Superposition of Waves | 波的叠加

📚 Superposition of Waves | 波的叠加

When two or more waves meet in the same region, their individual displacements combine at each instant. This single idea explains interference fringes, diffraction grating patterns, thin-film colours, and stationary waves on strings and in pipes.

当两列或更多波在同一区域相遇时，它们在每一瞬间的位移会相互叠加。这一基本思想可以解释干涉条纹、衍射光栅图样、薄膜颜色以及弦上和管中的驻波。

1. Principle of Superposition | 叠加原理

If two waves arrive at a point at the same time, the resultant displacement is the vector sum of the displacements that each wave would produce on its own. For displacements y₁ and y₂, the resultant is given by:

如果两列波同时到达某一点，该点的合位移等于每列波单独产生的位移的矢量之和。对于位移 y₁ 和 y₂，合位移由下式给出：

y = y₁ + y₂

This principle applies to both transverse and longitudinal waves, including water waves, sound waves, light waves, and waves on a stretched string. The medium is assumed to behave linearly, so doubling the amplitude of one wave simply doubles its contribution to the resultant.

该原理适用于横波和纵波，包括水波、声波、光波和拉伸弦上的波。通常假设介质是线性的，因此单独一列波的振幅加倍时，它对合位移的贡献也加倍。

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

When two waves meet in phase, a crest arrives with a crest and a trough arrives with a trough. The resultant amplitude is the sum of the individual amplitudes:

当两列波同相相遇时，波峰与波峰相遇，波谷与波谷相遇。合振幅等于两列波振幅之和：

A = A₁ + A₂

This is constructive interference. If the waves meet exactly in anti-phase, a crest arrives with a trough, and the resultant amplitude is the magnitude of the difference between the individual amplitudes:

这就是相长干涉。如果两列波完全反相相遇，波峰与波谷相遇，合振幅等于两列波振幅之差的绝对值：

A = |A₁ − A₂|

If the two waves have equal amplitude and are exactly in anti-phase, the resultant amplitude is zero. Energy is still conserved: it is redistributed into regions of constructive interference rather than being destroyed.

如果两列波振幅相等且完全反相，合振幅为零。能量仍然守恒：它只是被重新分配到相长干涉的区域，并没有被消灭。

3. Path Difference and Phase Difference | 路程差与相位差

For two waves of the same wavelength, a path difference of one complete wavelength λ corresponds to a phase difference of 2π radians. In general:

对于波长相同的两列波，路程差为一个完整波长 λ 时，对应的相位差为 2π 弧度。一般情况下：

phase difference = (2π / λ) × path difference

Constructive interference occurs when the path difference is an integer multiple of λ, written as nλ. Destructive interference occurs when the path difference is an odd multiple of ½λ, written as (n + ½)λ. In both cases n = 0, 1, 2, 3, …

当路程差是 λ 的整数倍时，即 nλ，发生相长干涉。当路程差是 ½λ 的奇数倍时，即 (n + ½)λ，发生相消干涉。两种情况中 n = 0, 1, 2, 3, ……

Path difference is a length, usually measured in metres, while phase difference is an angle, usually measured in radians or degrees. Exam questions often ask you to convert between them.

路程差是长度，通常以米为单位；相位差是角度，通常以弧度或度为单位。考试题常常要求你在两者之间进行换算。

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

In Young’s experiment, light from a single source is used to illuminate two narrow parallel slits separated by a small distance a. Diffraction at each slit spreads the light out, and the two overlapping waves act as coherent sources with a constant phase relationship.

在杨氏实验中，来自单一光源的光照射到两条相距很近的平行狭缝上，狭缝间距为 a。每条狭缝的衍射使光扩散开，两列重叠的光波成为具有恒定相位关系的相干光源。

On a distant screen, alternating bright and dark fringes are observed. Bright fringes occur where the path difference is nλ, and dark fringes occur where the path difference is (n + ½)λ.

在远处的屏幕上可以观察到明暗相间的条纹。路程差为 nλ 的位置出现亮纹，路程差为 (n + ½)λ 的位置出现暗纹。

5. Double-Slit Fringe Spacing | 双缝条纹间距

The separation between adjacent bright fringes, or adjacent dark fringes, is called the fringe spacing x. For small angles, x is given by:

相邻亮纹或相邻暗纹之间的距离称为条纹间距 x。在小角度近似下，x 由下式给出：

x = λD / a

Here λ is the wavelength of the light, D is the perpendicular distance from the double slit to the screen, and a is the separation between the two slits. This formula assumes D is much larger than a and that the fringes are observed near the central axis.

