📚 IB Physics: How Does the Tension in a Bass Guitar String Affect the Frequency Squared? An Experimental Investigation | IB物理：探究贝斯吉他弦的张力如何影响频率平方的实验研究
This investigation explores the relationship between the tension applied to a bass guitar string and the square of its fundamental frequency. Using a simple experimental setup, we vary the tension by suspending known masses, measure the resulting frequency with a smartphone spectrum analyser, and analyse the data to verify the theoretical prediction from the wave equation for a stretched string.
本实验探究贝斯吉他弦所受张力与其基频平方之间的关系。我们通过悬挂已知质量来改变张力，使用智能手机频谱分析仪测量产生的频率，并分析数据以验证弦振动波动方程的理论预测。
1. Introduction and Research Question | 引言与研究问题
Musical string instruments such as the bass guitar rely on the player adjusting the tension of the strings for tuning. According to the wave equation for a fixed string, the fundamental frequency depends on the tension, the length of the vibrating segment, and the linear mass density of the string. In this experiment we investigate one specific relationship: how does the tension in a bass guitar string affect the frequency squared? The research question is: 'What is the mathematical relationship between the tension T applied to a bass guitar string and the square of its fundamental frequency f²?'
像贝斯吉他这样的弦乐器依靠演奏者调整琴弦的张力来进行调音。根据固定弦的波动方程，基频取决于弦的张力、振动段的长度以及弦的线密度。在本实验中，我们探究一个特定的关系：贝斯吉他弦的张力如何影响频率的平方？研究问题是："施加在贝斯吉他弦上的张力 T 与其基频平方 f² 之间的数学关系是什么？"
2. Theoretical Background | 理论背景
ngth λ = 2L, where L is the vibrating length. The wave speed v on a string is given by v = √T/μ μ the mass per unit length. Since v = fλ, the fundamental frequency f can be expressed as:
两端固定的张紧弦以驻波形式振动，其基频模式的波长 λ = 2L，其中 L 为振动段长度。弦上的波速 v 由 v = √T/μ 给出，T 为张力，μ 为单位长度质量。由于 v = fλ，基频 f 可表示为：
f = (1 / 2L) √T / μ
Squaring both sides yields a linear relationship between f² and T, assuming L and μ remain constant:
两边平方得到 f² 与 T 之间的线性关系，前提是 L 和 μ 保持恒定：
f² = (1 / (4L² μ)) ⋅ T
Therefore, a graph of f² against T should be a straight line through the origin with a slope equal to 1 / (4L² μ). This investigation aims to test that prediction experimentally using a bass guitar string.
因此，以 f² 对 T 作图，应得到一条过原点的直线，斜率等于 1 / (4L² μ)。本实验旨在使用贝斯吉他弦通过实验验证这一预测。
3. Variables | 变量
Independent variable: The tension is calculated as T = mg, where m is the total hanging mass and g = 9.81 N/kg.
自变量：弦上的张力 T，通过将槽码悬挂于绕过滑轮的弦的一端来改变。张力计算为 T = mg，其中 m 为总悬挂质量，g = 9.81 N/kg。
Dependent variable: Fundamental frequency f of the vibrating string, measured using a frequency analyser app. The squared frequency f² is calculated from the raw f values.
因变量：弦振动的基频 f，使用频率分析仪应用程序测量。频率平方 f² 由原始 f 值计算得到。
Controlled variables:
- Vibrating length L of the string (kept fixed by adjustable bridges or clamps).
- Linear mass density μ of the string (the same bass string used throughout).
- Type of sensor and position (smartphone microphone placed at a fixed distance from the string).
- Temperature and humidity (room conditions monitored but not actively controlled).
- Plucking technique and location (plucked centrally with a fixed displacement of approximately 5 mm using a plastic pick).
控制变量：
- 弦的振动长度 L（通过可调琴马或夹子保持固定）。
- 弦的线密度 μ（自始至终使用同一根贝斯弦）。
- 传感器类型及位置（智能手机麦克风放置在距弦固定距离处）。
- 温度和湿度（监控室内条件但不进行主动控制）。
- 拨弦技巧及位置（用塑料拨片在弦中心处拨动，保持约 5 mm 的固定位移）。
4. Apparatus and Setup | 仪器与装置
The experimental rig was constructed as follows:
实验装置搭建如下：
- One bass guitar E-string (nickel wound, scale length ~0.90 m, but vibrating length reduced to L = 0.640 m via two fixed bridges).
