📚 Methods for Determining the Wavelength and Speed of Sound | 测定声波波长与声速的实验方法
). At room temperature (20°C), the accepted value is approximately 343 m/s. This article explores the most common and exam-relevant experimental techniques used in A-Level Physics to determine both the wavelength and the speed of sound.
声波在空气中的传播速度是一个基本的物理量，通常通过波长（λ）、频率（f）和波速（v = fλ）之间的关系来实验测定。在室温（20°C）下，公认值约为 343 m/s。本文探讨 A-Level 物理中最常见且最贴合考纲的测定声波波长与声速的实验方法。
1. Theoretical Foundation | 理论基础
Sound waves are longitudinal mechanical waves that require a medium to propagate. In a gas, the theoretical speed of sound is given by Newton–Laplace equation:
v = √γP/ρ
for air), P is the pressure, and ρ is the density of the gas. However, in experimental physics, we more frequently use the direct relationship:
v = fλ
where f is the frequency of the sound source (measured in Hz) and λ is the wavelength (measured in m). This equation forms the basis of every experimental method described below.
声波是纵波，需要介质才能传播。在气体中，声速的理论值由牛顿–拉普拉斯方程给出：
v = √γP/ρ
其中 γ 是绝热指数（空气约为 1.40），P 是压强，ρ 是气体密度。然而在实验物理中，我们更常使用直接的物理关系：
v = fλ
其中 f 是声源频率（单位 Hz），λ 是波长（单位 m）。这个方程是以下所有实验方法的理论基础。
2. Resonance Tube Method | 共鸣管法
The resonance tube method is the classic A-Level experiment for determining the speed of sound. A tuning fork of known frequency is held above a vertical tube partially filled with water. The water level is adjusted until resonance (a loud sound) is heard, indicating that a standing wave has formed.
共鸣管法是 A-Level 测定声速的经典实验。将已知频率的音叉置于部分充满水的竖直管上方，调节水位直到听到共鸣（响亮的声音），表明已形成驻波。
equals one-quarter of the wavelength. The second resonance occurs at L₂ = 3λ/4. Therefore:
λ = 2(L₂ − L₁)
ing into v = fλ gives the speed of sound. Note that the first resonance length includes an "end correction" term e ≈ 0.3d (where d is the tube diameter). The corrected equation is:
L₁ + e = λ/4, L₂ + e = 3λ/4
第一次共鸣时管长 L₁ 等于四分之一波长，第二次共鸣时 L₂ = 3λ/4。因此：
λ = 2(L₂ − L₁)
代入 v = fλ 即可求得声速。注意第一次共鸣长度包含"末端修正"项 e ≈ 0.3d（d 为管径），修正后的方程为：
L₁ + e = λ/4, L₂ + e = 3λ/4
By taking the difference L₂ − L₁, the end correction is automatically eliminated, improving the accuracy of the wavelength measurement.
通过取差值 L₂ − L₁，末端修正被自动消除，从而提高了波长测量的准确性。
3. Procedure for the Resonance Tube | 共鸣管实验步骤
The systematic procedure ensures reliable results:
系统的实验步骤可确保结果可靠：
Set up a tall glass tube (minimum 1 m) with a rubber hose connected to a water reservoir. Adjust the water level to fill the tube completely.
搭建一根高玻璃管（至少 1 m），通过橡胶管连接储水瓶，先将管内注满水。
Strike the tuning fork and hold it horizontally just above the open end of the tube, with one prong close to the opening.
敲击音叉，将其水平置于管口正上方，一叉臂靠近管口。
Slowly lower the water level until the sound is maximally loud (first resonance). Measure L₁ from the water surface to the top of the tube.
缓慢降低水位，直到声音最响（第一次共鸣），测量水面到管口的距离 L₁。
Continue lowering the water level to find the second resonance position. Measure L₂.
继续降低水位找到第二次共鸣位置，测量 L₂。
Record the temperature of the air in the tube and repeat for several different tuning forks (different frequencies).
记录管内空气温度，并使用多个不同频率的音叉重复实验。
For each frequency, plot a graph of L vs 1/f. The gradient is v/4, allowing the speed of sound to be determined from the slope.
对每个频率，绘制 L 对 1/f 的图像。斜率为 v/4，由此可从斜率求出声速。
4. Kundt's Tube Method | 孔脱管法
Kundt's tube is a historical but educationally valuable method that allows direct visualisation of standing sound waves. A glass tube contains a fine powder (such as lycopodium powder or cork dust). A piston attached to a loudspeaker drive on one end is driven at a known frequency.
孔脱管是一种经典但具有重要教学价值的方法，它允许直接观察声波驻波。玻璃管中装有细粉末（如石松粉或软木屑），一端连接扬声器驱动的活塞，以已知频率振动。
When the frequency is adjusted to produce resonance, the powder collects at the pressure antinodes (nodes of displacement), forming evenly spaced heaps. The distance between successive heaps is λ/2. Thus:
当频率调节到共振时，粉末在压力波腹（位移波节）处聚集，形成等间距的堆。相邻堆之间的距离为 λ/2。因此：
λ = 2 × (distance between adjacent heaps)
λ = 2 ×（相邻堆之间的距离）
he signal generator / counter, and v = fλ gives the speed of sound. The advantage of Kundt' standing wave pattern, which is especially helpful for understanding nodes and antinodes.
