4D1 Making and Hearing Sound | 4D1 声音的产生与听觉

📚 4D1 Making and Hearing Sound | 4D1 声音的产生与听觉

Understanding how sound is produced and perceived involves a rich interplay of physical principles and mathematical models. From the simple harmonic motion of a tuning fork to the complex waveforms decoded by the human ear, mathematics provides the essential language to describe vibrations, waves, frequency spectra, and the perception of loudness and pitch. This article explores the key mathematical concepts underpinning the making and hearing of sound, aligning with the 4D1 topic.

理解声音如何产生和感知，需要融合物理原理与数学模型。从音叉的简谐振动到人耳解码的复杂波形，数学为描述振动、波动、频谱以及响度与音调的感知提供了基本语言。本文探讨支撑声音产生与听觉的关键数学概念，对应 4D1 考点。

1. Simple Harmonic Motion: The Foundation of Sound | 简谐运动：声音的基础

Most sound sources, such as vocal cords or loudspeaker cones, vibrate to and fro. The simplest model of vibration is simple harmonic motion (SHM), where the displacement x from equilibrium follows x = A sin(ωt + φ). Here A is amplitude, ω is angular frequency (ω = 2πf), and φ is phase constant. The velocity and acceleration are given by v = dx/dt = ωA cos(ωt + φ) and a = d²x/dt² = –ω²x. SHM describes a pure tone, such as that generated by a tuning fork.

大多数声源，如声带或扬声器锥体，都会来回振动。最简单的振动模型是简谐运动（SHM），其离开平衡位置的位移 x 遵循 x = A sin(ωt + φ)。其中 A 为振幅，ω 为角频率（ω = 2πf），φ 为初相。速度和加速度分别为 v = dx/dt = ωA cos(ωt + φ) 和 a = d²x/dt² = –ω²x。简谐运动描述纯音，例如音叉产生的声音。

The period T = 1/f and the natural frequency of a mass-spring system is f = 1/(2π)√(k/m). In sound production, many vibrating systems are approximated as SHM for small amplitudes, and the restoring force is proportional to displacement. This linear relationship produces sinusoidal pressure variations in the air that our ears interpret as a steady pitch.

周期 T = 1/f，质量-弹簧系统的固有频率为 f = 1/(2π)√(k/m)。在声音产生中，许多振动系统在小振幅下可近似为简谐运动，回复力与位移成正比。这种线性关系在空气中产生正弦变化的压力，被我们的耳朵解读为稳定的音调。

2. Mathematical Description of a Sound Wave | 声波的数学描述

A sound wave travelling in one dimension can be expressed as a pressure variation or particle displacement. The general wave function is y(x,t) = A sin(kx – ωt + φ), where k = 2π/λ is the wave number, and the negative sign indicates propagation in the positive x-direction. The wave speed v is related by v = fλ = ω/k. In air at room temperature, sound speed is approximately 343 m/s.

一维传播的声波可以表示为压力变化或质点位移。一般波函数为 y(x,t) = A sin(kx – ωt + φ)，其中 k = 2π/λ 为波数，负号表示向正 x 方向传播。波速 v 满足 v = fλ = ω/k。在室温空气中，声速约 343 m/s。

The wave equation ∂²y/∂t² = v² ∂²y/∂x² governs the behaviour of small-amplitude sound waves. Any function of the form f(x ± vt) satisfies this equation. This linear partial differential equation allows superposition of solutions, which is crucial for analysing complex sounds.

波动方程 ∂²y/∂t² = v² ∂²y/∂x² 控制小幅声波的行为。任何形如 f(x ± vt) 的函数都满足该方程。这个线性偏微分方程允许解的叠加，这对分析复杂声音至关重要。

3. Frequency, Period, and Wavelength Relations | 频率、周期与波长的关系

The relationships between frequency f (Hz), period T (s), and wavelength λ (m) are fundamental: T = 1/f, λ = v/f. For a given medium, higher frequency corresponds to shorter wavelength. The human hearing range spans roughly 20 Hz to 20 000 Hz, meaning wavelengths from about 17 m to 1.7 cm in air. These numbers are derived directly from the wave equation.