其中 λ 是光波波长，D 是从双缝到屏幕的垂直距离，a 是两条狭缝之间的间距。该公式假设 D 远大于 a，并且条纹在中央轴线附近观察。

Since x is directly proportional to D and inversely proportional to a, increasing the screen distance spreads the fringes out, while increasing the slit separation makes them closer together.

由于 x 与 D 成正比，与 a 成反比，增大屏幕距离会使条纹分散得更开，而增大狭缝间距则会使条纹靠得更近。

6. Diffraction Gratings | 衍射光栅

A diffraction grating consists of many equally spaced parallel slits. The slit spacing d is the grating spacing, and constructive interference produces very sharp bright maxima at angles given by:

衍射光栅由许多等间距的平行狭缝组成。狭缝间距 d 称为光栅常数，相长干涉在以下角度产生非常锐利的亮条纹：

d sin θ = nλ

Here n is the order number: n = 0 gives the central maximum, n = 1 gives the first-order maximum on each side, and so on. θ is the angle between the incident beam direction and the diffracted beam direction.

其中 n 是级次：n = 0 对应中央极大，n = 1 对应两侧的一级极大，以此类推。θ 是入射光束方向与衍射光束方向之间的夹角。

Because sin θ cannot exceed 1, the highest possible order is found from n ≤ d / λ. Gratings are useful for measuring wavelength accurately because they produce widely separated, very sharp maxima.

因为 sin θ 不能超过 1，所以可观察到的最大级次由 n ≤ d / λ 决定。光栅可以产生间距较大且非常锐利的亮纹，因此常用于精确测量波长。

7. Stationary Waves Formation | 驻波的形成

A stationary wave is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose. This usually happens when a wave is reflected back along the same path.

当两列频率相同、振幅相同的行波沿相反方向传播并叠加时，就会形成驻波。这通常发生在波沿同一路径被反射回来时。

Nodes are points where the resultant displacement is always zero, and antinodes are points where the resultant displacement oscillates with maximum amplitude. Adjacent nodes are separated by half a wavelength, and adjacent antinodes are also separated by half a wavelength.

波节是合位移始终为零的点，波腹是合位移以最大振幅振动的点。相邻波节之间的距离为半个波长，相邻波腹之间的距离也为半个波长。

Unlike a progressive wave, a stationary wave does not transfer energy along the medium. Energy remains stored in the oscillating segments between nodes.

与行波不同，驻波不沿介质传递能量。能量被储存在波节之间的振动段内。

Progressive wave	Stationary wave
Energy is transferred in the direction of wave travel.	No net energy transfer along the medium.
All points have the same amplitude.	Amplitude varies from zero at nodes to maximum at antinodes.
Phase changes continuously along the wave.	Points between adjacent nodes oscillate in phase.

8. Harmonics in Strings and Pipes | 弦与管中的谐波

For a string fixed at both ends, both ends must be nodes. The fundamental mode has one antinode at the centre, so the length L equals half a wavelength: L = λ₁ / 2. The fundamental frequency is f₁ = v / (2L).

对于两端固定的弦，两端必须是波节。基频模式在弦中央有一个波腹，因此弦长 L 等于半个波长：L = λ₁ / 2。基频为 f₁ = v / (2L)。

All harmonic frequencies are integer multiples of the fundamental frequency:

所有谐波频率都是基频的整数倍：

fₙ = n f₁ = nv / (2L), n = 1, 2, 3, …

For a pipe open at both ends, both ends are antinodes. The harmonic series is the same as that of a stretched string, so fₙ = nv / (2L).

对于两端开口的管，两端都是波腹。其谐波序列与拉伸弦相同，因此 fₙ = nv / (2L)。

For a pipe closed at one end, the closed end must be a node and the open end must be an antinode. The fundamental mode has length L = λ₁ / 4, so f₁ = v / (4L). Only odd harmonics are possible:

对于一端封闭的管，封闭端必须是波节，开口端必须是波腹。基频模式满足 L = λ₁ / 4，因此 f₁ = v / (4L)。只有奇数谐波可以存在：

fₙ = n f₁ = nv / (4L), n = 1, 3, 5, …

9. Thin-Film Interference | 薄膜干涉

Thin-film interference arises when light reflects from the top and bottom surfaces of a thin transparent film, such as a soap bubble or a thin layer of oil on water. The two reflected waves overlap and interfere.

