📚 A-Level WJEC Physics: Waves – Key Concepts | A-Level WJEC 物理：波 考点精讲
Waves are fundamental to WJEC A-Level Physics, linking together ideas about energy transfer, interference, and the nature of light and sound. This revision guide walks you through every essential topic, from basic wave parameters to the Doppler effect, with clear explanations and key formulas you need for the exam.
波是 WJEC A-Level 物理的核心内容，它将能量传递、干涉以及光与声的本质紧密联系在一起。这份考点精讲带你逐一复习从基本波参数到多普勒效应的所有关键主题，提供清晰的解释和考试必用公式。
1. Wave Parameters and the Wave Equation | 波参数与波动方程
A wave transfers energy from one place to another without any net movement of matter. The displacement of a point on the wave is its distance from the undisturbed position, and the amplitude is the maximum displacement.
波将能量从一个地方传递到另一个地方，而不伴随物质的净移动。波上某点的位移是其到平衡位置的距离，振幅则是最大位移。
Key parameters include the wavelength λ (distance between two consecutive points in phase), the period T (time for one complete oscillation), and the frequency f (number of oscillations per second). The relationship between frequency and period is f = 1/T.
关键参数包括波长 λ（相邻两个同相位点之间的距离）、周期 T（一次完整振动所需的时间）和频率 f（每秒振动次数）。频率与周期的关系为 f = 1/T。
The speed of a wave v is linked to its frequency and wavelength by the wave equation:
波速 v 与频率、波长之间的关系由波动方程给出：
v = f λ
This equation holds for all types of waves, provided the medium does not change. When a wave moves from one medium to another, its frequency stays the same while its speed and wavelength change.
该方程适用于所有类型的波，前提是介质不变。当波从一种介质进入另一种介质时，其频率保持不变，而波速和波长会发生变化。
2. Transverse and Longitudinal Waves | 横波与纵波
In a transverse wave, the oscillations of particles are perpendicular to the direction of energy transfer. Examples include waves on a string, water ripples, and all electromagnetic waves.
在横波中，粒子的振动方向与能量传递方向垂直。例子包括弦上的波、水波涟漪及所有电磁波。
In a longitudinal wave, the oscillations are parallel to the direction of energy transfer. Sound waves in air and pressure waves are longitudinal. They consist of compressions (regions of high pressure) and rarefactions (regions of low pressure).
在纵波中，振动方向平行于能量传递方向。空气中的声波和压力波都是纵波。它们由压缩区（高压区）和稀疏区（低压区）组成。
Polarisation can only occur with transverse waves, which is a key piece of evidence for the nature of light as a transverse electromagnetic wave.
偏振现象只可能发生在横波中，这是光作为横电磁波这一本性的重要证据。
3. Polarisation | 偏振
Polarisation is the process of restricting the oscillations of a transverse wave to a single plane. An unpolarised wave has oscillations in many different planes perpendicular to the direction of travel.
偏振是将横波的振动限制在单一平面内的过程。非偏振波的振动分布在垂直于传播方向的多个不同平面内。
A polarising filter only allows oscillations in one specific plane to pass through. If a second filter (analyser) is placed after the first and rotated, the transmitted intensity changes according to Malus’s law:
偏振滤光片只允许特定平面的振动通过。若在第一个滤光片之后再放置第二个（检偏器）并旋转，透射光的强度将按马吕斯定律变化：
I = I₀ cos² θ
where I₀ is the intensity after the first polariser, and θ is the angle between the transmission axes of the two filters. When θ = 90°, the intensity drops to zero.
其中 I₀ 是经过第一个偏振片后的强度，θ 是两个滤光片透射轴之间的夹角。当 θ = 90° 时，强度降为零。
Polarisation provides clear evidence that light is a transverse wave; longitudinal waves cannot be polarised. Applications include Polaroid sunglasses, LCD screens, and stress analysis in materials.
偏振清楚地证明了光是横波；纵波无法被偏振。应用包括偏光太阳镜、液晶显示屏和材料应力分析。
4. Superposition and Coherence | 叠加与相干性
When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. This is the principle of superposition.
当两个或多个波在某点相遇时，合位移等于各波单独位移的矢量和——这就是叠加原理。
For a stable interference pattern to be observed, the sources must be coherent, meaning they emit waves with a constant phase difference and the same frequency. For maximum contrast, the amplitudes should also be similar.