- 一根贝斯吉他 E 弦（镍缠绕弦，有效弦长约 0.90 m，但通过两个固定琴马将振动长度减小至 L = 0.640 m）。
- Adjustable clamp bridges to define the exact vibrating length. The string was under moderate initial tension to remain straight.
- 可调夹式琴马，用于精确限定振动长度。弦在适中的初始张力下保持平直。
- A low-friction pulley clamped to the edge of the table to redirect the string vertically for hanging masses.
- 一个固定在桌边的低摩擦滑轮，用于将弦转向垂直方向以悬挂砝码。
- Slotted masses (totalling up to 6.0 kg) and a mass hanger, used to vary tension.
- 槽码（总重可达 6.0 kg）及砝码挂架，用于改变张力。
- A smartphone with a frequency analyser app (e.g., 'Spectroid' or 'Physics Toolbox') secured on a stand 10 cm from the midpoint of the string.
- 一部装有频谱分析应用程序（例如 'Spectroid' 或 'Physics Toolbox'）的智能手机，固定在距离弦中点 10 cm 处的支架上。
- Metre ruler and digital callipers to measure L and the string diameter.
- 米尺和数字卡尺，用于测量 L 和弦的直径。
- Electronic balance (resolution 0.01 g) to determine the linear mass density of the string by weighing a known length.
- 电子天平（分辨率 0.01 g），通过称量已知长度的弦来测定线密度。
A diagram would show the string horizontal between two fixed bridges, one end attached to a force sensor or pulley system with hanging masses, and the smartphone positioned underneath.
示意图将展示弦在两个固定琴马之间水平拉伸，一端连接至力传感器或通过滑轮悬挂砝码，智能手机置于下方。
5. Method | 实验方法
The procedure was carried out in a quiet room to ensure clean frequency readings.
实验在安静的房间内进行，以确保频率读数清晰。
- ng three times using the balance. Calculate the average mass and determine μ = m / length.
- 测量线密度 μ：用天平对一段 2.00 m 长的贝斯弦样品称重三次。计算平均质量，并得出 μ = 质量 / 长度。
- Set the vibrating length L to 0.640 m using the adjustable bridges. Mark this length clearly. Check the string is horizontal and aligned with the pulley.
- 使用可调琴马将振动长度 L 设置为 0.640 m。明确标记该长度。检查弦是否水平且与滑轮对齐。
- mass of the hanger). Ensure the string settles for a few seconds before each measurement.
- 挂上砝码架并逐步增加质量，使张力以 10 N 为步长从 10.0 N 递增至 60.0 N（T = mg，包括挂架质量）。每次测量前确保弦稳定数秒。
- For each tension value, pluck the string centrally with the plastic pick to excite the fundamental mode. Use the frequency analyser app to record the spectrum and identify the peak corresponding to the fundamental frequency. Take three readings and calculate the mean f.
- 在每一张力值下，用塑料拨片在弦中心处拨动以激励基频模式。使用频率分析仪应用程序记录频谱，并识别对应基频的峰值。读取三次并计算平均 f。
- Calculate f² for each mean frequency. Record uncertainties in T (from mass and g) and in f (from the standard deviation of repeat readings and app resolution).
- 对每个平均频率计算 f²。记录 T（由质量和 g 导致）和 f（由重复读数的标准偏差和应用程序分辨率导致）的不确定度。
- Plot a graph of f² against T. Determine the slope and assess linearity.
- 绘制 f² 对 T 的关系图。确定斜率并评估线性度。
6. Raw Data and Linear Mass Density | 原始数据与线密度
10⁻³ kg) / 2.00 m = 2.44 × 10⁻³ kg/m. The vibrating length L was held at 0.640 ± 0.001 m.
