One Dimensional Sound Fields

Sound in air that we hear with our ears consists of minute fluctuations in the ambient atmospheric air pressure. If we limit our attention to one dimensional spaces (like inside a narrow tube), we can write down a model for the fluctuating sound pressure. It helps to think of the total air pressure, P(t) as the sum of the static, ambient pressure \(P_0\) and the minute acoustic fluctuation, p(t), \(P(t)=P_0+p(t)\) . Of course, if the pressure changes at a point in space, even by a tiny fluctuating amount, we would also expect the density (or total mass of the molecules of the gas for a given volume) to also change depending on the change in pressure.