The Physics of Sound: Waves, Frequency, and Amplitude

Sound is all around us, from the melody of a favorite song to the cacophony of a bustling city street. But what exactly is sound, and how can we understand its properties using the language of mathematics? This chapter delves into the physics of sound, treating it as a mechanical wave that propagates through a medium. We will explore the fundamental parameters that define sound waves—frequency, amplitude, and waveform—and see how these correspond to the musical concepts of pitch, loudness, and timbre. By modeling sound with mathematical functions, we lay the groundwork for understanding the intricate relationship between physics and music.

Sound as a Mechanical Wave

At its core, sound is a vibration that travels through matter as a wave. Unlike light, which can propagate through a vacuum, sound requires a medium—such as air, water, or solid materials—to travel. This is because sound waves are mechanical waves: they rely on the collision and transfer of energy between particles in the medium. When an object vibrates, it disturbs the surrounding particles, creating regions of compression (where particles are pushed together) and rarefaction (where particles are pulled apart). These alternating pressure variations propagate outward from the source in the form of a longitudinal wave, meaning that the particle displacement is parallel to the direction of wave travel. Imagine a slinky stretched out on a table; if you push and pull one end, you create a wave that travels along the slinky, with coils bunching up and spreading apart—that’s analogous to how sound moves through air.

The speed of sound depends on the properties of the medium. In dry air at 20°C (68°F), sound travels at approximately 343 meters per second (about 767 mph). However, this speed increases with temperature and is generally faster in liquids and solids due to their higher density and elasticity. For example, sound travels about 1,480 m/s in water and 5,960 m/s in steel. This variation explains why you can hear an approaching train earlier by placing your ear on the rail track than by listening through the air—the solid metal transmits sound more efficiently.

A sound wave can be described by several key parameters: wavelength (λ), frequency (f), amplitude (A), and speed (v). These are related by the fundamental equation:

$$v = f \lambda$$

where *v* is the wave speed, *f* is the frequency (number of cycles per second, measured in hertz, Hz), and *λ* is the wavelength (distance between successive compressions or rarefactions). This relationship holds for all waves and is crucial for understanding how sound behaves in different environments. For instance, if you double the frequency, the wavelength halves, provided the speed remains constant.

Frequency and Pitch

Frequency is perhaps the most intuitive property of sound because it directly corresponds to the musical concept of pitch. A higher frequency produces a higher pitch, while a lower frequency yields a lower pitch. The human ear is sensitive to frequencies roughly between 20 Hz and 20,000 Hz (20 kHz), though this range shrinks with age and exposure to loud noises. Sounds below 20 Hz are called infrasound, and those above 20 kHz are ultrasound; both are inaudible to humans but can be perceived by other animals or detected with technology.

In music, the standard reference pitch is A4 = 440 Hz, meaning the A above middle C vibrates 440 times per second. This convention, adopted in the 20th century, allows instruments to tune consistently. But why do we perceive certain frequency ratios as harmonious? The answer lies in the simplicity of the ratio. When two frequencies have a simple integer ratio—such as 2:1 (octave), 3:2 (perfect fifth), or 4:3 (perfect fourth)—the combined waveform repeats regularly, creating a sense of consonance. For example, if one note is 440 Hz and another is 660 Hz (ratio 3:2), the combined sound wave has a fundamental period that is the least common multiple of their periods, resulting in a repeating pattern that our brains interpret as pleasant. More complex ratios, like 45:32 (tritone), produce dissonance because the waveform is less regular.

This mathematical relationship is the foundation of musical tuning systems. The ancient Greeks discovered that dividing a string into simple fractional lengths produced harmonious intervals. If a string of length *L* is stopped at half its length, the vibrating segment produces a note an octave higher (frequency doubled). Stopping at one-third yields a perfect fifth above the original (frequency multiplied by 3/2). These principles were formalized by Pythagoras and his followers, leading to the Pythagorean tuning based on pure fifths. However, such systems introduce small discrepancies when stacking intervals, which led to the development of equal temperament—a system where the octave is divided into 12 equal semitones, each with a frequency ratio of $2^{1/12} \approx 1.05946$. This compromise allows instruments to play in any key with acceptable harmony, at the cost of making intervals slightly impure compared to pure ratios.

To illustrate, consider the frequencies of the C major scale in equal temperament, based on A4 = 440 Hz:

• C4 (middle C): 261.63 Hz
• D4: 293.66 Hz
• E4: 329.63 Hz
• F4: 349.23 Hz
• G4: 392.00 Hz
• A4: 440.00 Hz
• B4: 493.88 Hz
• C5: 523.25 Hz

Notice that the ratio of C5 to C4 is exactly 2:1 (523.25/261.63 ≈ 2), confirming the octave. But the ratio of E4 to C4 is 329.63/261.63 ≈ 1.2599, which is close to the pure major third ratio of 5:4 = 1.25 but not exact. This tiny difference is the "tempering" that makes all keys usable.

Amplitude and Loudness

If frequency determines pitch, amplitude determines loudness. Amplitude is the maximum displacement of particles from their equilibrium position in a wave, or equivalently, the maximum pressure variation in a sound wave. Greater amplitude means more energy is carried by the wave, and we perceive this as a louder sound. However, the relationship between amplitude and perceived loudness is not linear; it is logarithmic. Our ears respond to sound pressure level (SPL) on a logarithmic scale measured in decibels (dB). The decibel scale is defined relative to a reference intensity, typically the threshold of human hearing at 1,000 Hz, which is $I_0 = 10^{-12} \text{ W/m}^2$. The sound level *L* in decibels is given by:

$$L = 10 \log_{10}\left(\frac{I}{I_0}\right)$$

where *I* is the sound intensity (power per unit area). Since intensity is proportional to the square of amplitude, a tenfold increase in intensity corresponds to a 10 dB increase, and a doubling of amplitude (which quadruples intensity) yields about a 6 dB increase.

Common sound levels illustrate this scale:

• Threshold of hearing: 0 dB
• Whisper: 30 dB
• Normal conversation: 60 dB
• Busy traffic: 85 dB
• Rock concert: 120 dB
• Threshold of pain: 130–140 dB

It’s important to note that loudness perception also depends on frequency. The human ear is most sensitive to frequencies around 2–5 kHz, so a 40 dB tone at 1,000 Hz may sound as loud as a 50 dB tone at 100 Hz. This is accounted for by weighting curves like A-weighting (dBA), which adjusts measurements to match human sensitivity.

In music, dynamics are indicated by terms like *piano* (soft), *forte* (loud), and their gradations (*mezzo-forte*, *crescendo*). These correspond to amplitude changes, but performers must consider that a louder note on a violin, for example, may also involve more bow pressure and speed, affecting the waveform beyond simple amplitude scaling.

Waveform and Timbre

Two sounds can have the same pitch and loudness but still sound different—think of a piano and a violin playing the same note at the same volume. This distinct quality is called timbre (or tone color), and it is determined by the waveform, specifically the harmonic content of the sound.

A pure tone, such as that from a tuning fork or a high-quality electronic oscillator, is a