📚 Wave Behavior | 波的行为
Wave behavior encompasses a range of phenomena that occur when waves interact with boundaries, other waves, or media with varying properties. From the reflection of light in mirrors to the interference patterns that reveal the wave nature of particles, understanding these behaviors is core to optics, acoustics, and modern physics. In this article, we will systematically explore reflection, refraction, diffraction, interference, and polarisation, linking each concept to key principles and equations commonly examined in IB and OCR A Level Physics.
波的行为涵盖了一系列现象，这些现象发生在波与边界、其他波或性质变化的介质相互作用时。从镜子中光的反射到揭示粒子波动性的干涉图样，理解这些行为是光学、声学和现代物理学的核心。本文将系统地探讨反射、折射、衍射、干涉和偏振，并将每个概念与IB和OCR A Level物理中常考的关键原理和方程联系起来。
1. The Nature of Waves | 波的本质
Waves transfer energy without transferring matter. In a transverse wave, particle displacement is perpendicular to the direction of energy propagation (e.g., light, water ripples). In a longitudinal wave, particle displacement is parallel to the direction of propagation (e.g., sound).
波传输能量而不传输物质。在横波中，质点位移垂直于能量传播方向（例如光波、水波）；在纵波中，质点位移平行于传播方向（例如声波）。
The speed v, frequency f, and wavelength λ of a wave are related by the fundamental wave equation. The period T is the reciprocal of frequency. When two points on a wave are separated by an integer multiple of λ, they are in phase; a separation of an odd multiple of half-wavelength gives a phase difference of π radians.
波速v、频率f和波长λ由基本波动方程关联。周期T是频率的倒数。当波上两点相距λ的整数倍时，它们同相；相距半波长的奇数倍时，相位差为π弧度。
v = fλ T = 1/f
| Property | Transverse | Longitudinal |
|---|---|---|
| Particle motion | Perpendicular to energy flow | Parallel to energy flow |
| Examples | Light, water waves, S-waves | Sound, P-waves |
| Can be polarised? | Yes | No |
2. Reflection of Waves | 波的反射
When a wave reaches a boundary between two media, at least part of its energy is reflected back into the original medium. The law of reflection states that the angle of incidence equals the angle of reflection, measured with respect to the normal.
当波到达两种介质的边界时，至少部分能量被反射回原介质。反射定律指出入射角等于反射角，两角均相对于法线测量。
If the boundary is fixed (e.g., a string tied to a rigid wall), the reflected wave undergoes a phase change of π radians (a 180° shift). At a free boundary (e.g., a loose end of a string), reflection occurs without phase inversion. For light waves, reflection at a boundary from a medium of higher refractive index can also introduce a phase change of π.
如果边界是固定的（例如系在刚性墙上的绳子），反射波会发生π弧度（180°）的相位突变。在自由边界（例如绳子的自由端），反射不发生相位反转。对于光波，在从折射率较高的介质界面反射时也会引入π相位变化。
The frequency and wavelength of the wave remain unchanged during reflection because the wave speed in the original medium is constant.
反射过程中频率和波长保持不变，因为波在原介质中的速度是恒定的。
3. Refraction of Waves | 波的折射
Refraction occurs when a wave passes from one medium to another and changes speed. Since frequency is determined by the source and remains constant, the wavelength must change. This causes the wave to change direction unless it strikes the boundary normally.
当波从一种介质进入另一种介质并改变速度时会发生折射。由于频率由波源决定且保持恒定，波长必然改变。这导致波的方向改变，除非垂直入射。
The degree of bending is described by Snell' ined as the ratio of the speed of light in vacuum to the speed in the material: n = c / v.
弯曲程度由斯涅尔定律描述，该定律将入射角和折射角与两种介质的折射率关联起来。材料的折射率n定义为真空中光速与材料中光速的比值：n = c / v。
n₁ sin θ₁ = n₂ sin θ₂
When light enters an optically denser medium (higher n), it slows down and bends towards the normal. Conversely, moving into a less dense medium causes the wave to speed up and bend away from the normal.
