4D3 Sound and Pitch | 4D3 声音与音高

📚 4D3 Sound and Pitch | 4D3 声音与音高

Sound is a physical wave that can be precisely described using mathematical functions. In Further Mathematics, the study of sound and pitch provides a rich context for applying trigonometric models, logarithms, Fourier series, and differential equations. This article explores the mathematical foundations behind pure tones, pitch perception, wave superposition, and the logarithmic scales that govern both musical tuning and loudness measurement.

声音是一种可以用数学函数精确描述的物理波动。在进阶数学中，对声音和音高的研究为三角函数建模、对数、傅里叶级数和微分方程提供了丰富的应用背景。本文探讨纯音、音高感知、波的叠加以及支配音乐调律和响度测量的对数标度背后的数学基础。

1. Mathematical Representation of Sound | 声音的数学表示

Any sound can be modelled as a pressure wave travelling through a medium. At a fixed point, the pressure variation over time is often approximated by a sinusoidal function for a pure tone. The simplest mathematical representation is y = A sin(2π f t + φ), where y is the displacement from equilibrium, A the amplitude, f the frequency in hertz, t time, and φ the phase angle. This model captures the periodic compression and rarefaction that characterise sound.

任何声音都可以建模为通过介质传播的压力波。在固定点上，纯音的压力随时间的变化通常用正弦函数近似。最简单的数学表示是 y = A sin(2π f t + φ)，其中 y 是偏离平衡位置的位移，A 是振幅，f 是以赫兹为单位的频率，t 是时间，φ 是相位角。这一模型刻画了声音特有的周期性压缩和稀疏。

2. Frequency, Period, and Angular Frequency | 频率、周期与角频率

The frequency f represents the number of cycles per second. Its reciprocal gives the period T, so T = 1/f. In many mathematical treatments, angular frequency ω = 2πf is used to simplify derivatives and integrals. A pure tone can then be written as y = A sin(ωt + φ). The choice of representation highlights the deep link between circular motion and oscillatory phenomena.

频率 f 表示每秒循环的次数。它的倒数给出周期 T，即 T = 1/f。在许多数学处理中，角频率 ω = 2πf 被用来简化导数和积分。纯音因而可以写成 y = A sin(ωt + φ)。这种表示的选择凸显了圆周运动与振荡现象之间的深层联系。

ω = 2πf, T = 1 / f

3. Pitch and Frequency: The Logarithmic Nature of Musical Scales | 音高与频率：音阶的对数本质

Pitch is the perceptual correlate of frequency. Western music divides the octave into twelve equal semitones, with a constant frequency ratio of 2^(1/12) between adjacent notes. Starting from A₄ = 440 Hz, the frequency of any note is given by f = 440 × 2^(n/12), where n is the number of semitones from the reference. This exponential relationship means pitch perception follows a logarithmic scale, making it an excellent application of indices and logarithms.

音高是频率的感知对应物。西方音乐将一个八度分成十二个相等的半音，相邻音符之间的频率比恒为 2^(1/12)。以 A₄ = 440 Hz 为基准，任意音符的频率为 f = 440 × 2^(n/12)，其中 n 是离开基准音的半音数。这种指数关系意味着音高感知遵循对数标度，这为指数和对数提供了一个极好的应用。

Semitone offset n	Note name	Frequency (Hz)
0	A₄	440.00
1	A♯₄ / B♭₄	466.16
12	A₅	880.00

4. Superposition of Sound Waves | 声波的叠加

Real sounds rarely consist of a single frequency. According to the principle of superposition, when two or more sound waves meet, the resultant displacement is the sum of the individual displacements. For two waves of the same frequency but different amplitude and phase, y₁ = A₁ sin(ωt + φ₁) and y₂ = A₂ sin(ωt + φ₂), the resultant can be expressed as a single sinusoid y = R sin(ωt + φ) using compound angle formulas.

