📚 Wave Phenomena | 波动现象
Wave phenomena describe the behaviours of waves as they propagate, interact with boundaries, superimpose with other waves, and travel through different media. These phenomena—reflection, refraction, diffraction, interference, standing waves, and the Doppler effect—form the foundation of much of classical and modern physics, from the design of optical instruments to the analysis of sound and light.
波动现象描述了波在传播、与边界相互作用、与其他波叠加以及在不同介质中传播时的行为。反射、折射、衍射、干涉、驻波和多普勒效应等现象，构成了从光学仪器设计到声光分析的经典和现代物理学基础。
1. Introduction and Huygens' Principle | 导论与惠更斯原理
Every wave carries energy without the bulk transport of matter. The shape and propagation of a wavefront can be predicted using Huygens' principle: every point on a wavefront acts as a source of secondary wavelets that spread out forwards with the same speed as the wave. The new wavefront is the tangent surface to all these wavelets.
每一种波都携带能量而不伴随物质的整体输运。波前的形状和传播可以用惠更斯原理预测：波前上的每一点都可以视为次级球面子波的波源，这些子波以与波相同的速度向前传播，新的波前是所有子波的公切面。
In a homogeneous medium, spherical wavelets from a point source produce a spherical wavefront; from a plane wave, they reconstruct a plane wavefront. This model elegantly explains why waves bend when entering a different medium or when passing through an aperture.
在均匀介质中，点源发出的球面子波形成球面波前；平面波的子波则重建出平面波前。这一模型巧妙地解释了波在进入不同介质或通过孔径时为什么会发生弯折。
2. Reflection, Refraction and Diffraction | 反射、折射与衍射
When a wave strikes a boundary between two media, part of its energy is reflected and part is transmitted. The law of reflection states that the angle of incidence equals the angle of reflection, measured with respect to the normal to the surface.
当波遇到两种介质的界面时，一部分能量被反射，一部分被透射。反射定律指出，入射角等于反射角，均相对于界面法线度量。
s speed upon crossing a boundary at an oblique angle, causing a change in direction. Snell' For light, this is usually written as n₁ sin θ₁ = n₂ sin θ₂, with n the refractive index.
当波以斜角穿过界面时，因速度改变而发生方向改变，这就是折射。斯涅尔定律将入射角、折射角与各介质中的波速联系起来：sin θ₁ / sin θ₂ = v₁ / v₂，其中 v 为波速。对光而言，通常写作 n₁ sin θ₁ = n₂ sin θ₂，n 为折射率。
Diffraction is the spreading of waves after passing through an aperture or around an obstacle. Huygens' principle shows that the wavelets at the edges of an opening propagate into regions that would be shadowed if waves travelled only in straight lines. Diffraction is most noticeable when the size of the aperture or obstacle is comparable to the wavelength.
衍射是波通过孔径或绕过障碍物后的扩展现象。惠更斯原理表明，孔边缘处的子波会传播到如果波仅沿直线传播将会被遮挡的区域。当孔径或障碍物的尺寸与波长可比拟时，衍射现象最为明显。
3. 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 principle of superposition leads to interference patterns where waves reinforce or cancel each other.
当两个或多个波在一点相遇时，合位移是各分位移的矢量和。这一叠加原理导致干涉图样，其中波相互增强或抵消。
For a stable interference pattern to be observed, the sources must be coherent: they must maintain a constant phase relationship and have the same frequency. In practice, this often means splitting a single wavefront (e.g., using a double slit) so that the two secondary sources automatically possess a fixed phase difference.
要观察到稳定的干涉图样，波源必须是相干的：它们必须保持恒定的相位关系并具有相同的频率。实践中，通常通过将单一波前分成两部分（例如用双缝），使两个次级波源自动具有固定的相位差。
Constructive: path difference Δ = nλ, n = 0, 1, 2, ...
相长干涉：光程差 Δ = nλ, n = 0, 1, 2, ...
Destructive: Δ = (n + ½)λ
相消干涉：Δ = (n + ½)λ
These path difference conditions hold for waves initially in phase. A half-wavelength shift introduces a phase change of π radians, switching constructive and destructive regions.
上述光程差条件适用于初相相同的波。半波长光程差引入 π 弧度的相位变化，使得相长和相消区域互换。
4. Young's Double‑Slit Interference | 杨氏双缝干涉
Young's double-slit experiment demonstrates the wave nature of light. A monochromatic source illuminates two narrow, closely spaced slits, which act as coherent sources. On a distant screen, alternating bright and dark fringes are observed.
