Describing sound

Before going any further, you will need to know how to describe a sound or sound field. The things we tend to notice when we hear a noise are its loudness (amplitude) and its pitch (frequency).

To describe the amplitude⁵ of a sound we usually use the root mean square (rms) pressure:

$$p_{\text{rms}} = \left(\overline{p^{2}}\right)^{1/2}$$

where the bar denotes `time average'. This is a useful measure but suffers from the problem that acoustic pressures of interest vary over a huge range. The threshold of human hearing is at $p_{\text{rms}}=20\mu$Pa while the threshold of pain and the onset of hearing damage are at about $p_{\text{rms}}=200$mPa, a difference of seven orders of magnitude. To keep the numbers manageable, we use a logarithmic scale. On this scale, the `difference' in sound pressure level between two pressures $p_{1}$ and $p_{2}$ is:

$$\Delta_{\text{SPL}} = 10\log\frac{\overline{p_{1}^{2}}}{\overline{p_{2}^{2}}}.$$

When we want to talk about only one noise, we use a standard reference pressure. Then the sound pressure level is

$$= 10\log\frac{\overline{p^{2}}}{\overline{p_{\text{ref}}^{2}}}.$$ (17)

The reference level is the nominal threshold of human hearing 20$\mu$Pa. The `units' of SPL are decibels, dB.

When we talk about the frequency of a sound, we do so with reference to a periodic wave. For a sinusoidal wave of frequency $f$, the pressure varies as $\exp[-\mathrm{j}2\pi f t]$. Because the wave advances in space at speed $c$, we can also characterize it by the spatial distance between two peaks or troughs, the wavelength. The wavelength $\lambda$ is related to the frequency by $\lambda=c/f$.