真实的声音很少只包含单一频率。根据叠加原理，当两个或更多声波相遇时，合位移是各分位移之和。对于频率相同但振幅和相位不同的两个波 y₁ = A₁ sin(ωt + φ₁) 和 y₂ = A₂ sin(ωt + φ₂)，可以利用复合角公式将合振动表示为一个单一正弦波 y = R sin(ωt + φ)。

R = √(A₁² + A₂² + 2A₁A₂ cos(φ₁ – φ₂)), tan φ = (A₁ sin φ₁ + A₂ sin φ₂) / (A₁ cos φ₁ + A₂ cos φ₂)

5. Interference and Beats | 干涉与拍频

When two waves have slightly different frequencies f₁ and f₂, their superposition produces a phenomenon known as beats. Using the identity sin α + sin β = 2 sin((α+β)/2) cos((α-β)/2), the sum y = A sin(2π f₁ t) + A sin(2π f₂ t) can be rewritten as 2A cos(π (f₁-f₂) t) sin(π (f₁+f₂) t). The amplitude modulation has a frequency equal to half the difference, but the perceived beat frequency—the number of loudness maxima per second—is |f₁ – f₂|.

当两个波的频率 f₁ 和 f₂ 略有不同时，叠加会产生拍频现象。利用恒等式 sin α + sin β = 2 sin((α+β)/2) cos((α-β)/2)，可将 y = A sin(2π f₁ t) + A sin(2π f₂ t) 改写为 2A cos(π (f₁-f₂) t) sin(π (f₁+f₂) t)。振幅调制的频率为差频的一半，但感知到的拍频——每秒响度最大值的次数——是 |f₁ – f₂|。

Beat frequency = |f₁ – f₂|

6. Fourier Series and Harmonics | 傅里叶级数与谐波

A fundamental concept in Further Mathematics is that any periodic waveform can be decomposed into a sum of sine and cosine terms with frequencies that are integer multiples of a fundamental frequency. For a periodic sound wave with period T, the Fourier series representation is y(t) = a₀/2 + Σ [aₙ cos(2πnft) + bₙ sin(2πnft)], where f = 1/T. The coefficients aₙ and bₙ are determined by integration over one period. The term with n=1 is the fundamental, while n>1 terms are harmonics that determine the timbre of the sound.

进阶数学中的一个基本概念是，任何周期波形都可以分解为一组正弦和余弦项之和，其频率是基频的整数倍。对于周期为 T 的周期声波，傅里叶级数表示为 y(t) = a₀/2 + Σ [aₙ cos(2πnft) + bₙ sin(2πnft)]，其中 f = 1/T。系数 aₙ 和 bₙ 通过对一个周期积分求得。n=1 的项是基频，而 n>1 的项是决定声音音色的谐波。

aₙ = (2/T) ∫₀ᵀ y(t) cos(2πnft) dt, bₙ = (2/T) ∫₀ᵀ y(t) sin(2πnft) dt

7. The Decibel Scale and Logarithms | 分贝标度与对数

Human perception of loudness is approximately logarithmic. Sound intensity level is measured in decibels (dB) using L = 10 log₁₀(I / I₀), where I is the intensity in W m⁻² and I₀ = 10⁻¹² W m⁻² is the threshold of hearing. This logarithmic scale compresses a vast range of intensities. The algebra of logs becomes essential when combining sound sources or calculating attenuation over distance.

人类对响度的感知近似对数关系。声强级用分贝 (dB) 度量，公式为 L = 10 log₁₀(I / I₀)，其中 I 是强度（W m⁻²），I₀ = 10⁻¹² W m⁻² 是听觉阈值。这种对数标度压缩了极宽的强度范围。在合并声源或计算距离衰减时，对数的运算变得至关重要。

8. Wave Speed and the Relationship v = fλ | 波速与关系式 v = fλ

For a travelling sound wave, the speed v, frequency f, and wavelength λ are linked by v = fλ. In air at room temperature, v ≈ 343 m s⁻¹. This simple algebraic relationship allows the calculation of wavelengths for audible frequencies. For instance, a 440 Hz note has a wavelength of about 0.78 m. When sound passes from one medium to another, frequency remains constant while speed and wavelength change, providing a rich source of proportional reasoning problems.