当光进入光密介质（较高n）时，速度减慢并靠近法线弯折；反之，进入光疏介质时，波加速并远离法线弯折。
4. Snell's Law and Total Internal Reflection | 斯涅尔定律与全内反射
When light travels from a medium of higher refractive index to one of lower refractive index (n₁ > n₂), there is a critical angle of incidence θc for which the refracted angle is 90°. For any angle of incidence greater than θc, all light is reflected internally, a phenomenon known as total internal reflection.
当光从高折射率介质射向低折射率介质（n₁ > n₂）时，存在一个使折射角达到90°的临界入射角θc。对于任何大于θc的入射角，所有光都被内反射，这一现象称为全内反射。
sin θc = n₂ / n₁ (with n₁ > n₂)
Total internal reflection is the operating principle behind optical fibres, where light signals are transmitted with very little loss. It is also observed in mirages and in sparkle of diamonds.
全内反射是光纤工作的原理，光信号在其中以极低损耗传输。还可在海市蜃楼和钻石的闪光中观察到。
5. Diffraction of Waves | 波的衍射
Diffraction is the spreading of waves when they pass through an aperture or around an obstacle. The extent of diffraction depends on the size of the gap relative to the wavelength: when the gap width is comparable to the wavelength, diffraction is most pronounced.
衍射是波通过缝隙或绕过障碍物时发生的扩展现象。衍射的程度取决于缝隙尺寸与波长的比较：当缝隙宽度与波长相近时，衍射最为显著。
or small angles, the angular half-width of the central maximum is approximately θ ≈ λ / b.
对于单缝衍射，中央亮纹两侧是强度逐渐减弱的亮纹。第一极小的条件为b sinθ = nλ，其中b为缝宽。对于小角度，中央亮纹的角半宽度约为θ ≈ λ / b。
Diffraction limits the resolution of optical instruments such as microscopes and telescopes because light from two close objects may overlap. The Rayleigh criterion defines the minimum resolvable angular separation.
衍射限制了显微镜和望远镜等光学仪器的分辨率，因为来自两个靠近物体的光可能重叠。瑞利判据定义了最小可分辨角间距。
6. 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 occurs when coherent waves overlap, producing regions of constructive and destructive interference.
叠加原理表明，当两个或多个波在一点相遇时，总的位移是各波位移的矢量和。当相干波重叠时发生干涉，产生相长和相消干涉区域。
For constructive interference, the path difference between the waves must be an integer multiple of the wavelength. For destructive interference, the path difference must be an odd multiple of half the wavelength. Coherent sources must have the same frequency and a constant phase relationship.
对于相长干涉，波程差必须是波长的整数倍；对于相消干涉，波程差必须是半波长的奇数倍。相干波源必须具有相同的频率和恒定的相位关系。
Δx = nλ (constructive) Δx = (n + ½)λ (destructive)
Examples include the two-slit experiment for light, interference of sound from two loudspeakers, and water wave patterns in a ripple tank.
例子包括光的双缝实验、两个扬声器发出的声音干涉以及水波盘中水波图样。
7. Young's Double-Slit Experiment | 杨氏双缝实验
Young's double-slit experiment provides clear evidence of the wave nature of light. A monochromatic light source illuminates two narrow slits, which act as coherent sources, producing an interference pattern of bright and dark fringes on a distant screen.
杨氏双缝实验提供了光波动性的明确证据。单色光源照亮两条窄缝，它们充当相干光源，在远处的屏幕上产生明暗相间的干涉条纹。
d. This relationship allows precise measurement of wavelength once Δx, D, and d are known.
亮条纹的条件为d sinθ = nλ，其中d为缝间距。如果屏幕与缝相距较远D，屏幕上条纹间距Δx由Δx = λD / d给出。这一关系使得在已知Δx、D和d时能够精确测量波长。
Δx = λD / d
White light produces a central white fringe flanked by spectra because different wavelengths have different fringe spacings, causing colour separation.
白光产生中央白色条纹，两侧为光谱，因为不同波长具有不同的条纹间距，导致色散。
8. Thin Film Interference | 薄膜干涉
Thin film interference arises when light reflects from the top and bottom surfaces of a thin layer of material, such as oil on water or a soap bubble. The two reflected beams can interfere constructively or destructively depending on the film thickness and the wavelength.
