference occurs when they are out of phase by 180° (π rad), leading to cancellation.
当两个或多个波相遇时，合位移是各波位移的矢量和——这就是叠加原理。当波同相时，发生相长干涉，振幅最大；当波反相（相差 180° 或 π rad）时，发生相消干涉，导致抵消。
For sustained interference patterns, sources must be coherent – that is, they maintain a constant phase difference and have the same frequency. The condition for constructive interference is a path difference Δx = nλ; for destructive interference it is Δx = (n + ½)λ, where n is an integer.
要获得稳定的干涉图样，波源必须是相干的——即保持恒定的相位差且频率相同。相长干涉的条件是波程差 Δx = nλ；相消干涉的条件是 Δx = (n + ½)λ，其中 n 为整数。
7. Young’s Double-Slit Experiment | 杨氏双缝实验
Young’s historic experiment demonstrates light interference using two narrow slits illuminated by a single coherent source. Bright fringes appear where path difference is an integer multiple of λ, dark fringes where it is a half-integer multiple. The fringe separation Δy is given by a simple geometric formula.
杨氏经典实验利用两个狭缝，由单一相干光源照明，展示了光的干涉。在波程差为 λ 的整数倍处出现亮纹，在半整数倍处出现暗纹。条纹间距 Δy 由一个简洁的几何公式给出。
Δy = λD / d
Here D is the distance from the slits to the screen and d is the slit separation. This equation is frequently tested, so pay attention to unit conversions and the fact that Δy is inversely proportional to d. It also allows determination of wavelength for unknown light.
其中 D 是双缝到屏幕的距离，d 是缝间距。这一方程式常被考查，注意单位转换以及 Δy 与 d 成反比。该公式还可用于测定未知光的波长。
8. Single-Slit Diffraction and Resolution | 单缝衍射与分辨率
When monochromatic light passes through a single narrow slit of width a, a central bright maximum dominates, flanked by dimmer fringes. Minima occur at angles given by a sinθ = nλ (n = 1, 2, 3…). The width of the central maximum is a measure of the spreading due to diffraction.
当单色光通过宽度为 a 的单狭缝时，形成一个占主导的中央亮纹，两侧是较暗的条纹。暗纹的位置满足 a sinθ = nλ (n = 1, 2, 3…)。中央亮纹的宽度反映了衍射引起的展宽程度。
a sinθ = nλ
Resolution of two point sources is limited by diffraction. The Rayleigh criterion states that two sources are just resolved when the first minimum of one coincides with the peak of the other. For a circular aperture of diameter b, the minimum resolvable angle is θ = 1.22λ / b; for a slit it is approximately λ / a.
两点光源的分辨率受到衍射的限制。瑞利判据指出，当一个源的第一暗纹恰好与另一个源的中央峰重合时，两个像刚好能被分辨。对于直径为 b 的圆孔，最小可分辨角为 θ = 1.22λ / b；对于单缝则约为 λ / a。
9. Standing Waves | 驻波
Standing waves form when two identical travelling waves move in opposite directions and superpose. They exhibit nodes (zero displacement) and antinodes (maximum displacement). Common examples are waves on a plucked string or in an air column inside a pipe.
当两列相同的行波沿相反方向传播并叠加时，就形成驻波。驻波显示出波节（位移为零）和波腹（位移最大）。常见例子如拨动的弦上的波，或管内气柱中的波。
For a string fixed at both ends, the allowed wavelengths are λ = 2L / n (n = 1,2,3…), giving frequencies f = n (v / 2L). In an open pipe, identical harmonic series apply; in a closed pipe, only odd harmonics exist: f = n (v / 4L) with n = 1,3,5…
对于两端固定的弦，允许的波长为 λ = 2L / n (n = 1,2,3…)，频率为 f = n (v / 2L)。在开管中，谐波系列相同；在闭管中，仅存在奇次谐波：f = n (v / 4L)，其中 n = 1,3,5…
fₙ = n × (v / 2L) (both ends fixed or open)
fₙ = n × (v / 4L) (one end closed, n odd)
10. Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency when a source and observer move relative to each other. For sound, the observed frequency f’ increases when source and observer approach and decreases when they recede. The general formula involves the speeds of source vₛ and observer vₒ.
多普勒效应是指当波源和观察者相对运动时，观测频率发生变化的现象。对于声波，当波源与观察者相互靠近时，观测频率 f’ 升高，远离时降低。一般公式涉及波源速度 vₛ 和观察者速度 vₒ。
f’ = f × (v ± vₒ) / (v ∓ vₛ)
For electromagnetic waves, the relativistic Doppler shift is different, but IB only requires a qualitative appreciation: light from receding galaxies is redshifted (longer λ), while approaching objects produce blueshift. The Doppler effect is used in speed cameras, echocardiography and astronomy.
对于电磁波，相对论多普勒频移有所不同，但 IB 只要求定性理解：来自退行星系的光发生红移（λ 变长），而靠近的物体产生蓝移。多普勒效应被应用于测速摄像机、心脏超声检查和天文学。
11. Polarisation | 偏振
Polarisation is a phenomenon exclusive to transverse waves, in which oscillations are restricted to a single plane. Unpolarised light can be polarised by a filter (Polaroid). Malus’s law describes the intensity transmitted through a second polariser at angle θ to the first.
偏振是横波独有的现象，指振动被限制在一个平面内。非偏振光可以通过偏振片（Polaroid）变成偏振光。马吕斯定律描述了透过与第一偏振片夹角为 θ 的第二偏振片后的光强。
I = I₀ cos²θ
Polarisation by reflection occurs at Brewster’s angle, where reflected and refracted rays are perpendicular. The condition is tanθB = n₂ / n₁. This concept often appears in HL optics questions alongside Malus’s law.
反射偏振发生在布儒斯特角，此时反射光线和折射光线互相垂直。条件为 tanθB = n₂ / n₁。这一概念在 HL 光学考题中常与马吕斯定律一同出现。
12. Electromagnetic Spectrum | 电磁波谱
All electromagnetic waves travel at speed c = 3.00 × 10⁸ m s⁻¹ in a vacuum and consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. The spectrum, in order of decreasing wavelength, includes radio, microwave, infrared, visible, ultraviolet, X-ray and gamma radiation.
所有电磁波在真空中以速度 c = 3.00 × 10⁸ m s⁻¹ 传播，并由相互垂直且垂直于传播方向的振荡电场和磁场构成。波谱按波长递减的顺序包括无线电波、微波、红外线、可见光、紫外线、X 射线和伽马射线。
| Region | Approximate Wavelength | Key Feature |
|---|---|---|
| Radio | > 0.1 m | Communication, MRI |
| Microwave | 1 mm – 0.1 m | Radar, heating |
| Infrared | 700 nm – 1 mm | Thermal imaging |
| Visible | 400 – 700 nm | Human vision |
| Ultraviolet | 10 – 400 nm | Sunburn, sterilisation |
| X-ray | 0.01 – 10 nm | 更多咨询请联系16621398022（同微信）
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