频率 f 从信号发生器/计数器读取，v = fλ 即可得到声速。孔脱管的优点是直观地展示了驻波图形，特别有助于理解波节和波腹的概念。
5. Double-Beam Oscilloscope Method | 双束示波器法
This modern and precise method uses two microphones connected to an oscilloscope. A loudspeaker emits a continuous sound wave of known frequency. The microphones are placed along the direction of propagation, separated by a known distance d.
这种现代且精确的方法使用两个连接示波器的麦克风。扬声器发出已知频率的连续声波，麦克风沿传播方向放置，相隔已知距离 d。
The oscilloscope displays both signals. When the microphone separation is changed, the phase difference between the two waveforms changes. The distance between consecutive positions of zero phase difference (when the two traces coincide) equals exactly one wavelength λ.
示波器同时显示两路信号。当麦克风间距改变时，两个波形之间的相位差也随之变化。相邻两次相位差为零（两条波形完全重合）时麦克风移动的距离恰好等于一个波长 λ。
Set the loudspeaker to a frequency such as 1.00 kHz and position both microphones close together.
将扬声器频率设为 1.00 kHz，两麦克风先靠拢放置。
Slowly move one microphone away from the other along the metre rule until the two traces on the oscilloscope are back in phase.
沿米尺缓慢移动一个麦克风远离另一个，直至示波器上两波形重新同相。
Measure the distance moved. This is one wavelength.
测量移动的距离，此即为一个波长。
Repeat several times, recording the distance for every successive in-phase position to reduce uncertainty.
重复多次，记录每隔一个同相位置的间距，以减少不确定性。
This method avoids the end-correction problems of the resonance tube and is quick to perform with good accuracy.
这种方法避免了共鸣管法中的末端修正问题，且实验快速、精度较高。
6. Lissajous Figure Method | 利萨如图形法
An alternative oscilloscope technique uses a single microphone and feeds its signal to the Y-input of the oscilloscope, while the signal generator providing the sound frequency is connected to the X-input. The resulting Lissajous figure (usually an ellipse) indicates the phase difference between the emitted and received waves.
另一种示波器技术只用一个麦克风，将其信号输入示波器 Y 轴，同时将声源信号发生器接 X 轴。所得利萨如图形（通常为椭圆）表示发射波与接收波之间的相位差。
als one wavelength λ. Record the frequency f from the signal generator and compute v = fλ.
将麦克风放在椭圆变为直线（相位差为 0° 或 180°）的位置，然后沿传播方向移动麦克风，直到直线以相同倾斜方向再次出现，此移动距离即等于一个波长 λ。记录信号发生器频率 f，计算 v = fλ。
Advantages of this method:
此方法的优点：
Only one microphone is needed, avoiding the need for matching two microphone responses.
只需一个麦克风，无需配对两个麦克风的响应。
The phase measurement is visually clear and straight-line positions are easy to identify.
相位测量清晰直观，直线位置容易判断。
7. Temperature Dependence of the Speed of Sound | 声速随温度的变化
The speed of sound in a gas increases with temperature. For air, the approximate relationship is:
声在气体中的速度随温度升高而增大。空气的近似关系为：
v = v₀√T/T₀ ≈ 331.3 + 0.61θ
K), and θ is the temperature in degrees Celsius. The linear approximation 331.3 + 0.61θ is valid for typical classroom temperatures (0–50°C).
其中 v₀ = 331.3 m/s 为 0°C（T₀ = 273.15 K）时的声速，θ 为摄氏温度。线性近似式 331.3 + 0.61θ 在通常课堂温度（0–50°C）下适用。
In any of the experiments above, the air temperature should be measured with a thermometer, and the experimental value of v should be compared to the theoretical value at the measured temperature. A typical question asks candidates to account for discrepancies between measured and theoretical values.
在上述任一实验中，都应用温度计测量空气温度，并将声速的实验值与所测温度下的理论值进行比较。典型考题会要求考生解释实验值与理论值之间的差异。
8. Data Analysis and Graph Plotting | 数据处理与作图
In the resonance tube method, the most examinable analysis is the graph of L (tube length at resonance) against 1/f. For the nth resonance:
在共鸣管法中，最常考的分析方法是绘制 L（共鸣时管长）对 1/f 的图像。对于第 n 次共鸣：
Lₙ + e = (2n − 1)λ/4 = (2n − 1)v/(4f)
Thus a graph of Lₙ vs 1/f is a straight line with:
因此 Lₙ 对 1/f 的图像为直线，其中：
Gradient = (2n − 1)v/4, so v = 4 × gradient / (2n − 1)
斜率 = (2n − 1)v/4，故 v = 4 × 斜率 / (2n − 1)
Intercept on the L-axis = −e, giving the end correction directly.