杨氏双缝实验证明了光的波动性。单色光源照射两条狭小且间距很近的缝，它们作为相干波源。在远处的屏幕上可以观察到明暗相间的条纹。
The fringe separation Δx on a screen placed at distance D from the slits is related to the slit separation d and the wavelength λ by:
放置在距双缝 D 处的屏幕上，条纹间距 Δx 与缝距 d 和波长 λ 的关系为：
Δx = λD / d
This formula assumes D ≫ ≈ tan θ ≈ θ holds. The angular position of the nth-order maximum is given by d sin θ = nλ.
该公式假设 D ≫ d，以便小角近似 sin θ ≈ tan θ ≈ θ 成立。第 n 级亮纹的角位置由 d sin θ = nλ 给出。
The intensity of the fringes is uniform when considering only two slits, but the presence of single-slit diffraction modulates the pattern (see Section 6). White light produces a central white fringe with coloured fringes on either side because different wavelengths interfere at different angles.
仅考虑双缝时，条纹强度是均匀的，但单缝衍射的存在会调制图样（见第6节）。白光产生一个中央白色条纹，两侧为彩色条纹，因为不同波长在不同角度处干涉。
5. The Diffraction Grating | 衍射光栅
A diffraction grating consists of a large number of equally spaced parallel slits (or lines). When illuminated by monochromatic light, it produces sharp, well-separated bright maxima at angles given by the grating equation:
衍射光栅由大量等距的平行狭缝（或刻线）组成。当单色光照射时，它会在由光栅方程给出的角度处产生锐利且间距很大的亮纹：
d sin θ = nλ, n = 0, 1, 2, ...
where d is the slit spacing (often expressed as 1/N for N lines per metre).
其中 d 为缝距（若每米有 N 条刻线，则 d = 1/N）。
The multiple beams from all slits interfere constructively, making the principal maxima very bright and sharp compared to the broad fringes from a double slit. Between principal maxima lie smaller secondary maxima and minima due to the multislit interference pattern.
来自所有缝的多光束发生相长干涉，使得主极大比双缝的宽条纹更亮、更锐利。在多缝干涉图样中，主极大之间还存在较小的次级极大和极小。
ws that higher orders and smaller d (more lines per mm) give better separation of colours.
光栅广泛用于光谱学中，将光分解为不同波长成分。角色散 dθ/dλ = n / (d cos θ) 表明，级次越高、d 越小（每毫米线数越多），颜色分离越好。
6. Single‑Slit Diffraction (HL) | 单缝衍射（高阶内容）
When a plane wave passes through a single slit of width a, each point in the slit acts as a wave source, producing a diffraction pattern consisting of a broad central maximum flanked by weaker, narrower secondary maxima. The intensity distribution is governed by the superposition of an infinite number of contributions across the slit.
当平面波通过宽度为 a 的单缝时，缝上每一点都作为一个波源，产生一个由宽阔的中央亮纹和两侧较暗、较窄的次级亮纹组成的衍射图样。强度分布由缝上无数点贡献的叠加决定。
The positions of the dark fringes (minima) are given by:
暗纹（极小）的位置由下式给出：
a sin θ = nλ, n = ±1, ±2, ±3, ...
imately 2λ / a. As a decreases, the pattern spreads out significantly, demonstrating that a narrower slit produces a wider central maximum.
中央亮纹的范围从 n = −1 到 n = +1，因此其角宽度约为 2λ / a。随着 a 减小，图样显著展宽，表明狭缝越窄，中央亮纹越宽。
le-slit minimum: d sin θ = nλ and a sin θ = mλ are satisfied simultaneously, so n/d = m/a.
在双缝实验中，单缝衍射包络调制了双缝干涉条纹。当某双缝极大值与单缝极小重合时，出现缺级现象：同时满足 d sin θ = nλ 和 a sin θ = mλ，即 n/d = m/a。
7. Resolution and the Rayleigh Criterion (HL) | 分辨率与瑞利判据（高阶内容）
The ability of an optical instrument to distinguish two closely spaced objects is limited by diffraction. The Rayleigh criterion states that two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other.
