Nature of Sound

Mechanical longitudinal pressure wave: what it is, how it propagates, and what characterises it.

Conceptual context

Sound is one of the most immediate physical experiences of everyday life, yet its nature is not always precisely understood. We hear, speak, and listen to music, but rarely ask what agitates the air between the source and our ear. This leaf answers that question from the ground up: sound is a mechanical disturbance that propagates through a material medium by transmitting energy from one layer to the next.

Unlike light, which can travel through a vacuum, sound requires a material medium to exist. There is no sound in outer space because there is no matter to transmit the oscillation. This dependence on the medium is, at the same time, the key that explains why sound travels at such different speeds in air, water, or steel, and why its properties change when it crosses from one medium to another.

The leaf on the nature of sound is the starting point of the entire sound block. Before discussing reflection, resonance, or the Doppler effect, it is essential to understand what propagates, how it propagates, and with which quantities that propagation is described.

🟢 Essential level

Imagine a column of air inside a tube. When a membrane vibrates at one end, it pushes the adjacent air layer, which compresses slightly. That compression pushes the next layer, and so on: the disturbance travels along the tube. Between compression and compression there is a zone of rarefaction, where the air is slightly less dense than at rest.

This is the mechanism of sound: a chain of compressions and rarefactions advancing in the same direction as the particle displacements. That is why sound is a longitudinal wave: the oscillation of particles is parallel to the propagation direction.

The quantities that characterize this phenomenon are acoustic pressure $p$ —the pressure variation from equilibrium—, frequency $f$ —how many oscillations occur per second—, wavelength $\lambda$ —the distance between two consecutive compressions— and propagation speed $v_s$, which depends on the medium, not on the source.

[!NOTE] The human ear detects pressure variations $p$ as small as 20 micropascals, equivalent to a particle displacement $\xi$ of the order of an atomic diameter. This illustrates the extraordinary sensitivity of the auditory system.

🔵 Formal level

Basic formulas

The most complete mathematical description of sound in its simplest case —a plane harmonic wave propagating in the positive x direction— is given by the acoustic pressure equation:

Plane harmonic pressure wave

where $p_0$ is the pressure amplitude, $f$ is the frequency, $\lambda$ is the wavelength, and $v_s$ is the propagation speed. The equation shows that the pressure at each point oscillates sinusoidally in time and varies sinusoidally in space. The argument of the cosine —called the phase— advances in time and decreases with distance, reflecting that the wavefront moves away from the source.

Real student questions

Why can sound not travel in a vacuum? Sound is a mechanical disturbance: it requires matter to compress and expand. Without particles to transmit the impulse from one layer to the next, there is no propagation mechanism. A bell vibrating in a vacuum emits no audible sound because there is no medium to carry the oscillation to the ear.

Why does sound travel faster in water than in air if water is denser? $v_s$ in a fluid depends on the ratio between the bulk modulus and the density. Water is far less compressible than air: its bulk modulus is about ten thousand times greater. Although its density is also about eight hundred times greater, the modulus "wins" and the speed is about four times higher than in air.

Why does $f$ not change when passing from one medium to another, but $\lambda$ does? $f$ is imposed by the source: it is the rate at which a loudspeaker membrane, for example, compresses and expands the fluid. That rate cannot change at the interface because if it did, particles at the boundary would receive more or fewer cycles than the source emits, violating continuity of motion. By contrast, $\lambda$ is the product of $v_s$ and the period, and since $v_s$ changes when the medium changes, $\lambda$ changes proportionally.

Final synthesis

Sound is a longitudinal mechanical wave: a succession of compressions and rarefactions propagating through matter, transporting energy without net mass transport. Its fundamental descriptive quantities —$p$, $f$, $\lambda$, $v_s$, and $I$— are precisely and verifiably related. $f$ belongs to the source; $v_s$ belongs to the medium; $\lambda$ results from both. $I$ grows as the square of $p_0$, which explains the logarithmic scale of sound perception and the utility of the decibel.