薄膜干涉是由光在透明薄膜的上表面和下表面发生反射而产生的，例如肥皂泡或水面上的薄油膜。两束反射光重叠并发生干涉。

For normal incidence and a film of thickness t and refractive index n, the optical path difference between the two reflected waves is approximately 2nt. An additional phase change of π occurs when a wave reflects at a boundary where the refractive index increases.

对于垂直入射、厚度为 t、折射率为 n 的薄膜，两束反射光之间的光程差约为 2nt。当波在折射率增大的界面上反射时，还会产生额外的 π 相位变化。

For a film in air, where the top reflection has a phase change of π and the bottom reflection does not, bright reflected light occurs when:

对于空气中的薄膜，如果上表面反射有 π 相位变化而下表面反射没有，则反射光出现亮纹的条件为：

2nt = (m + ½)λ, m = 0, 1, 2, …

Dark reflected light occurs when 2nt = mλ. These conditions explain why different colours are seen at different film thicknesses.

反射光出现暗纹的条件为 2nt = mλ。这些条件解释了为什么在不同薄膜厚度处会看到不同颜色。

10. Coherence and Wavefronts | 相干性与波前

Two sources are coherent if they emit waves with the same frequency and a constant phase difference. A laser is highly coherent, while light from an ordinary lamp must be passed through a single slit or a filter to improve coherence before it is used in a double-slit experiment.

如果两个波源发射频率相同且相位差恒定的波，则它们是相干波源。激光具有很高的相干性，而普通灯泡发出的光在用于双缝实验之前，必须先通过单缝或滤光片来提高相干性。

A wavefront is a surface connecting points of equal phase. For a point source, wavefronts are spherical; at a large distance they can be treated as approximately plane wavefronts. Rays show the direction of energy transfer and are perpendicular to wavefronts.

波前是连接相位相同各点的面。对于点波源，波前是球面；在距离很远时，可以把它们近似看作平面波前。光线表示能量传播的方向，并且与波前垂直。

11. Exam Skills: Common Pitfalls | 考试技巧与常见误区

A common mistake is to write the superposition principle as if waves bounce off each other. They do not: they pass through each other and their displacements add only while they overlap.

一个常见误区是把叠加原理描述成波会相互弹开。实际上它们不会：波会相互穿过，只是在重叠区域内位移发生叠加。

Students often confuse path difference with phase difference. Always state whether the answer is in wavelengths, metres, radians, or degrees, and use the conversion factor 2π per wavelength.

学生经常混淆路程差和相位差。作答时要明确说明单位是波长、米、弧度还是度，并使用每个波长对应 2π 的换算关系。

In double-slit calculations, do not confuse the slit separation a with the fringe spacing x or the screen distance D. Always convert all lengths to metres before substituting into x = λD / a.

在双缝计算中，不要把狭缝间距 a 与条纹间距 x 或屏幕距离 D 混淆。代入 x = λD / a 之前，要先把所有长度单位换算成米。

For stationary waves, remember that a closed end of a pipe is a displacement node, while an open end is a displacement antinode. A string fixed at both ends has nodes at both ends, not antinodes.

对于驻波，要记住管的封闭端是位移波节，而开口端是位移波腹。两端固定的弦在两端都是波节，而不是波腹。

12. Linking Superposition to Real Phenomena | 叠加原理与实际现象的联系

Superposition explains a wide range of observations: the bright and dark bands in Young’s experiment, the colours of soap bubbles and oil films, the resonant notes produced by musical instruments, and the sharp spectra produced by diffraction gratings in spectrometers.

叠加原理可以解释大量现象：杨氏实验中的明暗条纹、肥皂泡和油膜的颜色、乐器产生的共鸣音，以及光谱仪中衍射光栅产生的锐利光谱。

When solving problems, identify first whether the system involves two interfering progressive waves, or a progressive wave and its reflection forming a stationary wave. This distinction determines the relevant equations and boundary conditions.

解题时，首先要判断系统是两列行波发生干涉，还是一列行波与其反射波形成驻波。这一区别决定了使用哪些方程和边界条件。

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Superposition of Waves | 波的叠加