薄膜干涉发生在光从薄层材料（例如水面上的油膜或肥皂泡）的上表面和下表面反射时。两束反射光依据薄膜厚度和波长发生相长或相消干涉。
The optical path difference is approximately 2 n t, where n is the refractive index of the film and t its thickness. Additional phase changes of π may occur upon reflection at a boundary with a higher refractive index. This is used in anti-reflection coatings on lenses.
光程差大约为2 n t，其中n是薄膜折射率，t是厚度。在折射率较高的边界反射时可能额外产生π相位变化。这一原理被用于透镜的增透膜。
2 n t = mλ (constructive, with appropriate phase correction)
In Newton's rings, a curved lens on a flat glass plate creates an air film of varying thickness, producing concentric interference rings.
在牛顿环装置中，一块曲面透镜放在平板玻璃上形成厚度变化的空气薄膜，产生同心干涉环。
9. Polarisation of Waves | 波的偏振
Polarisation is a property unique to transverse waves. In an unpolarised transverse wave, oscillations occur in many planes perpendicular to the direction of propagation. A polarising filter transmits only the component oscillating in a specific plane, producing plane-polarised light.
偏振是横波独有的性质。在非偏振横波中，振动方向分布在垂直于传播方向的多个平面内。偏振片只允许在特定平面内振动的分量通过，产生平面偏振光。
When unpolarised light passes through an ideal polariser, its intensity is reduced by half because only one component of the oscillating electric field is transmitted. Polarised light is used in LCD screens, sunglasses, and stress analysis in materials.
非偏振光通过理想偏振片时强度减半，因为只有电场振荡的一个分量通过。偏振光用于液晶显示屏、太阳镜和材料的应力分析。
Polarisation by reflection occurs when light reflects off a non-metallic surface at Brewster's angle, where reflected light is completely plane-polarised. Scattering by the atmosphere also causes polarisation.
反射偏振发生在光以布儒斯特角从非金属表面反射时，此时反射光完全成为平面偏振光。大气散射也会产生偏振。
10. Malus's Law | 马吕斯定律
Malus' ty I is given by I = I₀ cos² θ, where I₀ is the intensity of the incident polarised light.
马吕斯定律描述了平面偏振光通过第二个偏振片（检偏器）后的强度，此时检偏器透振轴与入射光偏振方向成θ角。透射强度I为I = I₀ cos² θ，其中I₀是入射偏振光的强度。
I = I₀ cos² θ
ompletely blocked. This law is verified experimentally by rotating the analyser and measuring the output intensity.
当θ = 0°时，cos² θ = 1，强度最大；当θ = 90°时，光被完全阻挡。通过旋转检偏器并测量输出强度，可以实验验证该定律。
Combining Malus's law with the fact that an ideal polariser halves the intensity of unpolarised light gives a complete description of light intensity through sequential polarisers, important in optical communication and sensor technology.
将马吕斯定律与理想偏振片使非偏振光强度减半的事实结合，可完整描述光通过多个偏振片后的强度变化，这在高通光学和传感器技术中很重要。
11. Standing Waves | 驻波
A standing wave is formed when two identical waves travelling in opposite directions superpose. Unlike travelling waves, standing waves do not transfer energy. They are characterised by nodes (points of zero displacement) and antinodes (points of maximum displacement).
驻波由两列相同但传播方向相反的波叠加形成。与行波不同，驻波不传输能量。其特征是波节（位移为零的点）和波腹（位移最大的点）。
the string length. Higher harmonics satisfy L = n(λ/2). The wave speed on a string is v = √T/μ, where T is tension and μ is mass per unit length.
在两端固定的弦上，驻波只能存在于特定频率，形成谐波模式。基频对应λ = 2L，其中L为弦长。高次谐波满足L = n(λ/2)。弦上的波速v = √T/μ，T为张力，μ为单位长度质量。
fₙ = n(v / 2L) n = 1,2,3…
Standing waves are also found in air columns (open or closed pipes), where they produce musical notes. In closed pipes, the displacement node at the closed end leads to a quarter-wavelength resonance for the fundamental.
驻波也存在于空气柱（开管或闭管）中，产生乐音。在闭管中，封闭端处的位移波节使得基频对应四分之一波长共振。
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