对于行进的声波，速度 v、频率 f 和波长 λ 满足 v = fλ。在室温空气中，v ≈ 343 m s⁻¹。这个简单的代数关系可以计算可听频率的波长。例如，440 Hz 的音符波长约为 0.78 m。当声音从一种介质进入另一种介质时，频率保持不变，而速度和波长发生变化，这为比例推理问题提供了丰富的来源。

λ = v / f

9. The Doppler Effect and Rational Functions | 多普勒效应与有理函数

When a source or observer moves, the perceived frequency shifts. For a source moving with speed vₛ towards a stationary observer, the observed frequency is f’ = f × v / (v – vₛ). If the observer moves towards a stationary source, f’ = f × (v + vₒ) / v. These equations form rational functions of velocity and provide practice in algebraic manipulation, asymptotics, and the domain restrictions that vₛ < v to avoid infinite or negative frequencies.

当声源或观察者移动时，感知到的频率会发生偏移。对于以速度 vₛ 向静止观察者移动的声源，观测频率为 f’ = f × v / (v – vₛ)。如果观察者以速度 vₒ 向静止声源移动，f’ = f × (v + vₒ) / v。这些方程构成速度的有理函数，为代数运算、渐近行为以及避免频率无限或负数的定义域限制 vₛ < v 提供了练习。

10. Pythagoras, Equal Temperament, and Continued Fractions | 毕达哥拉斯调律、平均律与连分数

Historically, tuning systems relied on simple integer ratios. Pythagorean tuning uses the perfect fifth ratio 3:2. Stacking twelve fifths almost returns to the starting note after seven octaves, but (3/2)¹² ≠ 2⁷. This discrepancy is the Pythagorean comma, approximately 1.01364. Equal temperament resolves this by making the ratio of a semitone exactly 2^(1/12), an irrational number. The quest to approximate this irrational by rationals leads to continued fraction expansions, a beautiful intersection of number theory and music.

历史上，调律体系依赖于简单的整数比。毕达哥拉斯调律使用纯五度比 3:2。连续堆叠十二个五度几乎能在七个八度后回到起始音，但 (3/2)¹² ≠ 2⁷。这个差异就是毕氏音差，约为 1.01364。平均律通过将半音比精确设为 2^(1/12)（一个无理数）来解决这一问题。用有理数逼近这个无理数的探索引向了连分数展开，这是数论与音乐的一个优美交汇。

(3/2)¹² ≈ 129.746, 2⁷ = 128

11. Modelling Loudspeakers and Resonance | 扬声器与共振建模

The motion of a loudspeaker cone can be modelled by a driven damped harmonic oscillator: m x” + c x’ + k x = F₀ sin(ωt). Solving this second-order linear differential equation yields a particular integral whose amplitude depends on the driving frequency. The resonant frequency occurs where the amplitude is maximised, given by ω_r = √(k/m – c²/(2m²)). This requires techniques from Further Pure Mathematics, including complementary functions, particular integrals, and analysis of the amplitude-frequency graph.

扬声器锥盆的运动可以用受驱阻阻尼谐振子来建模：m x” + c x’ + k x = F₀ sin(ωt)。求解这个二阶线性微分方程可以得到一个特解，其振幅取决于驱动频率。振幅最大处的共振频率为 ω_r = √(k/m – c²/(2m²))。这需要用到进阶纯数学中的技巧，包括余函数、特解以及幅频图的分析。

12. Summary and Further Links | 总结与延伸链接

The mathematical study of sound and pitch weaves together trigonometry, sequences, logarithms, Fourier analysis, wave equations, and differential equations. These topics are central to the Further Mathematics syllabus, and sound offers an intuitive context to explore abstract concepts. Mastering these applications strengthens problem-solving skills and prepares for advanced study in engineering, acoustics, and signal processing.

声音与音高的数学研究将三角学、数列、对数、傅里叶分析、波动方程和微分方程融为一体。这些主题是进阶数学课程大纲的核心，而声音为探索抽象概念提供了一个直观的语境。掌握这些应用可以增强解决问题的能力，并为工程学、声学及信号处理等领域的深入研习做好准备。

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