📚 How Does Tension in a Bass Guitar String Affect the Frequency Squared? | 贝斯吉他弦的张力如何影响频率的平方？
When a bass guitarist plucks a string, the pitch produced depends on several physical properties: the string’s length, its mass per unit length, and the tension to which it is tuned. This article explores the relationship between tension and the frequency of vibration, specifically examining why the square of the frequency is directly proportional to the tension. We derive the wave equation for a stretched string, connect it to standing waves, and outline how one can experimentally verify the linear relationship between frequency squared and tension.
当贝斯手拨动琴弦时，发出的音高取决于几个物理特性：弦的长度、单位长度的质量以及调音时所施加的张力。本文探讨张力与振动频率之间的关系，特别是解释为什么频率的平方与张力成正比。我们将推导张紧弦的波动方程，将其与驻波联系起来，并概述如何通过实验验证频率平方与张力之间的线性关系。
1. Standing Waves on a Stretched String | 张紧弦上的驻波
A plucked bass string vibrates in a standing wave pattern, with nodes at the fixed ends (bridge and nut) and an antinode near the centre for the fundamental frequency. The string’s motion can be described as a superposition of normal modes, each with a wavelength determined by the string length L. For the fundamental mode, the wavelength λ is twice the length: λ = 2L. Understanding this boundary condition is the first step in linking tension to frequency.
被拨动的贝斯弦以驻波模式振动，其两端（琴桥和琴枕）为节点，基频的波腹大致在中心位置。弦的运动可描述为各简正模式的叠加，每个模式的波长由弦长 L 决定。对于基频模式，波长 λ 等于弦长的两倍：λ = 2L。理解这一边界条件是建立张力与频率关系的第一步。
2. Wave Speed on a String | 弦上的波速
The speed v of a transverse wave travelling along a perfectly flexible string depends on the tension T and the linear density μ (mass per unit length). By applying Newton’s second law to a small element of the string, we obtain the fundamental relationship:
v = √(T / μ)
This equation shows that increasing the tension raises the wave speed, while a heavier string (larger μ) slows it down. For a bass guitar, the thicker strings have higher μ, requiring greater tension to produce low-frequency notes.
横波在完全柔韧的弦上传播的速度 v 取决于张力 T 和线密度 μ（单位长度的质量）。对弦的一小段应用牛顿第二定律，可得到基本关系式：
v = √(T / μ)
该公式表明，增大张力会提升波速，而更粗重的弦（μ 较大）则会使波速变慢。对贝斯而言，较粗的弦具有较大的 μ，因此需要更大的张力才能发出低频音符。
3. Linking Frequency to Wave Speed and Wavelength | 频率与波速和波长的关联
For any periodic wave, the frequency f is related to wave speed v and wavelength λ by the universal wave equation:
v = f λ
Combining this with the standing-wave condition for the fundamental mode (λ = 2L) and the expression for wave speed gives:
f = v / (2L) = (1 / 2L) √(T / μ)
This is the Mersenne’s law for a vibrating string. It tells us that frequency increases with the square root of tension, decreases with length, and decreases with the square root of linear density.
对于任何周期性波，频率 f 与波速 v 和波长 λ 通过通用波动方程关联：
v = f λ
将此式与基频驻波条件（λ = 2L）以及波速表达式相结合，可得：
f = v / (2L) = (1 / 2L) √(T / μ)
这就是弦振动的梅森定律。它告诉我们，频率与张力的平方根成正比，与弦长成反比，并与线密度的平方根成反比。
4. Why Frequency Squared Matters: Linearising the Relationship | 为什么关注频率平方：将关系线性化
The raw relationship f ∝ √T is not linear, which makes it difficult to analyse directly. By squaring both sides of Mersenne’s law, we obtain a linear equation:
f² = (1 / (4L² μ)) × T
Since L and μ are constants for a given string setup, f² is directly proportional to T. Plotting f² against T yields a straight line through the origin, whose gradient equals 1/(4L²μ). This linearisation allows experimental verification and easy extraction of the string’s linear density or validation of the theoretical model.
原始的 f ∝ √T 关系并非线性，这使得直接分析较为困难。将梅森定律两边平方后，我们得到一个线性方程：
f² = (1 / (4L² μ)) × T
由于对于给定的琴弦设置，L 和 μ 均为常数，因此 f² 与 T 成正比。绘制 f² 对 T 的图像将得到一条通过原点的直线，其斜率等于 1/(4L²μ)。这种线性化方法使得实验验证成为可能，也便于提取弦的线密度或检验理论模型。
5. Experimental Setup for Measuring Tension and Frequency | 测量张力与频率的实验装置
To investigate how tension affects frequency squared, one can mount a bass string over two fixed bridges (defining L) and attach one end to a force meter or hang known masses over a pulley. Plucking the string produces a sound whose fundamental frequency can be measured using a smartphone frequency analyser app or an oscilloscope connected to a magnetic pickup. Tension is varied by changing the hanging mass, and for each tension value the frequency is recorded several times to reduce random error.
为了探究张力如何影响频率平方，可以将贝斯弦安装于两个固定琴码之间（从而确定 L），一端连接测力计，或通过滑轮悬挂已知质量的重物。拨动琴弦会产生声音，其基频可通过智能手机频谱分析软件或连接磁拾音器的示波器来测量。通过改变悬挂质量来改变张力，对于每个张力值多次记录频率以减小随机误差。
6. Data Collection and Calculation | 数据收集与计算
For each added mass m, the tension is T = mg (g = 9.81 m s⁻²). The fundamental frequency is obtained from the spectrum by identifying the lowest strong peak. After recording f, compute f². An organised data table helps to identify patterns and is often laid out as follows:
对于每一悬挂质量 m，张力为 T = mg（g = 9.81 m s⁻²）。基频通过频谱中最低的强峰来识别。记录 f 后，计算 f²。一张有序的数据表有助于发现规律，其通常格式如下：
| Mass m / kg | Tension T / N | Frequency f / Hz | f² / Hz² |
|---|---|---|---|
| 0.50 | 4.91 | 41.2 | 1697 |
| 1.00 | 9.81 | 58.3 | 3399 |
| 1.50 | 14.72 | 71.4 | 5098 |
| 2.00 | 19.62 | 82.5 | 6806 |
These hypothetical values illustrate a clear quadratic growth of f with T, meaning f² grows linearly with T.
这些假设数据表明 f 随 T 呈显著的二次方增长，即 f² 随 T 线性增长。
7. Graph Plotting and Interpretation | 绘图与解释
A graph of f² (y-axis) against T (x-axis) is plotted. Based on the equation f² = (1/(4L²μ)) T, we expect a straight line passing through the origin. The gradient of the best-fit line can be used to calculate the linear density μ of the string if L is known, or to confirm the consistency of the experimental setup. Any deviation from linearity could indicate that the string is not perfectly flexible, that the tension is not uniform, or that the amplitude of vibration is too large, introducing non-linear effects.
绘制 f²（纵轴）对 T（横轴）的图。根据公式 f² = (1/(4L²μ)) T，我们预期得到一条通过原点的直线。最佳拟合线的斜率可用于计算弦的线密度 μ（已知 L 的情况下），或用以确认实验设置的一致性。任何偏离线性的现象都可能表明弦并非完全柔韧、张力不均匀，或振幅过大而引入了非线性效应。
8. Error Analysis and Improvements | 误差分析与改进
Common sources of error include friction at the bridge and pulley, inaccurate measurement of L, and difficulty in determining the exact fundamental frequency due to overtones. Using a magnetic pickup directly connected to an oscilloscope improves frequency resolution. The string should be plucked gently and consistently near the centre to favour the fundamental mode. Repeating measurements and applying the method of least squares for the gradient provides more reliable results.
常见的误差来源包括琴码和滑轮处的摩擦、L 的测量不准确，以及由于泛音存在而难以精确确定基频。使用直接连接示波器的磁拾音器可以提高频率分辨率。应在弦的中间位置轻柔且一致地拨动，以利于产生基频模式。重复测量并对斜率应用最小二乘法能提供更可靠的结果。
9. Connecting to the Bass Guitar in Practice | 联系实际贝斯吉他演奏
When a bassist tunes their instrument, they adjust the tension of each string using the tuning pegs. Tightening a string increases T, raising the pitch. The relationship f² ∝ T means that a small increase in tension results in a noticeable rise in pitch, particularly for thinner strings already under high tension. Understanding this physics helps players anticipate how much to turn the peg to achieve the desired note, and explains why changing string gauge (μ) also demands a new tension setting for the same pitch.
当贝斯手调音时，他们通过弦钮调节每根弦的张力。拧紧琴弦会增大 T，从而提高音高。f² ∝ T 的关系意味着张力的微小增加会引起音高的明显变化，特别是对于已处于高张力下的细弦。理解这一物理原理有助于乐手预估需要转动弦钮的幅度以达到目标音符，也解释了为什么更换不同规格的琴弦（改变 μ）后，要获得相同的音高就必须重新调整张力设定。
10. Extending the Idea: Harmonics and Overtones | 拓展概念：谐波与泛音
While the fundamental frequency dominates what we hear as the note, a bass string also vibrates in higher harmonics. The frequency of the nth harmonic is:
fₙ = n f₁ = (n / 2L) √(T / μ)
Thus, fₙ² = n² f₁², and the same linear dependence on T holds. Exploring harmonics reinforces that the relationship between tension and frequency squared applies across all normal modes, forming the basis for the instrument’s rich timbre.
尽管基频主导了我们所听到的音符，但贝斯弦也会在更高的谐波处振动。第 n 次谐波的频率为：
fₙ = n f₁ = (n / 2L) √(T / μ)
因此，fₙ² = n² f₁²，与张力 T 同样保持线性关系。对谐波的探索进一步强化了张力与频率平方之间的关系适用于所有简正模式这一事实，也为乐器丰富的音色奠定了基础。
Published by TutorHao | IB Physics Revision Series | aleveler.com
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