The Physics of Sound: Why Music Is Mathematics in Motion / 声音的物理学：音乐为何是运动中的数学

Chapter 1: The Physics of Sound — Why Music Is Mathematics in Motion

第 1 章：声音的物理学——音乐为何是运动中的数学

> Chapter Question / 本章核心问题 > What is the physical essence of an audible sound, and how can we describe it precisely in the language of mathematics? > 可听声音的物理本质是什么？我们如何用数学语言精确地描述它？

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Strike a tuning fork. Hold it to your ear. You hear a clear, unwavering tone — a single pitch, neither rising nor falling. Pick up a guitar and pluck a string. You hear something more complex: a fundamental note surrounded by a halo of brighter, quieter tones that give a guitar its characteristic "guitar-ness." Sing a vowel. Hum it. Whisper it. The same word can be a chord or a single line, depending on how your vocal cords vibrate.

/ 用音叉敲击一下，然后贴近耳朵。你听到一个清晰、稳定的音——一个单独的音高，不升不降。拿起吉他拨动一根弦，你听到更复杂的声音：一个基音，被一圈更明亮、更轻柔的泛音环绕，正是这些泛音赋予了吉他独特的"吉他感"。唱一个元音——大声唱、低声哼、轻声说——同一个词，可以是和弦，也可以是单线，取决于你的声带如何振动。

What is the difference between these three acts? At the level of physics, the difference is small. At the level of mathematics, the difference is enormous. And the bridge between those two statements — between the air pressure pushing against your eardrum and the equation on a mathematician's blackboard — is the subject of this chapter.

/ 这三种行为之间的差异是什么？在物理层面，差异很小。在数学层面，差异巨大。而这两种陈述之间的桥梁——耳膜所承受的气压与数学家黑板上的方程之间的桥梁——正是本章的主题。

The argument we will develop in the following pages is simple to state, but its consequences run through every chapter that follows:

/ 我们将在接下来的章节中展开的论证，表述简单，但其影响贯穿本书的每一章：

> Sound is not a thing. It is a pattern of pressure variations in a medium, and every audible pattern can be written as a sum of sine waves. > 声音不是一种"物"，而是介质中压力变化的模式，每一个可听的模式都可以写成若干正弦波之和。

If that sentence seems technical, set it aside for a moment. By the end of the chapter you will see exactly what it means, you will be able to compute it, and you will hear it with your own ears when you run a short program in Python. The single fact above is the foundation on which all of music theory — and all of music itself — is built.

/ 如果这句话显得过于技术性，请暂时搁置。在本章末尾，你将准确理解它的含义，能够亲自计算它，并能在运行一段简短的 Python 程序时用耳朵"听到"它。上面这一单一事实，就是所有音乐理论——以及音乐本身——所建立的基础。

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1.1 The Physical Nature of Sound: A Pressure Wave in Three Dimensions / 声音的物理本质：三维空间中的压力波

1.1.1 What is actually happening when you "hear" a sound / 当你"听到"一个声音时，究竟发生了什么

The most common misconception about sound is that it is a kind of substance that travels from a source to a listener. It is not. Sound is a *disturbance* that propagates through an already-existing medium — typically air, but also water, metal, wood, or even the bones of your skull.

/ 关于声音最常见的误解是：声音是从声源传到听者的某种"物质"。并非如此。声音是 *扰动*，它通过一种已经存在的介质传播——通常是空气，也可以是水、金属、木头，甚至你颅骨的骨骼。

When the cone of a loudspeaker pushes outward, it compresses the air molecules immediately in front of it. Those molecules push against their neighbors, which push against theirs, and so on — a chain reaction of compressions that travels outward at approximately 343 meters per second in air at 20 °C. When the cone pulls back inward, it leaves a region of lower pressure (a *rarefaction*), and that low-pressure region also propagates outward. What reaches your ear is not a chunk of air from the speaker, but a *pattern* of high- and low-pressure regions traveling through the air that was already there.

/ 当扬声器的锥盆向外推时，它压缩紧贴前方的空气分子。这些分子又挤压它们的邻居，邻居再挤压下一层，以此类推——一连串的压缩以大约 343 米/秒（20°C 的空气中）的速度向外传播。当锥盆向内回拉时，会留下一个低压区（即 *稀疏*），这个低压区也会向外传播。到达你耳朵的并不是从扬声器飞来的一团空气，而是穿过原本就存在于那里的空气而传播的一个 *高/低压交替的模式*。

This is what physicists mean by a *longitudinal mechanical wave*: a wave in which the medium's particles oscillate in the same direction as the wave's propagation, and which requires a material medium to exist. In a vacuum, there is no sound. (This is not a metaphor: in the silent vacuum of space, no amount of shouting would reach another astronaut across the cabin — they would have to touch helmets, or use radio, which is an electromagnetic wave, a fundamentally different phenomenon.)

