What is $a(x,t)$?

The wave phenomena is found in may different situations, and $a(x,t)$ represents a different physical quantity in each situation. For example, it is well known that disturbances in air propagate from one point to another as waves and are perceived by us as sound. Any source of sound (eg. a loud speaker) produces compressions and rarefactions in the air, and the patterns of compressions and rarefactions propagate from one point to another. Using to denote the air density , we can express this as $\rho(x,t)=\bar{\rho}+\Delta \rho(x,t)$ where $\bar{\rho}$ is the density in the absence of the disturbance and $\Delta \rho(x,t)$ is the change due to the disturbance. We can use equation (6.1) to represent a sinusoidal sound wave if we identify $a(x,t)$ with $\Delta \rho(x,t)$.

The transverse vibrations of a stretched string is another examples. In this situation $a(x,t)$ corresponds to $y(x,t)$ which IS the displacement of the string shown in Figure 6.1.

Figure 6.1: