Interference of Waves: A-Level Physics Explained | 波的干涉：A-Level 物理精讲

📚 Interference of Waves: A-Level Physics Explained | 波的干涉：A-Level 物理精讲

Interference occurs when two or more waves superpose at a point to produce a resultant wave. In A-Level Physics, interference is central to understanding how light and sound waves interact. This guide covers superposition, coherence, path difference, Young’s double-slit experiment, the diffraction grating, and thin-film interference, with a focus on CIE examination skills.

当两个或多个波在空间某点相遇时，它们会发生叠加，这种现象就是波的干涉。在 A-Level 物理中，干涉是理解光波和声波相互作用的核心内容。本指南将系统讲解叠加原理、相干性、光程差、杨氏双缝实验、衍射光栅和薄膜干涉，并重点介绍 CIE 考试所需的解题技巧。

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1. The Principle of Superposition | 波的叠加原理

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. It applies to all wave types, including light, sound, water waves, and microwaves. If two crests or two troughs meet, they reinforce to give a larger amplitude. If a crest meets a trough, they cancel partially or completely.

当两个或多个波在某一点相遇时，合位移等于各个波单独产生的位移的矢量和，这就是波的叠加原理。它适用于所有类型的波，包括光波、声波、水波和微波。如果两个波峰或两个波谷相遇，它们相互增强，合振幅增大；如果一个波峰遇到一个波谷，它们会部分或完全抵消。

Mathematically, the resultant displacement y at any instant is y = y₁ + y₂, where y₁ and y₂ are the displacements due to the individual waves. This linear addition is valid as long as the wave amplitudes are not so large that the medium behaves non-linearly. The superposition principle is the foundation of interference, diffraction, and standing waves.

从数学上看，任意时刻的合位移可以写成 y = y₁ + y₂，其中 y₁ 和 y₂ 分别是两个波单独引起的位移。只要波的振幅不是大到使介质产生非线性响应，这种线性叠加就成立。叠加原理是干涉、衍射和驻波等现象的理论基础。

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2. Constructive and Destructive Interference | 相长干涉与相消干涉

Constructive interference occurs when two waves arrive at a point in phase. Their displacements add together, so the resultant amplitude is the sum of the individual amplitudes. In terms of path difference, this happens when the path difference is a whole number of wavelengths: path difference = 0, λ, 2λ, 3λ, and so on. Bright fringes in light interference correspond to constructive interference.

相长干涉发生在两个波同相到达某一点时。它们的位移相互叠加，因此合振幅等于两个分振幅之和。从光程差的角度看，当光程差为波长的整数倍时发生相长干涉，即光程差 = 0、λ、2λ、3λ 等。光的干涉图样中的亮条纹就对应相长干涉。

Destructive interference occurs when two waves arrive in antiphase, meaning a phase difference of 180° or π radians. The resultant amplitude is the difference between the individual amplitudes. If the waves have equal amplitude and are exactly in antiphase, complete cancellation occurs. For light, dark fringes are produced by destructive interference. The path difference is then an odd number of half-wavelengths: path difference = λ/2, 3λ/2, 5λ/2, and so on.

相消干涉发生在两个波反相到达某一点时，即相位差为 180° 或 π 弧度。合振幅等于两个分振幅之差。如果两波振幅相等且完全反相，就会完全抵消。对光波来说，暗条纹就是相消干涉的结果。此时光程差为半波长的奇数倍，即光程差 = λ/2、3λ/2、5λ/2 等。

It is important to understand that interference does not create or destroy energy. Instead, it redistributes energy in space. The energy missing from dark fringes appears as extra energy in bright fringes. The total energy averaged over the whole interference pattern remains the same as the sum of the energies of the individual waves.

需要特别理解的是，干涉并不会产生或消灭能量，它只是在空间中重新分配能量。暗条纹处减少的能量会出现在亮条纹处。对整个干涉图样取平均后，总能量仍等于各列波单独存在时能量之和。

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3. Coherence and Path Difference | 相干性与光程差

For a stable, observable interference pattern, the two sources must be coherent. Coherent sources have the same frequency and a constant phase difference. This means their phase relationship does not change with time. A laser is an excellent coherent source. Ordinary light from a filament lamp is not coherent because its atoms emit light in random bursts, so the phase difference changes too rapidly to observe stable fringes.

要获得稳定、可观察的干涉图样，两个光源必须相干。相干光源具有相同的频率和恒定的相位差，也就是说它们的相位关系不随时间改变。激光是非常好的相干光源。普通白炽灯发出的光不相干，因为原子随机地发出光脉冲，相位差变化太快，无法观察到稳定的条纹。

Path difference is the difference in distance travelled by two waves from their sources to a particular point. It is often written as Δx. Path difference can be expressed in metres or in number of wavelengths. Constructive interference occurs when the path difference is a whole number of wavelengths. Destructive interference occurs when the path difference is an odd multiple of half a wavelength.

光程差是两列波从各自波源到某一点的路程之差，通常记为 Δx。光程差可以用米表示，也可以用波长数表示。当光程差为波长的整数倍时发生相长干涉；当光程差为半波长的奇数倍时发生相消干涉。

Type of interference	Path difference condition
Constructive / bright fringe	Δx = nλ, n = 0, 1, 2, 3…
Destructive / dark fringe	Δx = (n + ½)λ, n = 0, 1, 2, 3…

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4. Relationship Between Path Difference and Phase Difference | 光程差与相位差的关系

Phase difference φ is related to path difference Δx by the equation:

相位差 φ 与光程差 Δx 之间的关系由以下公式给出：

φ = (2π/λ) × Δx

This equation states that a path difference of one whole wavelength corresponds to a phase difference of 2π radians, which is 360°. A path difference of half a wavelength gives a phase difference of π radians, which is 180°. This relationship is useful when a question asks you to convert between the spatial description of interference and the angular description.

这个公式表明，一个完整波长的光程差对应 2π 弧度的相位差，也就是 360°。半个波长的光程差对应 π 弧度的相位差，也就是 180°。当题目要求你在干涉的空间描述和角度描述之间进行转换时，这个关系非常有用。

For example, if the path difference at a point is 3λ/2, the phase difference is 3π radians. Since this corresponds to an odd multiple of π, the waves arrive in antiphase and destructive interference occurs. If the path difference is 2λ, the phase difference is 4π radians, and constructive interference occurs because 4π is a whole multiple of 2π.

例如，如果某点的光程差为 3λ/2，则相位差为 3π 弧度。由于这是 π 的奇数倍，两列波反相到达，因此发生相消干涉。如果光程差为 2λ，则相位差为 4π 弧度，由于 4π 是 2π 的整数倍，因此发生相长干涉。

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5. Young’s Double-Slit Experiment | 杨氏双缝实验

Young’s double-slit experiment is a classic demonstration of light interference. Monochromatic light is first passed through a single narrow slit. This single slit acts as a source of light that illuminates two closely spaced slits. Because the light reaching the two slits comes from the same original wavefront, the two slits behave as coherent sources.

杨氏双缝实验是演示光波干涉的经典实验。单色光首先通过一个狭缝，这个狭缝作为光源照亮两条相距很近的双缝。由于到达双缝的光来自同一个原始波前，因此这两条缝可以看作相干光源。

At each of the two slits, diffraction spreads the light waves outwards. The overlapping light beams from the two slits interfere on a distant screen. Bright fringes appear where light from the two slits arrives in phase, and dark fringes appear where it arrives in antiphase. The central bright fringe is formed where the path difference is zero. On either side, bright fringes occur at path differences of λ, 2λ, 3λ, while dark fringes occur at λ/2, 3λ/2, 5λ/2, and so on.

光在每个缝处发生衍射并向周围展开，来自两个缝的光束在远处的屏上重叠并发生干涉。到达时同相的位置出现亮条纹，反相的位置出现暗条纹。中央亮条纹在光程差为零处形成。两侧的亮条纹分别对应光程差为 λ、2λ、3λ 的位置，暗条纹对应光程差为 λ/2、3λ/2、5λ/2 等的位置。

The fringe pattern near the centre consists of equally spaced bright and dark fringes. However, the bright fringe intensity is not perfectly uniform; it is modulated by the single-slit diffraction envelope. In CIE examination answers, it is usually sufficient to state that the bright fringes are equally spaced and that the central fringe is the brightest.

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