From Traveling Waves to Standing Waves | 从行波到驻波的演化过程

📚 From Traveling Waves to Standing Waves | 从行波到驻波的演化过程

When two traveling waves of the same frequency and amplitude pass through the same medium in opposite directions, they interfere to produce a remarkable pattern: a standing wave. This article walks through each step of the evolution from a simple traveling wave to a standing wave — a core idea in the CIE A-Level Physics syllabus.

当两列频率相同、振幅相等的行波在同一种介质中沿相反方向传播时，它们相互干涉，形成一种令人惊叹的图景——驻波。本文将逐步梳理从简单行波到驻波的演化过程，这是 CIE A-Level 物理考纲中的核心内容。

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1. Traveling Waves: The Basics | 行波基础

A traveling wave (also called a progressive wave) transfers energy from one point of space to another. It is described by the equation:

y = A sin(kx − ωt)

where y is the displacement, A is the amplitude, k = 2π/λ is the wave number, and ω = 2πf is the angular frequency. The quantity (kx − ωt) is known as the phase.

行波（又称前进波）将能量从空间中的一点传递到另一点。它的位移方程为：

y = A sin(kx − ωt)

其中 y 为位移，A 为振幅，k = 2π/λ 为波数，ω = 2πf 为角频率。表达式 (kx − ωt) 被称为相位。

In a traveling wave, every particle in the medium vibrates with the same amplitude A, while the wave crest moves forward with speed v = fλ. At any fixed position, the phase changes continuously with time, so the wave pattern appears to “travel” through the medium.

在行波中，介质中的每个质点都以相同的振幅 A 振动，而波峰以速度 v = fλ 向前移动。在任意固定位置处，相位随时间连续变化，因此波形看起来在介质中”穿行”。

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2. The Principle of Superposition | 叠加原理

The principle of superposition states that when two or more waves meet at the same point in a medium, the resultant displacement is the vector sum of the individual displacements:

y = y₁ + y₂

叠加原理指出：当两列或多列波在介质中的同一点相遇时，合位移等于各列波单独在该点产生的位移的矢量和：

y = y₁ + y₂

This principle applies to all linear wave systems, including sound waves in air, waves on a string, and electromagnetic waves. It is the physical foundation on which standing-wave formation is built.

这一原理适用于所有线性波动系统，包括空气中的声波、弦上的波和电磁波。它是驻波形成的物理基础。

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3. Two Waves in Opposite Directions | 两列相向传播的波

Now consider two identical traveling waves moving directly toward each other:

y₁ = A sin(kx − ωt)   (traveling to the right / 向右传播)

y₂ = A sin(kx + ωt)   (traveling to the left / 向左传播)

The first wave travels in the +x direction and the second in the −x direction. They have the same amplitude A, the same wavelength λ, and the same frequency f. As they overlap, the superposition principle tells us to add their displacements point by point.

第一列波沿 +x 方向传播，第二列波沿 −x 方向传播。二者振幅同为 A，波长同为 λ，频率同为 f。当它们在空间重叠时，叠加原理要求我们在每一点将各自的位移逐点相加。

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4. The Mathematics of Standing Waves | 驻波的数学推导

Applying superposition to the two waves gives:

y = y₁ + y₂ = A sin(kx − ωt) + A sin(kx + ωt)

Using the trigonometric identity sin α + sin β = 2 sin[(α + β)/2] cos[(α − β)/2], we substitute α = kx − ωt and β = kx + ωt. This yields the clean result:

y = 2A sin(kx) cos(ωt)

根据叠加原理合成两列波：

y = y₁ + y₂ = A sin(kx − ωt) + A sin(kx + ωt)

利用三角恒等式 sin α + sin β = 2 sin[(α + β)/2] cos[(α − β)/2]，令 α = kx − ωt，β = kx + ωt，可得简明的结果：

y = 2A sin(kx) cos(ωt)

This equation is the signature of a standing wave. The spatial factor sin(kx) and the temporal factor cos(ωt) are now separated. Every particle vibrates in simple harmonic motion with the same frequency, but the amplitude of each particle, 2A|sin(kx)|, is fixed by its position x.

该方程正是驻波的特征方程。空间因子 sin(kx) 与时间因子 cos(ωt) 已经分离。每个质点都以相同频率做简谐振动，但每个质点的振幅 2A|sin(kx)| 由其位置 x 决定。

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5. Nodes and Antinodes | 波节与波腹

Positions for which sin(kx) = 0 give y = 0 at every instant. These are called nodes. They occur when:

kx = nπ → x = nλ/2,  n = 0, 1, 2, …

At a node, the particles of the medium do not move at all.

凡是 sin(kx) = 0 的位置，任意时刻均有 y = 0，这些点称为波节。它们出现在：

kx = nπ，即 x = nλ/2，n = 0, 1, 2, …

在波节处，介质质点完全不动。

Positions for which |sin(kx)| = 1 give the maximum amplitude 2A. These are called antinodes and are located at:

kx = (n + ½)π → x = (2n + 1)λ/4

Two consecutive nodes (or two consecutive antinodes) are separated by λ/2, while the distance between a node and its adjacent antinode is λ/4. These fixed positions never change with time — the pattern appears to “stand still.”

凡是 |sin(kx)| = 1 的位置具有最大振幅 2A，称为波腹，位于：

kx = (n + ½)π，即 x = (2n + 1)λ/4

相邻两个波节（或相邻两个波腹）之间的距离为 λ/2，而波节与相邻波腹之间的距离为 λ/4。这些位置固定不变——波形看起来”静止”了。

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6. Wavelength, Frequency and Harmonics | 波长、频率与谐波

Since the node-to-node spacing is λ/2, a standing wave can fit onto a string of length L only when L equals an integer number of half-wavelengths.

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

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