L 轴截距 = −e，直接给出末端修正。
lotting multiple data points also averages out random errors, and the use of error bars enables the calculation of the uncertainty in the final value of v.
使用第一次共鸣数据（n = 1）时，声速即为 v = 4 × 斜率。绘制多个数据点可平均消除随机误差，而使用误差棒可以计算最终 v 值的不确定度。
9. Sources of Error and Improvements | 误差来源与改进
Every experimental method has inherent limitations. Below are the most common errors and the corresponding improvements.
每种实验方法都存在固有限制。以下是常见误差及相应的改进方法。
| Error | 误差 | Improvement | 改进 |
| Judging the loudest sound by ear is subjective | 用耳朵判断最大音量具有主观性 | Use a microphone connected to an oscilloscope or sound level sensor to locate the maximum amplitude objectively | 使用连接示波器或声级传感器的麦克风，客观确定最大振幅位置 |
| End correction not accounted for | 未考虑末端修正 | Use the difference (L₂ − L₁) to eliminate e, or plot the L vs 1/f graph and read the intercept | 利用(L₂ − L₁)差值消去 e，或绘制 L–1/f 图并读取截距 |
| Parallax error in reading the ruler | 读取刻度尺时产生视差 | Use a set-square to align the eye perpendicular to the ruler; use a travelling microscope for finer readings | 使用直角尺确保视线垂直于刻度尺，或使用读数显微镜获得更精确数值 |
| Frequency of tuning fork drifts or has tolerance | 音叉频率漂移或存在公差 | Use a signal generator with a digital frequency counter for higher accuracy | 使用带数字频率计的信号发生器以获得更高精度 |
| Temperature not constant during the experiment | 实验过程中温度不稳定 | Record temperature at the start and end, take the average; avoid breathing into the tube near the opening | 实验前后各记录一次温度并取平均值；避免对着管口呼吸 |
10. Safety Considerations | 安全注意事项
Although sound wave experiments are generally low-risk, the following precautions should be observed:
虽然声波实验整体风险较低，但仍应注意以下安全事项：
Avoid prolonged exposure to loud volumes from the loudspeaker, which may cause hearing damage. Use headphones or stand at a distance.
避免长时间暴露在扬声器的大音量下，以免损伤听力。可佩戴耳机或保持距离。
Glass tubes should be handled with care to avoid breakage; report any cracks immediately.
玻璃管需小心拿放，防止破裂；发现裂纹应立即报告。
Water reservoirs in the resonance tube setup should be stable and placed away from electrical equipment.
共鸣管装置中的储水瓶应放置稳固，并远离电气设备。
When using a signal generator, ensure that connections are correctly made and that the equipment is in good condition.
使用信号发生器时，确保连接正确且设备状态良好。
11. Worked Example | 例题分析
A tuning fork of frequency 512 Hz is used with a resonance tube. The first and second resonance positions are at 16.2 cm and 50.4 cm respectively. Calculate the speed of sound and the end correction.
使用频率为 512 Hz 的音叉配合共鸣管实验，第一和第二次共鸣位置分别在 16.2 cm 和 50.4 cm 处。求声速和末端修正。
Solution | 解答：
Wavelength:
波长：
λ = 2(L₂ − L₁) = 2 × (50.4 − 16.2) = 2 × 34.2 = 68.4 cm = 0.684 m
Speed of sound:
声速：
v = fλ = 512 × 0.684 = 350.2 m/s
End correction from first resonance:
由第一次共鸣求末端修正：
e = λ/4 − L₁ = 0.171 − 0.162 = 0.009 m = 0.9 cm
ntal value of v = 350 m/s is slightly higher than the accepted value at room temperature (~ ng warmer than 20°C, or systematic timing errors in locating the exact resonance position.
实验值 v = 350 m/s 略高于室温下的公认值（约 343 m/s）。这可能是因为空气温度高于 20°C，或在确定准确共鸣位置时存在系统误差。
12. Summary | 总结
Determining the wavelength and speed of sound is a core practical skill in A-Level Physics. The resonance tube method remains the most commonly examined experimental technique, requiring careful measurement of the first and second resonance positions. Kundt's tube provides visual evidence of standing waves, while oscilloscope-based methods offer greater precision and speed. Understanding the underlying equations, graph analyses, and error sources is essential for tackling both practical and theoretical exam questions.
测定声波波长与声速是 A-Level 物理的核心实验技能。共鸣管法仍是最常考查的实验技术，需要仔细测量第一、二次共鸣位置。孔脱管提供驻波的直观证据，而基于示波器的方法则精度更高、速度更快。理解基本方程、图像分析和误差来源，对于应对实操题和理论题都至关重要。
Whichever method is used, always record the temperature of the air, compare your experimental value with the theoretical expectation, and estimate the uncertainty in your final answer. This is the hallmark of a well-conducted physics experiment.
无论使用哪种方法，都要记录空气温度、将实验值与理论期望值比较，并估算最终答案的不确定度。这是高质量物理实验的标志。
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