📚 Determining the Wavelength and Speed of Sound | 测定声音的波长与速度
In A-Level Physics, understanding how to determine the wavelength and speed of sound is a fundamental practical skill. Sound waves are longitudinal mechanical waves that travel through a medium by compressions and rarefactions. This article explores experimental methods for measuring the speed of sound in air and the wavelength of sound waves using resonance tubes, oscilloscopes, and interference techniques.
在A-Level物理中，学会测定声音的波长与速度是一项基本实验技能。声波是通过介质传播的纵波，由疏密相间的振动形成。本文将探讨使用共振管、示波器和干涉技术等方法测量空气中声速和声波波长的实验方法。
1. Introduction to Sound Waves | 声波简介
Sound waves are longitudinal waves that require a medium to travel. They consist of alternating high-pressure compressions and low-pressure rarefactions. The speed of sound in air at room temperature is approximately 340 m s⁻¹, but it varies with temperature and humidity. Key characteristics include frequency f, wavelength λ, and speed v, which are linked by the wave equation.
声波是需要介质传播的纵波，由交替的高压压缩区和低压稀疏区组成。室温下空气中的声速约为 340 m s⁻¹，但会随温度和湿度变化。声波的关键特征包括频率 f、波长 λ 和速度 v，它们由波动方程联系起来。
2. The Wave Equation | 波动方程
peed of sound, we can measure the frequency and wavelength of a sound wave and apply this equation. Thus any experiment that yields λ and f will allow the calculation of v. Typical frequencies for experiments are a few hundred hertz to a few kilohertz, giving wavelengths in the metre to centimetre range.
联系波速 (v)、频率 (f) 和波长 (λ) 的基本关系是 v = fλ。要测定声速，可以测量声波的频率和波长，并应用此公式。因此，任何能得出 λ 和 f 的实验都能算出 v。典型实验频率在几百赫兹到几千赫兹之间，对应的波长在米到厘米量级。
3. Direct Measurement Using a Microphone and Oscilloscope | 使用麦克风和示波器直接测量
One direct method involves connecting a microphone to an oscilloscope. A loudspeaker driven by a signal generator produces a sound wave at a known frequency f. The microphone picks up the wave at a measured distance d from the speaker. By observing the phase difference on the oscilloscope, we can determine the wavelength. For instance, moving the microphone until the trace shifts by exactly one complete cycle corresponds to a distance of one wavelength λ.
一种直接方法是连接麦克风与示波器。由信号发生器驱动的扬声器产生已知频率 f 的声波。麦克风在距离扬声器 d 处接收声波。通过观察示波器上的相位差可以确定波长。例如，移动麦克风直到轨迹恰好移动一个完整周期，对应的距离就是一个波长 λ。
t. This method requires precise distance and time measurements and careful calibration of the oscilloscope's time base.
或者，使用两个麦克风检测相继的波前并测量时间延迟，通过 v = d / Δt 计算速度。这种方法需要精确的距离和时间测量，以及示波器时基的仔细校准。
4. Resonance Tube Experiment (Closed-End) | 闭管共振实验
A classic experiment uses a resonance tube, which is a vertical tube partially filled with water, creating an adjustable air column with one closed end (the water surface). A tuning fork of known frequency f is held above the open end. The length of the air column L is adjusted by changing the water level until resonance occurs, indicated by a loud sound. At resonance, a standing wave forms with a node at the closed end and an antinode at the open end.
经典实验使用共振管，这是一根部分注水的竖直管，形成一端封闭（水面）的可调节空气柱。将已知频率 f 的音叉置于开口端上方。通过改变水位调节空气柱长度 L，直到产生共振，表现为响亮的声音。共振时，管内形成驻波，封闭端为波节，开口端为波腹。
. By finding two successive resonance lengths L₁ and L₂ corresponding to different harmonics, we can eliminate e and calculate λ.
对于基频模式（第一谐波），长度 L ≈ λ/4。然而必须考虑管口校正 e，因此有效长度为 L + e = λ/4。通过找到对应于不同谐波的两次相继共振长度 L₁ 和 L₂，可以消除 e 并计算 λ。
5. Determining Wavelength from Resonance Tube Data | 由共振管数据测定波长
3), L₂ + e = 3λ/4. Subtracting gives L₂ - L₁ = λ/2. Therefore, the wavelength λ = 2(L₂ - L₁). This eliminates the end correction, giving a more accurate result. Often the third and fifth harmonics can also be used to improve precision.
在闭管中，第 n 次共振（n 为奇数）的条件是 Lₙ + e = nλ/4。对于第一次共振 (n=1)，L₁ + e = λ/4。第二次 (n=3)，L₂ + e = 3λ/4。两式相减得 L₂ - L₁ = λ/2。因此波长 λ = 2(L₂ - L₁)。这就消除了管口校正，得到更准确的结果。也常用第三和第五次谐波来进一步提高精度。
6. Calculating the Speed of Sound | 计算声速
d multiple trials improve precision. Alternatively, by measuring the frequency and the length difference directly, v = 2f(L₂ - L₁). This method typically yields values around 340 m s⁻¹ for air at 20 °C.
一旦确定了波长 λ，且已知音叉频率 f，声速 v 即可通过 v = fλ 求出。仔细测量和多次重复可提高精度。或者，直接测量频率和长度差，得到 v = 2f(L₂ - L₁)。此方法在 20 °C 空气中通常得到约 340 m s⁻¹ 的值。
7. The Kundt's Tube Method | 昆特管法
Kundt's tube visualises standing waves in a transparent horizontal tube with a fine powder (e.g., lycopodium). A loudspeaker connected to a signal generator creates sound waves in the tube. The powder gathers at the displacement nodes, forming dust heaps. The distance between adjacent heaps is half the wavelength (λ/2). This method is useful for measuring the speed of sound in gases other than air or at different temperatures.
