📚 A-Level OCR Physics: Waves Key Points Explained | A-Level OCR 物理：波 考点精讲
Waves are a cornerstone of the OCR A-Level Physics specification, appearing in Module 4 and underpinning topics from sound to quantum physics. A thorough grasp of wave terminology, behaviour, and the mathematical models that describe them is essential for both the written papers and practical endorsements. This revision guide unpacks every key concept—from fundamental wave properties and phase relationships to interference, stationary waves, polarisation, and the behaviour of light at slits and gratings.
波是 OCR A-Level 物理考纲（模块 4）的核心内容，也是声学、光学乃至量子物理的基础。透彻理解波的术语、行为及其数学描述，对笔试和实验考核都至关重要。这份考点精讲将逐一拆解关键概念——从基本的波特性和相位关系，到干涉、驻波、偏振以及光在狭缝和光栅中的行为。
1. Wave Fundamentals: Types and Properties | 波的基础：类型与特性
A wave transfers energy without transferring matter. In a transverse wave, such as light or a ripple on a string, the oscillations are perpendicular to the direction of energy travel. In a longitudinal wave, such as sound, the oscillations are parallel to the direction of energy travel, producing compressions and rarefactions.
波传递能量而不传递物质。在横波（如光或弦上的涟漪）中，振动方向与能量传播方向垂直。在纵波（如声波）中，振动方向与能量传播方向平行，形成疏密区域。
The displacement–distance graph shows a snapshot of the wave profile at one instant, allowing direct measurement of wavelength λ. The displacement–time graph shows the motion of a single particle over time, giving the period T. Frequency f = 1/T. The speed of a wave is related to frequency and wavelength by the wave equation.
位移-距离图是某一瞬间波形的快照，可直接测量波长 λ。位移-时间图显示单个质点随时间的变化，给出周期 T。频率 f = 1/T。波速与频率和波长的关系由波动方程描述。
v = f λ
Amplitude A is the maximum displacement from the equilibrium position. Phase describes the fraction of a cycle a wave has completed at a given point or instant, often expressed in radians or degrees (1 full cycle = 2π rad = 360°).
振幅 A 是质点偏离平衡位置的最大位移。相位描述波在某一位置或时刻完成了一个周期的几分之几，常用弧度或度表示（一个完整周期 = 2π rad = 360°）。
Key quantities, their symbols and standard units must be memorised: displacement/m, amplitude/m, wavelength/m, period/s, frequency/Hz, speed/ m s⁻¹, phase/rad.
必须熟记各物理量及其符号与单位：位移/m、振幅/m、波长/m、周期/s、频率/Hz、速度/ m s⁻¹、相位/rad。
2. Phase and Phase Difference | 相位与相位差
Phase difference Δφ between two points on a wave, or between two waves, measures the fraction of a cycle by which one leads or lags the other. It is calculated from the path difference Δx and wavelength.
相位差 Δφ 描述波上两点或两个波之间一个超前或滞后另一个的周期比例。它可由路径差 Δx 与波长计算。
Δφ = (2π × Δx) / λ
Waves are in phase when their phase difference is 0 or a multiple of 2π rad (0°, 360°, 720°…); they interfere constructively. Waves are in antiphase when Δφ = π, 3π, 5π … rad (180°, 540°…); they interfere destructively. Phase difference is a crucial concept for understanding superposition, interference and standing waves.
波同相时相位差为 0 或 2π rad 的整数倍（0°、360°、720°…），发生相长干涉。波反相时 Δφ = π、3π、5π … rad（180°、540°…），发生相消干涉。相位差是理解叠加、干涉和驻波的关键概念。
When analysing oscilloscope traces or displacement–time graphs, the time shift Δt can be converted to phase difference: Δφ = 2π (Δt / T). For a stationary wave, points between adjacent nodes vibrate in phase with each other but in antiphase with points in the next loop.
分析示波器波形或位移-时间图时，可将时间偏移 Δt 转换为相位差：Δφ = 2π (Δt / T)。在驻波中，相邻波节间的点同相振动，但与相邻波腹中的点反相。
3. Polarisation | 偏振
Polarisation is a property exclusive to transverse waves. An unpolarised wave vibrates in multiple planes perpendicular to the direction of propagation. A polarising filter transmits only the component of vibration along its transmission axis, producing a plane-polarised wave.
