Sound and Acoustics: Sound Propagation
Sound Propagation in One Dimension
We begin by considering a longitudinal compression wave traveling in a fluid in the -direction. An infinitesimal element with initial dimensions and is shown in Figure 12.1. This fluid element will deform as the wave passes and a deformed configuration of the fluid element is shown highlighted in gray. Here represents the displacement of the fluid in the -direction while is the acoustic overpressure in the fluid (i.e. the pressure above the local atmospheric (equilibrium) pressure). At the left end of the deformed fluid element in Figure 12.1 the material has displaced an amount and is acted upon by a pressure . At the right end, the displacement and pressure are
Application of Newton’s law in the -direction results in
We assume that the pressure (above atmospheric) is related to the volumetric strain by
where is the bulk modulus of the fluid. Since the only motion of the fluid element is in the -direction, we see that
so that 12.2 becomes
and 12.1 becomes
This is the wave equation which governs the displacement of the fluid. To consider the response of the pressure, we begin by differentiating 12.1 with respect to
Using the result from 12.3 this becomes
which can be rearranged to give
Comparing 12.4 and 12.5 we see that both the displacement and the pressure satisfy the same wave equation. The wave speed in both cases is given by
This speed is known as the speed of sound in the medium.
Speed of Sound
Generally in acoustics we are concerned with the transmission of sound through air. Under most conditions of interest, air can be treated as an ideal gas so that
Here is the total pressure of the air given by
where is the atmospheric pressure and is the acoustic overpressure. in 12.7 is the absolute temperature and is the ideal gas constant which for air is
As we saw in 12.2, the change in pressure of a gas is related to the volumetric strain according to
Pressure fluctuations of this sort are an adiabatic (no heat transfer) process which for an ideal gas follow
where is the ratio of specific heats ( ). Differentiating 12.10 with respect to volume shows that
Comparing 12.9 and 12.11 we see that for an ideal gas the bulk modulus is given by
From 12.6 we know that the speed of sound is related to the bulk modulus and density by
which using 12.12 becomes
However, from the ideal gas law we know that
so that 12.13 becomes
With and , this gives
in m/s where is the absolute temperature.
For a solid, the bulk modulus of a material is
where is Young’s Modulus and is Poisson’s ratio. This represents how much a solid is compressed if acted upon by a hydrostatic stress state. Note that if , which is typical of many engineering materials, then the bulk modulus is approximately the same as Young’s Modulus. Table 12.1 shows the speed of sound in several media.
Frequency and Wavelength
The speed of sound is related to \itspos frequency and wavelength by
Human hearing is nominally within the range of 20 Hz to 20 kHz. For air at 15 °C, ( = 340 m/s) this corresponds to wavelengths ranging from 17 m to 17 mm.
Out of interest we can compare this to human vision. The visible light spectrum ( = 3 m/s) nominally corresponds to
Notice that for the visible light spectrum the ratio of the maximum frequency to the minimum frequency is less than two. For human hearing this ratio is about 1000.