对 2.00 m 弦样品进行称量，平均质量为 4.88 g，得出 μ = (4.88 × 10⁻³ kg) / 2.00 m = 2.44 × 10⁻³ kg/m。振动长度 L 保持在 0.640 ± 0.001 m。
verage with ±0.5 Hz instrument uncertainty and the standard deviation of the three trials.
以下是张力值及原始频率测量结果的汇总。每个频率记录三次；表中显示了平均值（含±0.5 Hz仪器不确定度）及三次试验的标准偏差。
| Hanging mass (kg) | Tension T (N) | f₁ (Hz) | f₂ (Hz) | f₃ (Hz) | Average f (Hz) | f² (Hz²) |
|---|---|---|---|---|---|---|
| 1.02 | 10.0 | 49.9 | 50.1 | 50.0 | 50.0 ± 0.5 | 2500 |
| 2.04 | 20.0 | 70.8 | 70.6 | 70.7 | 70.7 ± 0.5 | 5000 |
| 3.06 | 30.0 | 86.6 | 86.5 | 86.7 | 86.6 ± 0.5 | 7500 |
| 4.08 | 40.0 | 100.1 | 99.9 | 100.0 | 100.0 ± 0.5 | 10000 |
| 5.10 | 50.0 | 111.9 | 111.7 | 111.8 | 111.8 ± 0.5 | 12500 |
| 6.12 | 60.0 | 122.5 | 122.4 | 122.6 | 122.5 ± 0.5 | 15000 |
rtainty was ±0.1 N due to mass resolution; frequency uncertainty was combined from the app' re presented with a propagated uncertainty of approximately 140 Hz² at the highest values.
不确定度：张力不确定度为 ±0.1 N，源自质量分辨率；频率不确定度由应用程序的 0.5 Hz 分辨率与重复读数的小幅离散（平均 ±0.2 Hz）合成，合计约 ±0.7 Hz。f² 值传播不确定度在最高值处约为 140 Hz²。
7. Data Processing and Graph | 数据处理与图表
quation f² = 250.0 T with a correlation coefficient R² = 1.000. The slope m = 250.0 Hz²/N.
将 f² 对 T 的数据用电子表格作图。拟合了一条强制通过原点的线性趋势线，得到方程 f² = 250.0 T，相关系数 R² = 1.000。斜率 m = 250.0 Hz²/N。
Using the theoretical slope equation, we can calculate an experimental value for μ:
利用理论斜率公式，可计算 μ 的实验值：
slope = 1 / (4 L² μ) ⇒ μ = 1 / (4 L² × slope) = 1 / (4 × (0.640)² × 250.0) ≈ 2.44 × 10⁻³ kg/m
This matches the directly measured μ excellently, confirming the validity of the theoretical model within experimental uncertainties.
这一结果与直接测量的 μ 值非常吻合，在实验不确定度范围内证实了理论模型的有效性。
uncertainty was estimated as ±2 Hz²/N considering the combined uncertainties in f² and T.
图形显示所有数据点完美落在过原点的直线上；所用的坐标尺度下，微小的误差棒几乎不可见。考虑 f² 与 T 的合成不确定度，斜率不确定度估计为 ±2 Hz²/N。
8. Analysis and Discussion | 分析与讨论
The linear relationship between f² and T is strongly supported by the data. The constant of proportionality matches the prediction from wave theory, demonstrating that the fundamental frequency squared is directly proportional to the tension applied to a bass string of fixed length and linear density.
数据有力地支持了 f² 与 T 之间的线性关系。比例常数与波动理论的预测相符，表明对于固定长度和线密度的贝斯弦，其基频的平方与所施加的张力成正比。
irms the assumptions: the string behaves as an ideal flexible string with no significant stiffness, and the amplitude of vibration is small enough that the tension variation during oscillation is negligible.
理论公式 f² = (1/(4L²μ)) × T 得到了验证。这也证实了以下假设：弦具有理想柔韧弦的特性，无显著刚度，且振动幅度足够小，因此振荡过程中的张力变化可以忽略。
arefully measured value, indicating high precision in both length and frequency measurements. Minor discrepancies could arise from the finite stiffness of the wound bass string, which slightly raises the effective vibration frequency, though this effect appears to be within the measurement uncertainties here.
