📚 C.4 Standing Waves and Resonance | C.4 驻波与共振
Standing waves and resonance are fundamental concepts in wave physics, appearing in musical instruments, engineering structures, and even quantum mechanics. In this section, we explore how standing waves form from the superposition of travelling waves, identify nodes and antinodes, and derive the harmonic series for strings and air columns. We also examine resonance, where a system oscillates with maximum amplitude at its natural frequency, and discuss practical consequences of resonance.
驻波和共振是波动物理学中的基本概念，出现在乐器、工程结构甚至量子力学中。在本节中，我们探讨驻波如何由行波叠加形成，识别波节和波腹，并推导弦和空气柱的谐波序列。我们还研究共振，即系统在其固有频率处以最大振幅振荡的现象，并讨论共振的实际影响。
1. Introduction to Standing Waves | 驻波简介
A standing wave is a wave pattern that results from the interference of two identical waves travelling in opposite directions. Unlike a travelling wave, a standing wave does not transfer energy from one point to another; instead, energy is stored in the wave pattern with alternating regions of constructive and destructive interference.
驻波是由两列相同且反向传播的波干涉形成的波动模式。与行波不同，驻波不会将能量从一点传递到另一点；相反，能量储存在波型中，并具有交替的相长干涉和相消干涉区域。
The simplest way to produce a standing wave is to reflect a wave from a boundary, causing the incident and reflected waves to superpose. This is commonly seen on a stretched string when a wave reflects at a fixed end and interferes with the incoming wave.
产生驻波最简单的方法，是让波从边界反射，使入射波和反射波叠加。这种现象常见于拉伸的弦上：波在固定端反射，并与入射波发生干涉。
A standing wave appears to be stationary, but it is still made of two travelling waves moving in opposite directions. It is a powerful model for musical notes, laser cavities, and electron orbitals in quantum physics.
驻波看起来静止不动，但它仍然由两列反向传播的行波组成。它是解释音符、激光谐振腔以及量子物理中电子轨道的重要模型。
2. Formation by Superposition and Reflection | 叠加与反射形成
The principle of superposition states that when two or more waves meet, their displacements add vectorially. For a stable standing wave, the two waves must have the same frequency, wavelength, amplitude, and speed, and they must travel in opposite directions.
叠加原理指出，当两列或多列波相遇时，它们的位移按矢量相加。要形成稳定的驻波，两列波必须具有相同的频率、波长、振幅和波速，并且沿相反方向传播。
Reflection at a boundary is the usual source of the second wave. At a fixed end, a transverse wave reflects with a phase change of π, or half a wavelength, which helps create a node at the boundary. At a free end, reflection occurs without phase inversion, so an antinode can form.
边界的反射通常是第二列波的来源。在固定端，横波反射时发生 π 的相位变化，即半个波长的变化，这有助于在边界处形成波节。在自由端，反射不发生相位反转，因此可能形成波腹。
If the two waves have equal amplitudes, the nodes are points of zero displacement and the antinodes have maximum displacement. If the amplitudes are unequal, the nodes are not perfectly zero, giving a partial standing wave.
如果两列波振幅相等，波节处的位移为零，波腹处的位移最大。如果振幅不相等，波节处不会完全为零，形成部分驻波。
3. Nodes, Antinodes and Energy | 波节、波腹与能量
A node is a point in a standing wave where the displacement is always zero. An antinode is a point where the displacement varies with maximum amplitude. The distance between adjacent nodes is half a wavelength, and the distance from a node to the nearest antinode is one quarter of a wavelength.
波节是驻波中位移始终为零的点。波腹是位移变化幅度最大的点。相邻波节之间的距离是半个波长，波节到最近波腹的距离是四分之一波长。
On a stretched string, nodes occur at fixed ends and antinodes occur at positions of maximum movement. In an air column, we describe displacement nodes and displacement antinodes; pressure nodes and pressure antinodes are in opposite positions.
在拉伸的弦上，固定端出现波节，最大运动位置出现波腹。在空气柱中，我们描述位移波节和位移波腹；压力波节和压力波腹的位置则相反。
Although a standing wave does not transfer net energy along the wave, energy is not lost. In a string, energy oscillates between kinetic energy and elastic potential energy; in an air column, energy is associated with pressure and displacement variations.
虽然驻波不沿波传播方向传递净能量，但能量并没有消失。在弦上，能量在动能和弹性势能之间振荡；在空气柱中，能量与压力和位移变化有关。
4. Standing Waves on a Stretched String | 弦上的驻波
wavelengths, so the allowed wavelengths are given by L = n λₙ / 2, where n = 1, 2, 3, ...
对于两端固定的弦，边界条件要求两端都是波节。长度 L 必须包含整数个半波长，因此允许的波长为 L = n λₙ / 2，其中 n = 1, 2, 3, ...
nsverse wave on a string depends on tension T and mass per unit length μ, according to v = √T / μ. Therefore the allowed frequencies are fₙ = n v / (2L) = n f₁.
整理可得 λₙ = 2L / n。弦上横波的速度取决于张力 T 和单位长度质量 μ，即 v = √T / μ。因此允许的频率为 fₙ = n v / (2L) = n f₁。
v = √T / μ
fₙ = n v / (2L) = n f₁
The lowest frequency is called the fundamental frequency, f₁ = (1 / 2L)√T / μ. The higher frequencies are called harmonics or overtones. A string fixed at both ends supports all integer harmonics, not just odd ones.
最低频率称为基频，f₁ = (1 / 2L)√T / μ。更高频率称为谐音或泛音。两端固定的弦支持所有整数次谐波，而不仅仅是奇数次谐波。
5. Harmonic Series on a String | 弦的谐波序列
The harmonic series on a string is f₁, 2f₁, 3f₁, 4f₁, ... The nth harmonic has n loops, where each loop is one half wavelength. The number of antinodes equals the harmonic number n.
弦上的谐波序列为 f₁, 2f₁, 3f₁, 4f₁, ... 第 n 次谐波有 n 个波环，每个波环是一个半波长。波腹的数量等于谐波次数 n。
ops, then λ = 2L / n and f = n v / (2L). This method works for strings and for open pipes.
在考试题目中，数出可见的波环或波腹数量即可确定 n。如果有 n 个波环，则 λ = 2L / n，f = n v / (2L)。这种方法适用于弦和开管。
40 / (2 × 0.80) = 150 Hz. The second harmonic is 300 Hz, and the third harmonic is 450 Hz.
例如，长度为 0.80 m 的弦，波速为 240 m s⁻¹。其基频为 f₁ = 240 / (2 × 0.80) = 150 Hz。第二次谐波为 300 Hz，第三次谐波为 450 Hz。
6. Standing Waves in Air Columns | 空气柱中的驻波
Air columns support longitudinal standing waves. At an open end, air molecules can move freely, so there is a displacement antinode and a pressure node. At a closed end, air molecules cannot move freely, so there is a displacement node and a pressure antinode.
空气柱支持纵驻波。在开口端，空气分子可以自由移动，因此存在位移波腹和压力波节。在封闭端，空气分子不能自由移动，因此存在位移波节和压力波腹。
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