偏振是横波独有的特性。非偏振波在与传播方向垂直的多个平面内振动。偏振片只允许沿其透振轴方向的振动分量通过，从而产生平面偏振波。
For an ideal polariser, the transmitted intensity I of initially unpolarised light is halved: I = I₀/2. When two polarisers are crossed at an angle θ, Malus’s law applies to the second filter after the first has already produced plane-polarised light.
对于理想偏振片，初始非偏振光通过后强度减半：I = I₀/2。当两个偏振片以夹角 θ 放置时，马吕斯定律适用于已通过第一个偏振片的平面偏振光入射第二个偏振片。
I = I₀ cos² θ
Rotating the analyser varies the intensity from a maximum (θ = 0°) to zero (θ = 90°). This observation provides direct evidence that light is a transverse wave. Applications include glare-reducing sunglasses, liquid crystal displays (LCDs) and stress analysis using photoelasticity.
旋转检偏器会使光强从最大（θ = 0°）变为零（θ = 90°）。该现象是光为横波的直接证据。应用包括防眩太阳镜、液晶显示器（LCD）和利用光弹性进行应力分析。
Polarisation of microwaves can be demonstrated using a metal grid: when the grid wires are parallel to the E-field, the wave is reflected or absorbed; when perpendicular, it passes through. Sound, a longitudinal wave, cannot be polarised.
微波的偏振可用金属栅格演示：当栅格线平行于电场方向时，波被反射或吸收；垂直时则可透过。声波是纵波，无法被偏振。
4. Superposition and Interference | 叠加与干涉
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. Interference effects arise when coherent waves—waves with the same frequency and a constant phase difference—overlap.
叠加原理指出，当两个或多个波在一点相遇时，合位移等于各列波位移的矢量和。当相干波（频率相同且相位差恒定的波）重叠时，就会发生干涉现象。
Constructive interference occurs where the waves are in phase, giving a resultant amplitude equal to the sum of the individual amplitudes. Destructive interference occurs where the waves are in antiphase, producing a minimum resultant amplitude (complete cancellation if amplitudes are equal). The condition can be expressed in terms of path difference p.d.
相长干涉发生在波同相处，合振幅等于各振幅之和。相消干涉发生在波反相处，合振幅最小（若振幅相等则完全抵消）。条件可用路径差（p.d.）表示。
- Constructive / 相长: p.d. = nλ (n = 0, 1, 2…)
- Destructive / 相消: p.d. = (n + ½)λ (n = 0, 1, 2…)
Coherence can be achieved by using a single source split into two paths (e.g. Young’s slits) or by synchronised signal generators for sound/microwaves. Two separate sources are seldom coherent unless they are locked by modern techniques.
相干性可通过将单个光源分成两路（如杨氏双缝）实现，或使用同步信号发生器产生声波/微波。两个独立光源通常不具相干性，除非采用现代锁频技术。
5. Young’s Double-Slit Experiment | 杨氏双缝实验
Young’s double-slit experiment provides classic evidence for the wave nature of light. Monochromatic, coherent light passing through two narrow parallel slits produces an interference pattern of equally spaced bright and dark fringes on a distant screen.
杨氏双缝实验是光波动性的经典证据。单色相干光通过两条平行狭缝后，在远处屏幕上形成等间距的明暗相间干涉条纹。
The central bright fringe corresponds to zero path difference. Bright fringes appear where the path difference is nλ; dark fringes where it is (n + ½)λ. The fringe spacing Δy (distance between adjacent bright or adjacent dark fringes) depends on wavelength λ, slit separation d and distance D from slits to screen.
中央亮纹对应零路径差。亮纹出现在路径差为 nλ 处；暗纹出现在路径差为 (n + ½)λ 处。条纹间距 Δy（相邻亮纹或相邻暗纹间距）取决于波长 λ、双缝间距 d 以及缝到屏幕的距离 D。
Δy = λD / d
This formula is only valid for small angles (Δy ≪ D). Experiments often measure Δy for several fringes and divide by the number of fringe separations to reduce uncertainty. Using a laser gives a clear, stable pattern; a white light source yields a spectrum with a white central fringe and overlapping colours on each side.
该公式仅在小角度下成立（Δy ≪ D）。实验常通过测量多条条纹间距后再除以间隔数来减小不确定度。使用激光可得到清晰稳定的图样；使用白光则产生白色中央亮纹及两侧重叠的彩色条纹。
6. Diffraction and Diffraction Gratings | 衍射与衍射光栅
Diffraction is the spreading of waves when they pass through an aperture or around an obstacle. The effect is most significant when the gap size is comparable to the wavelength. In a single-slit diffraction pattern, a broad central maximum is flanked by much weaker secondary maxima.