要观察到稳定的干涉图样，波源必须相干，即它们发出的波具有恒定的相位差和相同的频率。为获得最大对比度，振幅也应尽可能相近。
Constructive interference occurs when the path difference is an integer multiple of the wavelength (Δ = nλ), giving a resultant amplitude equal to the sum of the individual amplitudes.
当路程差为波长的整数倍 (Δ = nλ) 时发生相长干涉，合振幅等于各振幅之和。
Destructive interference occurs when the path difference is an odd multiple of half a wavelength (Δ = (n + ½)λ), giving a minimum (ideally zero) amplitude.
当路程差为半波长的奇数倍 (Δ = (n + ½)λ) 时发生相消干涉，振幅最小（理想情况为零）。
The phase difference φ is related to the path difference Δ by φ = (2π/λ) × Δ. A phase difference of 2π radians corresponds to one whole wavelength.
相位差 φ 与路程差 Δ 之间的关系为 φ = (2π/λ) × Δ。2π 弧度的相位差对应一个完整的波长。
5. Young’s Double-Slit Experiment | 杨氏双缝干涉实验
Young’s double-slit experiment demonstrates the interference of light and allows the wavelength of light to be measured. Monochromatic light illuminates two narrow, parallel slits, which act as coherent sources.
杨氏双缝实验展示了光的干涉现象，并可用于测量光的波长。单色光照射两条平行的窄缝，它们作为相干光源。
On a screen placed at a distance D, a pattern of equally spaced bright and dark fringes is observed. The fringe separation Δx (distance between adjacent bright or dark fringes) is given by:
在距离为 D 的屏幕上可观察到等间距的明暗条纹。条纹间距 Δx（相邻亮纹或暗纹之间的距离）由下式给出：
Δx = λ D / d
where d is the separation between the two slits. This equation is valid only when D ≫ d and for small angles.
其中 d 为双缝间距。该等式仅在 D ≫ d 且角度很小时成立。
The pattern can be explained by the path difference between waves from the two slits. Bright fringes occur where the path difference is nλ, dark fringes where it is (n + ½)λ.
图样可通过两缝波的路径差来解释：路程差为 nλ 处出现亮纹，(n + ½)λ 处出现暗纹。
Replacing the double slits with a diffraction grating improves precision, as the maxima are much sharper and brighter.
用衍射光栅替代双缝可提高精度，因为极大更尖锐、更明亮。
6. The Diffraction Grating | 衍射光栅
A diffraction grating consists of many equally spaced parallel slits. When monochromatic light passes through, the transmitted waves interfere to produce a pattern of sharp principal maxima at specific angles.
衍射光栅由大量等间距的平行狭缝构成。单色光通过时，透射波相互干涉，在特定角度产生锐利的主任意极大。
The condition for a bright fringe of order n is given by the grating equation:
第 n 级明纹的条件由光栅方程给出：
d sin θ = nλ
where d is the slit spacing (grating constant), θ is the angle of the nth-order maximum, and λ is the wavelength. n can be 0, ±1, ±2, …
其中 d 为缝距（光栅常数），θ 为第 n 级极大的衍射角，λ 为波长。n 可取 0, ±1, ±2 …
If a grating has N lines per metre, then d = 1/N. A grating produces a smaller line spacing and therefore larger angular separation between orders than a double slit, making it ideal for measuring wavelengths in spectral analysis.
若光栅每米有 N 条刻线，则 d = 1/N。与双缝相比，光栅的线间距更小，因而级次间的角分离更大，非常适用于光谱分析中的波长测量。
When white light is used, each order (except n=0) spreads into a continuous spectrum, with violet deviated least and red deviated most. This is how spectrometers separate light into its component wavelengths.
使用白光时，除零级外各级次都会展开成连续光谱，其中紫光偏折最小，红光偏折最大。这就是光谱仪将光分成不同波长成分的原理。
7. Stationary Waves | 驻波
A stationary (or standing) wave is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose. Unlike a progressive wave, there is no net transfer of energy along a stationary wave.
当两列频率和振幅相同、传播方向相反的行波相遇叠加时，便形成驻波（定常波）。与行波不同，驻波没有净能量传递。
The waveform shows points of zero displacement called nodes, and points of maximum amplitude called antinodes. The distance between adjacent nodes (or antinodes) is λ/2.
波形中出现位移为零的波节和振幅最大的波腹。相邻波节（或波腹）之间的距离为 λ/2。
For a string fixed at both ends, the allowed wavelengths and frequencies are:
对于两端固定的弦，允许的波长和频率为：
λₙ = 2L / n, fₙ = n v / (2L)
where L is the string length, v is the wave speed, and n = 1, 2, 3 … represents the harmonic number. The fundamental frequency (n=1) is the lowest.
