Speed of Sound

The speed of sound is the same in an acoustic tube as in the open air. In this section, we will explain some formulae relating to this quantity.

When sound propagates in an acoustic tube, there are two key quantities we need to keep track of:

• the pressure in the tube $p(x,t)$ at a given point $x$ along the tube's length and time $t$, measured in Pa or N/m, and
• the volume velocity in the tube $u(x,t)$ at the same point $x$ along the tube's length and time $t$, measured in $\mathrm{m}^3$/s.

It may be shown that the two quantities above obey the wave equation, and thus each quantity may be decomposed into right-traveling and left-traveling wave components as

$$p(x,t) = p^+\left(x - \frac{c}{t}\right) + p^-\left(x + \frac{c}{t}\right)$$ (1)

and

$$u(x,t) = u^+\left(x - \frac{c}{t}\right) + u^-\left(x + \frac{c}{t}\right),$$ (2)

as discussed in the traveling waves laboratory assignment. What the equations above state is that any pressure and volume velocity functions that may realistically exist in the tube can each be decomposed into a pair of simpler functions, one a waveform traveling to the right (e.g., $p^+(x - c/t)$), and the other a waveform traveling to the left (e.g., $p^-(x + c/t)$). The speed at which these waveforms travel, $c$, is the speed of sound in the tube.

As mentioned earlier, the speed of sound $c_\mathrm{air}$ in an air-filled acoustic tube is the same as the speed of sound in the open air. As a result, the radius of the tube has no effect on $c_\mathrm{air}$. In order to derive a formula for $c_\mathrm{air}$, it may first be shown that, based on classical fluid mechanics,

$$c^2 = \frac{\partial p}{\partial \rho},$$ (3)

where $p$ denotes the fluid pressure, and $\rho$ denotes the fluid density.

Next we need to note an important and non-obvious property about sound propagation. It turns out that sound propagates approximately adiabatically. What this means is that as the fluid pressure fluctuates rapidly during sound propagation, the temperature of the fluid also fluctuates, but no heat is gained or lost due to this fluctuation. If heat were gained or lost, we would not have an adiabatic process. The key equation describing an adiabatic process relates pressure $p$ and density $\rho$ as follows:

$$p \rho^{- \gamma} = C_0,$$ (4)

where $C_0$ is a constant, and $\gamma$ is the adiabatic index. For diatomic gases such as air, $\gamma_\mathrm{dia} = 1.4$. Using the equations above, it may be shown that

$$c_\mathrm{air} = \sqrt{\frac{\gamma_\mathrm{dia} p_\mathrm{air}}{\rho_\mathrm{air}}}.$$ (5)

Next, the ideal gas law ($pV = nRT$) may be used to derive the following formula:

$$c_\mathrm{air} = \sqrt{\frac{\gamma_\mathrm{air} R T}{M_\mathrm{air}}},$$ (6)

where $R$ is the universal gas constant ($= 8.314$ J/[mol K]), $T$ is the absolute temperature in degrees Kelvin, and $M$ is the molar mass of the fluid (in kg/mol). For dry air, $M = 0.02895$ kg/mol.