衍射是波通过孔隙或绕过障碍物时发生扩展的现象。当缝隙尺寸与波长相当时，衍射效应最显著。单缝衍射图样中，中央亮纹宽而亮，两侧是强度低得多的次级亮纹。
A diffraction grating consists of many equally spaced slits. It produces very sharp, bright interference maxima at angles θ that satisfy the grating equation. The grating has N lines per metre, so the slit spacing d = 1/N.
衍射光栅由大量等距狭缝组成。在满足光栅方程的角位置 θ 处，产生非常尖锐明亮的干涉极大。光栅每米有 N 条刻线，缝距 d = 1/N。
d sin θ = nλ
Here n is the order number (n = 0, 1, 2…). For a given order, longer wavelengths diffract at larger angles. A spectrometer with a diffraction grating can be used to measure λ precisely. The number of orders visible is limited when sin θ > 1, yielding maximum order nₘₐₓ ≤ d/λ.
n 为级数（n = 0, 1, 2…）。对同一级，波长越长，衍射角越大。使用配备衍射光栅的光谱仪可精确测量 λ。可见的级数受 sin θ ≤ 1 限制，最大级数 nₘₐₓ ≤ d/λ。
Comparing a grating to a double slit: grating maxima are much sharper and brighter, allowing higher resolution of closely spaced wavelengths. Grating patterns are used in spectroscopy to identify elements via their emission or absorption spectra.
光栅与双缝对比：光栅的极大更锐利、更明亮，可分辨波长更接近的谱线。光栅图样用于光谱学，通过发射或吸收光谱识别元素。
7. Stationary (Standing) Waves | 驻波
A stationary wave is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose. Energy is not transmitted along a stationary wave; it is trapped between nodes (points of zero displacement) and antinodes (points of maximum displacement).
驻波由两列频率和振幅相同、传播方向相反的波叠加形成。驻波不传递能量，能量被局限在波节（位移为零的点）和波腹（位移最大的点）之间。
Adjacent nodes are separated by half a wavelength, λ/2. The phase difference between oscillations in two adjacent loops is π rad (180°); all points within the same loop oscillate in phase. Standing waves can be set up on strings, in air columns, and using microwaves or sound waves.
相邻波节间距为半波长 λ/2。相邻两段中各点振动的相位差为 π rad（180°）；同一段内的各点同相振动。驻波可在弦上、空气柱中以及利用微波或声波建立。
On a string fixed at both ends, resonant frequencies correspond to integer multiples of the fundamental frequency f₀. The wavelength of the fundamental is λ = 2L, where L is the string length. Harmonics satisfy: λₙ = 2L/n, and fₙ = n f₀, giving the harmonic series.
对于两端固定的弦，共振频率对应基频 f₀ 的整数倍。基频波长为 λ = 2L（L 为弦长）。各次谐波满足 λₙ = 2L/n，fₙ = n f₀，构成谐音序列。
fₙ = n × (v / 2L)
In a tube closed at one end, only odd harmonics are present. The fundamental has λ = 4L, and fₙ = n f₁ where n = 1, 3, 5…. Practical investigation of stationary waves often uses a vibration generator, a pulley and slotted masses to change string tension.
在一端封闭的管中，只有奇次谐波。基频波长 λ = 4L，fₙ = n f₁，其中 n = 1, 3, 5…。驻波的实验研究常使用振动发生器、滑轮和槽码来改变弦的张力。
8. Wave Speed on a String | 弦上的波速
The speed of a transverse wave on a perfectly flexible string is determined by the tension T and the mass per unit length μ (linear density). This relationship is vital for understanding how string instruments are tuned.
理想柔弦上横波的波速由张力 T 和单位长度的质量 μ（线密度）决定。这一关系对理解弦乐器的调音至关重要。
v = √(T / μ)
Tension is the force applied along the string, measured in newtons. Linear density μ is the mass divided by the length (kg m⁻¹). Combining this with v = fλ allows prediction of the harmonic frequencies. To increase the pitch of a string instrument, a musician increases tension (or uses a lighter string, or shortens the vibrating length).
张力是施加在弦上的力，单位为牛顿。线密度 μ 是质量除以长度（kg m⁻¹）。结合 v = fλ 可预测各谐波频率。若要提高弦乐器的音调，演奏者可增大张力（或使用较轻的弦，或缩短振动长度）。
A standard experiment measures μ by weighing a known length of string, then uses a signal generator and vibration transducer to find the fundamental frequency for different tensions. Plotting f² against T should yield a straight line passing through the origin, confirming the relationship.