频率 f（Hz）、周期 T（s）和波长 λ（m）之间的关系是基础：T = 1/f，λ = v/f。对于给定介质，频率越高波长越短。人耳的可听范围大约从 20 Hz 到 20 000 Hz，对应空气中的波长约为 17 m 到 1.7 cm。这些数值直接源于波动方程。

Pitch is the perceptual correlate of frequency. A mathematical understanding of frequency allows us to construct musical scales. In equal temperament, the frequency of a note a semitone higher is multiplied by 2^(1/12) ≈ 1.0595, ensuring a doubling of frequency every octave.

音高是频率的感知对应。对频率的数学理解使我们能够构建音阶。在十二平均律中，高一个半音的音符频率乘以 2^(1/12) ≈ 1.0595，确保每升高一个八度频率翻倍。

4. Superposition and Interference of Waves | 波的叠加与干涉

When two or more sound waves meet, they superpose linearly. The resultant displacement is the algebraic sum of individual displacements. Constructive interference occurs when waves are in phase, leading to maximum amplitude, while destructive interference occurs when they are out of phase by π radians, minimising amplitude. The principle of superposition is used in noise-cancelling headphones, where an inverted waveform is added to cancel ambient noise.

当两个或多个声波相遇时，它们线性叠加。合位移是各分位移的代数和。当波同相时发生相长干涉，振幅最大；当波相位相差 π 弧度时发生相消干涉，振幅最小。叠加原理用于降噪耳机，通过添加反相波形来消除环境噪声。

Beats are a common example of superposition. Two tones with slightly different frequencies f₁ and f₂ produce a resultant wave whose amplitude varies at the beat frequency f_beat = |f₁ – f₂|. This is modelled by the trigonometric identity: sin(2πf₁t) + sin(2πf₂t) = 2 cos(π(f₁–f₂)t) sin(π(f₁+f₂)t). The beat frequency is the magnitude of the difference.

拍是叠加的常见例子。两个频率略有不同的 f₁ 和 f₂ 的纯音产生的合成波，其振幅以拍频 f_beat = |f₁ – f₂| 变化。这可用三角恒等式描述：sin(2πf₁t) + sin(2πf₂t) = 2 cos(π(f₁–f₂)t) sin(π(f₁+f₂)t)。拍频即为两频率差值的绝对值。

5. Fourier Series: Decomposing Complex Sounds | 傅里叶级数：分解复杂声音

Real-world sounds are rarely pure sine waves. French mathematician Joseph Fourier showed that any periodic waveform can be expressed as a sum of sine and cosine terms: f(t) = a₀/2 + Σₙ₌₁∞ (aₙ cos(nωt) + bₙ sin(nωt)), where ω = 2π/T. This Fourier series decomposes a sound into its harmonic components. A clarinet and a violin playing the same note sound different because their Fourier coefficients differ, defining timbre.

现实中的声音很少是纯正弦波。法国数学家傅里叶证明，任何周期波形都可以表示为正弦和余弦项之和：f(t) = a₀/2 + Σₙ₌₁∞ (aₙ cos(nωt) + bₙ sin(nωt))，其中 ω = 2π/T。傅里叶级数将声音分解为各次谐波分量。单簧管和小提琴演奏同一音符听起来不同，是因为它们的傅里叶系数不同，从而决定了音色。

For a square wave of amplitude A, the series contains only odd harmonics: f(t) = (4A/π)(sin ωt + (1/3) sin 3ωt + (1/5) sin 5ωt + …). The relative strengths of harmonics give each instrument its unique voice. The human ear performs a biological Fourier analysis via the basilar membrane in the cochlea.

对于振幅为 A 的方波，级数仅含奇次谐波：f(t) = (4A/π)(sin ωt + (1/3) sin 3ωt + (1/5) sin 5ωt + …)。各次谐波的相对强度赋予每种乐器独特的音色。人耳通过耳蜗中的基底膜进行生物性的傅里叶分析。

6. Harmonics and Timbre: A Mathematical Signature | 谐波与音色：数学特征

A vibrating string or air column produces a fundamental frequency f₀ and integer multiples: 2f₀, 3f₀, … called harmonics or overtones. In a string fixed at both ends, the allowed wavelengths are λₙ = 2L/n, so fₙ = n(v/(2L)). The set of amplitudes {Aₙ} constitutes the spectrum. The timbre—the quality that distinguishes a piano from a trumpet—is entirely determined by this harmonic spectrum.

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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