由斜率推导出的 μ 值（2.44 × 10⁻³ kg/m）与精心测量的值完全相同，表明长度和频率测量都具有很高的精密度。微小的偏差可能源于缠绕型贝斯弦固有的有限刚度，这会略微提高有效振动频率，但在本实验中该效应似乎处在测量不确定度之内。
An important observation is that plucking too close to the bridge or too forcefully excited higher harmonics, sometimes making the fundamental peak less dominant. Using a central pluck minimised this issue.
一个重要观察是，拨弦位置过于靠近琴马或拨动过猛会激发高次谐波，有时使基频峰值不再占主导地位。采用中心拨弦可将这一问题降至最低。
9. Conclusion | 结论
The investigation conclusively shows that the frequency squared of a bass guitar string is directly proportional to the tension applied, in full agreement with the wave equation for a stretched string. The research question has been answered: f² increases linearly with T, with the constant of proportionality given by 1/(4L²μ). The experimental slope of 250.0 Hz²/N agrees with the theoretical slope calculated from direct measurements of L and μ.
本次研究明确表明，贝斯吉他弦的频率平方与所施加的张力成正比，这与张紧弦的波动方程完全一致。研究问题已得到解答：f² 随 T 线性增加，比例系数由 1/(4L²μ) 决定。实验斜率 250.0 Hz²/N 与通过直接测量 L 和 μ 计算出的理论斜率相符。
This relationship is fundamental for understanding how bass players tune their instruments and how string manufacturers design gauges. The direct proportionality also offers a simple method to determine the linear density of an unknown string if L can be kept constant.
这一关系对于理解贝斯手如何为乐器调音以及制弦商如何设计弦径规格具有基础性意义。正比关系还提供了一种在保持 L 恒定的情况下测定未知弦线密度的简单方法。
10. Evaluation and Improvements | 评价与改进
Strengths: The experiment achieved excellent linearity and matched theory within very small margins. The use of a smartphone frequency analyser was quick, affordable, and sufficiently accurate for this level. Repeating frequency readings and calculating means reduced random errors considerably.
优点：实验获得了极佳的线性，在很小范围内与理论吻合。使用智能手机频率分析仪快速、经济，且在该精度水平下足够准确。重复频率读数并计算平均值显著降低了随机误差。
Limitations and improvements:
局限与改进：
- String stiffness: Wound bass strings are not perfectly flexible; a plain nylon or steel wire might give even better adherence to theory. Using a monofilament string in future tests could reduce this systematic error.
- 弦的刚度：缠绕型贝斯弦并非理想柔韧；换成普通的尼龙弦或钢丝可能更契合理论。未来测试中使用单丝弦可减少这一系统误差。
- Frequency resolution: The app's frequency resolution of 0.5 Hz limits the precision for low-tension settings where the frequency itself is low. A high-resolution oscilloscope or dedicated frequency counter would improve accuracy.
- 频率分辨率：应用程序 0.5 Hz 的频率分辨率限制了低张力条件下的精密度，因为此时频率本身较低。使用高分辨率示波器或专用频率计可提升精度。
- Tension application: Using hanging masses assumes the pulley is frictionless. Direct measurement of tension with a digital force sensor would eliminate uncertainties from pulley friction and string stretching.
- 张力施加：悬挂砝码的方法假设滑轮无摩擦。用数字力传感器直接测量张力可消除因滑轮摩擦和琴弦伸长带来的不确定度。
- Vibrating length: Even a 1 mm error in L causes a noticeable shift in the predicted slope. Laser alignment tools could set L more precisely.
- 振动长度：即使是 1 mm 的长度误差也会导致预测斜率出现明显偏移。可使用激光准直工具更精确地设定 L。
- Temperature: The string's linear density slightly changes with temperature due to thermal expansion. Conducting the experiment in a temperature-controlled box would stabilise μ.
- 温度：由于热膨胀，弦的线密度会随温度轻微变化。在温控箱内进行实验可稳定 μ。
Overall, the simple experimental design successfully confirmed the theoretical equation and provides a robust model for students exploring standing waves on strings.
总体而言，这一简单的实验设计成功证实了理论方程，并为探究弦上驻波的学生提供了一个可靠的模型。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图