标准实验先称量一段定长弦以测 μ，然后用信号发生器和振动传感器测定不同张力下的基频。绘制 f² 对 T 图，应得到一条过原点的直线，验证该关系。
9. Electromagnetic Waves and the EM Spectrum | 电磁波与电磁波谱
Electromagnetic waves are transverse oscillations of electric and magnetic fields, in phase with each other and perpendicular to the direction of propagation. They all travel at the speed of light in a vacuum, c = 3.00 × 10⁸ m s⁻¹, satisfying c = fλ.
电磁波是电场和磁场的横向振荡，两者同相且垂直于传播方向。它们在真空中均以光速 c = 3.00 × 10⁸ m s⁻¹ 传播，满足 c = fλ。
The electromagnetic spectrum, in order of increasing frequency (decreasing wavelength), is: radio waves, microwaves, infrared, visible light (red to violet), ultraviolet, X-rays, gamma rays. All exhibit typical wave behaviours: reflection, refraction, diffraction and interference.
电磁波谱按频率递增（波长递减）为：无线电波、微波、红外线、可见光（红到紫）、紫外线、X 射线、伽马射线。所有电磁波均表现出反射、折射、衍射和干涉等典型波动行为。
Different regions are distinguished by their production mechanisms and interactions with matter. For example, radio waves are produced by accelerating electrons in an aerial, while X-rays are produced by high-energy electron bombardment of a metal target. The wave nature of X-rays can be demonstrated using crystal diffraction (Bragg’s law is beyond A-Level but the concept of wavelength-scale spacing is important).
各波段通过其产生机制和与物质的相互作用加以区分。例如，无线电波由天线中加速的电子产生，X 射线由高能电子轰击金属靶产生。X 射线的波动性可通过晶体衍射验证（布拉格定律超出 A-Level 范围，但波长量级间距的概念很重要）。
In the double-slit and grating equations, wavelength λ refers to the electromagnetic wavelength. The visible spectrum ranges approximately from 380 nm (violet) to 750 nm (red).
在双缝和光栅方程中，波长 λ 指电磁波的波长。可见光谱范围大约从 380 nm（紫光）到 750 nm（红光）。
10. Reflection, Refraction and Total Internal Reflection | 反射、折射与全内反射
At a boundary between two media, a wave can be reflected, transmitted (refracted) or absorbed. The law of reflection states that the angle of incidence equals the angle of reflection, measured relative to the normal. Refraction obeys Snell’s law.
在两种介质的界面处，波可能被反射、透射（折射）或吸收。反射定律指出入射角等于反射角，均相对于法线测量。折射遵循斯涅耳定律。
n₁ sin θ₁ = n₂ sin θ₂
The refractive index n of a medium is the ratio of the speed of light in a vacuum c to speed in the medium v: n = c/v. It is a dimensionless number ≥ 1. When light passes from a medium of higher n to one of lower n, it bends away from the normal.
介质的折射率 n 是真空光速 c 与该介质中光速 v 之比：n = c/v。它是一个无量纲数，≥ 1。当光从折射率较高的介质射向较低的介质时，光线远离法线偏折。
Total internal reflection (TIR) occurs when the angle of incidence in the denser medium exceeds the critical angle θc. At the critical angle, the angle of refraction is 90°; sin θc = n₂/n₁, where n₂ < n₁. For glass-to-air, θc ≈ 42°.
当光密介质中的入射角大于临界角 θc 时，发生全内反射（TIR）。在临界角处，折射角为 90°，sin θc = n₂/n₁，其中 n₂ < n₁。对于玻璃到空气，θc ≈ 42°。
Optical fibres rely on TIR to transmit light signals along a thin glass or plastic core surrounded by cladding of lower refractive index. Applications include high-speed data transmission (internet cables) and endoscopy in medicine.
光纤依靠 TIR 沿细玻璃或塑料纤芯传输光信号，纤芯外围包裹着较低折射率的包层。应用包括高速数据传输（互联网线缆）和医用内窥镜检查。
Wave effects at boundaries also include phase changes upon reflection: a wave reflected at a fixed end (or from a denser medium) undergoes a phase change of π rad; reflection from a free end (or less dense medium) yields no phase change. This is essential for explaining the formation of nodes and antinodes in standing waves.
波在边界还有相位突变：在固定端（或从光密介质）反射时，波经历 π rad 的相位变化；从自由端（或光疏介质）反射时无相位变化。这对解释驻波中波节和波腹的形成至关重要。
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