其中 L 为弦长，v 为波速，n = 1, 2, 3 … 表示谐波次数。基频 (n=1) 是最低的频率。
For a pipe closed at one end, only odd harmonics are present: fₙ = n v / (4L) with n = 1, 3, 5 … For a pipe open at both ends, the harmonics follow the same relationship as a string: fₙ = n v / (2L).
对于一端封闭的管，只存在奇数谐波：fₙ = n v / (4L)，n = 1, 3, 5 …。对于两端开口的管，谐波关系与弦相同：fₙ = n v / (2L)。
Stationary waves explain the sound produced by musical instruments as well as resonance phenomena in columns of air and stretched strings.
驻波解释了乐器发声以及空气柱和张紧弦中的共振现象。
8. Refraction | 折射
Refraction is the change in direction of a wave when it passes from one medium into another due to a change in speed. The frequency remains constant, but the wavelength and speed alter.
折射是波在从一种介质进入另一种介质时因速度改变而发生的方向变化。频率保持不变，但波长和波速改变。
Snell’s law governs the angles of incidence θ₁ and refraction θ₂, linking them to the refractive indices n₁ and n₂ of the two media:
斯涅尔定律描述了入射角 θ₁ 和折射角 θ₂ 之间的关系，并联系两种介质的折射率 n₁ 和 n₂：
n₁ sin θ₁ = n₂ sin θ₂
The refractive index n of a medium is defined as n = c / v, where c is the speed of light in a vacuum and v is the speed in the medium. It is always ≥ 1.
介质的折射率 n 定义为 n = c / v，其中 c 是真空中的光速，v 是介质中的光速。折射率始终 ≥ 1。
When light travels from a less dense to a more dense medium (n₂ > n₁), it bends towards the normal. When going from a denser to a less dense medium, it bends away from the normal.
光从光疏介质进入光密介质 (n₂ > n₁) 时，折射光线偏向法线；从光密介质进入光疏介质时，则偏离法线。
9. Total Internal Reflection | 全内反射
Total internal reflection (TIR) can occur when light travels from a medium of higher refractive index to one of lower refractive index. If the angle of incidence is greater than a critical angle θc, all light is reflected back into the denser medium, with no transmission.
全内反射（TIR）发生在光从较高折射率介质射向较低折射率介质时。若入射角大于临界角 θc，则全部光线反射回光密介质，没有透射。
The critical angle is given by:
临界角由下式给出：
sin θc = n₂ / n₁
When the second medium is air (n₂ ≈ 1), this simplifies to sin θc = 1 / n₁. For glass with n ≈ 1.5, the critical angle is about 42°.
当第二种介质为空气 (n₂ ≈ 1) 时，公式简化为 sin θc = 1 / n₁。对于折射率约为 1.5 的玻璃，临界角约为 42°。
Optical fibres exploit total internal reflection to transmit light signals over long distances with very little loss. The core of the fibre has a higher refractive index than the surrounding cladding, so light entering at a suitable angle stays within the core by repeated TIR.
光纤利用全内反射以极低损耗远距离传输光信号。纤芯的折射率高于包层，因此以合适角度进入的光线通过连续的全内反射被限制在纤芯内。
TIR is also responsible for the sparkle of diamonds and the use of prisms as reflectors in binoculars and periscopes.
全内反射也解释了钻石的闪耀以及棱镜在双筒望远镜和潜望镜中用作反射器的原理。
10. The Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency of a wave when there is relative motion between the source and the observer. It applies to sound, light, and other waves.
多普勒效应是指波源与观测者之间存在相对运动时观测频率发生变化的现象。这适用于声波、光波及其他波动。
For sound waves, when a source moves towards a stationary observer, the wavefronts are compressed, resulting in a higher observed frequency (higher pitch). When the source moves away, the frequency is lowered.
对于声波，当波源向静止的观察者运动时，波前被压缩，观测频率变高（音调升高）；波源远离时，频率降低。
The observed frequency f ‘ is given by:
观测频率 f ‘ 由下式给出：
f ‘ = f (v ± vₒ) / (v ± vₛ)
where f is the source frequency, v is the wave speed in the medium, vₒ is the observer’s speed, and vₛ is the source’s speed. Signs are chosen based on direction: use + when observer moves towards source or source moves towards observer effectively reducing distance.
其中 f 为波源频率，v 为波在介质中的速度，vₒ 为观察者速度，vₛ 为波源速度。符号根据方向选择：
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