昆特管利用透明水平管内细粉末（如石松粉）使驻波可视化。连接信号发生器的扬声器在管内产生声波。粉末聚集在位移波节处，形成粉尘堆。相邻粉尘堆之间的距离为半波长 (λ/2)。此方法可用于测量非空气气体或不同温度下的声速。
ator, and v = fλ gives the speed. This method allows a clear visual demonstration of the wave pattern.
通过测量多个波节间的距离，得到平均 λ/2，从而 λ = 2 × 平均距离。从信号发生器读取频率 f，即可由 v = fλ 计算速度。该方法可以清晰直观地展示波动图样。
8. Two-Source Interference with Sound | 声波双源干涉
e connected to an oscilloscope is moved along a line parallel to the speakers. Constructive interference (maxima) occurs when the path difference is nλ, and destructive interference (minima) when path difference = (n + ½)λ. By measuring the separation between adjacent maxima y, and the distance from sources to detection line D, and the speaker separation a, the wavelength can be found using λ = ay / D. This is analogous to Young's double-slit experiment for light.
声波干涉图样也可用于测量。两个相同的扬声器连接到同一信号发生器，发出相干声波。一个连接示波器的麦克风沿平行于扬声器连线的直线移动。当路程差为 nλ 时产生相长干涉（极大值），路程差为 (n + ½)λ 时产生相消干涉（极小值）。通过测量相邻极大值间距 y，波源到检测线的距离 D，以及扬声器间距 a，可由 λ = ay / D 确定波长。这与光的杨氏双缝实验类似。
9. Using a Signal Generator and Loudspeakers | 使用信号发生器和扬声器
oscope method, the time base of the oscilloscope can be calibrated to measure period T = 1/f, or frequency can be verified. Combining with a measured λ gives v. Also, measuring the phase shift between two fixed microphones separated by distance x gives δφ = 2πx/λ. Hence λ can be deduced if the phase difference is measured.
此实验通常使用信号发生器产生已知频率 f 的正弦信号。扬声器发出声音。在麦克风-示波器方法中，可校准示波器的时基来测量周期 T = 1/f，或验证频率。配合测出的 λ 可得 v。另外，测量相距 x 的两固定麦克风之间的相位差 δφ = 2πx/λ，即可推导出 λ。
10. Standing Waves in Open Tubes | 开管驻波
, and harmonics are integer multiples. For a tube of fixed length L, by adjusting the driving frequency until resonance, λ can be determined from λ = 2L/n, where n is the harmonic number. Combined with the known frequency, the speed v = fλ can be calculated. This method also requires an end correction, but for high-Q resonances, it can be precise.
开管（两端开口）中形成驻波时，开口端均为波腹。基频满足 L = λ/2，谐波为整数倍。对于固定长度 L 的管道，调节驱动频率直至共振，可由 λ = 2L/n 确定波长，其中 n 是谐波次数。结合已知频率可计算声速 v = fλ。该方法也需考虑管口校正，但高 Q 值共振可以很精确。
11. Accuracy, Errors, and Precautions | 准确性、误差与注意事项
Common sources of error include: parallax error when reading the water level or distances; temperature variations affecting the speed of sound; end corrections not properly accounted for; background noise affecting the detection of resonance maxima; and frequency instability of the signal generator. Minimising errors involves using a metre rule with fine markings, measuring lengths at room temperature and noting temperature, taking repeat readings, and using a sensitive microphone or good sound insulation.
常见误差来源有：读取水位或距离时的视差；温度变化影响声速；未正确考虑管口校正；背景噪声影响共振极大值的判断；信号发生器频率不稳定。减小误差的方法包括使用刻度精细的米尺，在室温下测量长度并记录温度，重复读数，以及使用灵敏麦克风或良好隔音。
The following table summarises the methods and their typical error sources.
下表总结了各种方法及典型的误差来源。
| Method | Measured Quantity | Speed v Equation | Main Error Sources |
|---|---|---|---|
| Resonance tube (closed) | L₂ - L₁ = λ/2 | v = 2f(L₂ - L₁) | Parallax, end correction, temperature |
| Kundt's tube | Dust heap spacing = λ/2 | v = fλ | Measuring node positions, uniform powder |
| Oscilloscope & microphone | Phase shift or time delay | v = fλ or v = d/Δt | Time base calibration, distance accuracy |
| Two-source interference | Fringe spacing y | λ = ay/D, then v = fλ | Path difference, alignment of detectors |
All methods require independent measurement of frequency, either from a calibrated signal generator or by using a known tuning fork. The temperature should be recorded as v is proportional to √T, where T is the absolute temperature.
所有方法均需独立测量频率，可使用校准的信号发生器或已知频率的音叉。应记录温度，因为 v 与 √T 成正比（T 为绝对温度）。
12. Summary | 总结
ve equation v = fλ to experimental data. Classic methods such as the resonance tube, Kundt' trol of temperature are essential for accurate results. These experiments not only reinforce the understanding of standing waves and interference but also develop essential practical skills for A-Level Physics.
测定声音的波长和速度需要将波动方程 v = fλ 应用于实验数据。经典方法如共振管、昆特管和干涉法均可提供可靠数值。关注实验细节、认真处理管口校正并控制环境温度是获得准确结果的关键。这些实验不仅加深了对驻波和干涉的理解，也培养了A-Level物理重要的实验技能。
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