光学仪器分辨两个靠得很近的物体的能力受衍射限制。瑞利判据指出，当一个衍射图样的中央极大恰好与另一个的第一极小重合时，两个点光源刚好能被分辨。
For a circular aperture of diameter b, the minimum angular separation θ (in radians) for resolution is:
对于直径为 b 的圆孔，可分辨的最小角间距 θ（以弧度计）为：
θ ≈ 1.22 λ / b
The factor 1.22 arises from the Bessel function description of circular aperture diffraction. This criterion applies to telescopes, microscopes, and the human eye. Larger apertures (larger b) yield better resolution; shorter wavelengths also improve resolution.
系数 1.22 来源于圆孔衍射的贝塞尔函数描述。该判据适用于望远镜、显微镜和人眼。孔径越大（b 越大），分辨率越好；较短的波长也能提高分辨率。
8. Thin‑Film Interference (HL) | 薄膜干涉（高阶内容）
Thin-film interference arises when light reflects off the two surfaces of a thin transparent film, producing two coherent beams that superpose. The path difference is approximately 2 n t, where n is the refractive index of the film and t its thickness. Phase changes upon reflection must be accounted for: a π shift occurs when light reflects from a medium of higher refractive index.
薄膜干涉是光在透明薄膜的两个表面反射后，产生两束相干光叠加的现象。光程差约为 2 n t，其中 n 是薄膜的折射率，t 是膜厚。必须考虑反射时的相位变化：光从较高折射率介质反射时发生 π 相移。
For a film in air (nfilm > nair λ (m = 0, 1, 2, ...). The conditions reverse if there is a phase change at both surfaces.
对于空气（nfilm > nair）中的薄膜，第一次反射有 π 相变，第二次反射无相变。因此，反射光相长干涉的条件为 2 n t = (m + ½) λ，相消干涉条件为 2 n t = m λ。如果两个表面都发生相变，条件则互换。
This principle explains the vibrant colours seen in soap bubbles and oil slicks. Varying the thickness t across the film causes different wavelengths to interfere constructively at different positions, producing colourful patterns.
这一原理解释了肥皂泡和油膜上看到的鲜艳色彩。膜厚 t 在各处变化，导致不同位置满足不同波长的相长干涉条件，从而产生多彩的图样。
9. Standing Waves | 驻波
A standing wave is formed by the superposition of two identical waves travelling in opposite directions. It is characterised by nodes (points of zero displacement) and antinodes (points of maximum displacement). No net energy is transmitted along the medium.
驻波由两列相同、反向行进的波叠加形成。其特征为波节（位移恒为零的点）和波腹（位移最大的点）。沿介质方向没有净能量传输。
onds to the harmonic number. The fundamental frequency is f₁ = v/2L with v the wave speed.
对于两端固定的弦，允许的波长由长度 L 决定：驻波必须在两端形成波节，因此 L = n λ/2，其中 n = 1, 2, 3, ... 对应谐波序号。基频为 f₁ = v/2L，v 为波速。
a string. For a pipe closed at one end, L = (2n−1) λ/4, so only odd harmonics are present.
在空气柱中，开口端对应于波腹（位移波腹/压力波节），而闭口端是位移波节。对于两端开口的管，L = n λ/2，与弦相同；对于一端闭口的管，L = (2n−1) λ/4，因此只有奇数谐波。
10. The Doppler Effect | 多普勒效应
The Doppler effect is the change in observed frequency when a wave source and an observer are in relative motion. For sound waves, the observed frequency f' is given by:
多普勒效应是当波源与观察者之间存在相对运动时，观察到的频率发生改变的现象。对声波而言，观测频率 f' 由下式给出：
f' = f × (v ± vₒ) / (v ∓ vₛ)
where f is the source frequency, v the speed of sound, vₒ the observer's speed, and vₛ the source's speed. The signs are chosen so that f' > f when source and observer approach each other.
其中 f 为波源频率，v 为声速，vₒ 为观察者速度，vₛ 为波源速度。当波源与观察者相互靠近时，选择符号使 f' > f。
is the relative velocity of approach and c the speed of light. This effect is the basis for redshift and blueshift in astronomy and for police speed radar.
对于电磁波，高速情况下必须使用相对论多普勒公式，但低速时频移比 Δf/f ≈ v/c，其中 v 为相对接近速度，c 为光速。这一效应是天文学中红移和蓝移以及警用测速雷达的基础。
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