Longitudinal waves (sound waves)

A sound wave is disturbance consisting of a succession of compressions and rarefactions traveling through a material medium. Sound waves can be one-dimensional, two-dimensional and three-dimensional. In this page we will focus on the study of one-dimensional sound waves.

In order to see how a sound wave is produced and how it is described mathematically, we will consider the situation represented in the figure below. A cylinder of infinite length with a movable piston at one end contains a gas. Initially the gas is at equilibrium at pressure p0.

As the piston moves in and out of the cylinder undergoing simple harmonic motion of amplitude sm as shown in red in the figure, the gas particles will undergo simple harmonic motion as well, and therefore there will be inside the cylinder certain zones where pressure will be higher than p0, called compressions (in dark blue) and others where pressure will be lower than p0, called rarefactions (in light blue). These zones will go back and forth from high pressure to low pressure. In the figure is represented the state of the gas for a fixed instant of time.

The wave thus generated is harmonic because it is produced by a simple harmonic oscillation, and longitudinal because the gas particles oscillate from left to right (along the x-axis) parallel to the direction of wave propagation.

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Mathematically, the wave function describing such waves can be written in two different ways: as the displacement of a small volume element inside the gas with respect to its equilibrium position or as a function describing the pressure variation inside the cylinder.

The function describing the displacement s of a small volume element with respect to its equilibrium position is formally identical to the one describing a transverse harmonic wave :

Where ω is the angular frequency of the wave and k its wavenumber and whose values are respectively:

T(s) is the period of the harmonic wave, defined as the time it takes for a volume element to complete one cycle,

λ (m) is the wavelength of the harmonic wave, defined as the distance between two successive compressions or rarefactions,

sm is the amplitude of the sound wave: is the maximum displacement of the volume element from its equilibrium position.

In terms of the pressure, the wave function of the sound wave is given by:

where Δpm is the maximum variation of pressure above its equilibrium value p0.

Both wave functions have a phase difference of π/2. That means that, when the displacement of a volume element is zero, its pressure variation reaches its maximum. This can be understood by analyzing the particle displacement inside the volume element compared to pressure differences.

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In the figure below the compressions and rarefactions of the gas are shown. You can also see the motion state of the gas particles located at points where the pressure is p0.

As you can see, in zones where pressure is p0, the average particle displacement is maximum (in absolute value) because all particles are moving in the same direction. On the opposite hand, within rarefactions (R) and compressions (C) the average displacement is zero because positive and negative displacements are cancelled out.

In the figure below both wave functions are plotted:

The relationship between the amplitude of both functions is given by:

where ρ is the gas density and v is the wave speed:

The speed of sound in air is about 340 meters per second.

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