/ 这就是物理学家所说的 *纵波*：介质粒子沿波传播方向振荡的机械波，并且需要物质介质才能存在。在真空中，没有声音。（这不是比喻：在太空的寂静真空中，无论你怎么喊叫，对面的宇航员都不会听见——他们必须让头盔相碰，或者使用无线电，而无线电是电磁波，是本质上完全不同的现象。）

1.1.2 The three numbers that describe any sound / 描述任何声音的三个数字

Although the air pressure at your eardrum is a continuous function of time, every audible sound can be described — to an astonishing degree of precision — by just three physical parameters, and combinations of them. These parameters are the alphabet of acoustics, and they deserve careful attention.

/ 尽管你耳膜处的气压是时间的连续函数，但每一种可听的声音都可以用——以惊人的精度——仅三个物理参数及其组合来描述。这些参数是声学的字母表，值得我们仔细关注。

1. Frequency $f$ (measured in hertz, Hz). Frequency is the number of complete pressure cycles per second. One hertz means one cycle per second; 440 Hz means 440 cycles per second. The standard tuning pitch, the A above middle C, is defined to be 440 Hz. Frequency is what your brain ultimately interprets as *pitch* — how "high" or "low" a note sounds.

/ 1. 频率 $f$（单位：赫兹 Hz）。 频率是每秒完整压力循环的次数。1 赫兹即每秒一个循环；440 赫兹即每秒 440 个循环。标准调音音高（中 C 上方的 A）被定义为 440 Hz。频率是大脑最终解读为 *音高* 的物理量——音符听起来"高"或"低"的程度。

2. Amplitude $A$ (or equivalently, sound pressure level, measured in decibels, dB). Amplitude is the maximum deviation of the pressure from its average value. Doubling the amplitude does not double the perceived loudness; human loudness perception is roughly logarithmic, which is why sound engineers use the decibel scale: $$\text{SPL (dB)} = 20 \log_{10}\left(\frac{p}{p_0}\right)$$ where $p_0 = 20\,\mu\text{Pa}$ is the threshold of human hearing. A 10 dB increase is perceived as roughly a doubling of loudness. A jet engine at 30 m is around 140 dB; normal conversation is about 60 dB; a quiet library is around 30 dB.

/ 2. 振幅 $A$（或等价的声压级，单位：分贝 dB）。 振幅是压力相对于其平均值的最大偏移量。振幅翻倍并不会让"响度感觉"翻倍；人耳的响度感知大致是对数的，这也是音响工程师使用分贝标度的原因： $$\text{SPL (dB)} = 20 \log_{10}\left(\frac{p}{p_0}\right)$$ 其中 $p_0 = 20\,\mu\text{Pa}$ 是人耳的听阈。增加 10 dB 大致被感知为响度翻倍。30 米外的喷气发动机约 140 dB；正常交谈约 60 dB；安静的图书馆约 30 dB。

3. Phase $\varphi$ (measured in radians or degrees). Phase describes *where in its cycle* a wave is at a chosen reference time. Two waves with identical frequency and amplitude can add constructively (pressure peaks align) or destructively (peak aligns with trough, producing silence), depending solely on their phase relationship. Phase is the parameter you will not hear as directly as pitch or loudness — but it is essential for understanding timbre, interference, and the geometry of resonance, as we shall see.

/ 3. 相位 $\varphi$（单位：弧度或度）。 相位描述波在某一选定参考时刻 *处于循环的哪个位置*。两个频率、振幅完全相同的波，根据它们的相位关系，可以相加（压力峰对齐，产生建设性干涉）也可以相消（峰与谷对齐，产生寂静）。相位是这三个参数中你不会像音高和响度那样直接"听到"的——但它对理解音色、干涉和谐振的几何至关重要，我们稍后会看到。

A fourth, derived quantity is also worth knowing: wavelength $\lambda$, related to frequency by $\lambda = v / f$, where $v$ is the speed of sound in the medium. For a 440 Hz tone in air, $\lambda \approx 343/440 \approx 0.78$ m, about the length of a guitar. For a low note at 55 Hz (the A below the bass clef), $\lambda \approx 6.2$ m, the length of a small room. Wavelength is not an independent parameter — it follows from $f$ and $v$ — but it is the parameter that determines how a wave interacts with physical objects and enclosures, and it is the key to understanding resonance in the next section.

/ 还有一个派生的物理量值得了解：波长 $\lambda$，由 $\lambda = v / f$ 与频率相关，其中 $v$ 是介质中的声速。在空气中，440 Hz 音的 $\lambda \approx 343/440 \approx 0.78$ 米，约等于一把吉他的长度。对于低音 55 Hz（低音谱号下方的 A），$\lambda \approx 6.2$ 米，约等于一个小房间的长度。波长不是独立参数——它由 $f$ 和 $v$ 决定——但它决定了波如何与物理物体和封闭空间相互作用，也是理解下一节中谐振的关键。

1.1.3 The human hearing range: why 20 Hz to 20 kHz / 人耳听觉范围：为什么是 20 Hz 到 20 kHz

A healthy young human ear can detect pressure waves from approximately 20 Hz to 20,000 Hz (20 kHz). Below 20 Hz we feel the vibration more than we hear it; above 20 kHz we hear nothing at all, although dogs, cats, and bats can. The upper limit declines with age — a 60-year-old may hear nothing above 12 kHz — which is why the marketing of "high-resolution" audio above 20