Sound Wave Solutions

Sound waves can be characterized one of two ways: as organized fluctuations in the position of the molecules of the fluid as they oscillate around an equilibrium displacement or as organized fluctuations in the pressure of the fluid as molecules are crammed closer together or are diven farther apart than they are on average in the quiescent fluid.

Sound waves propagate in one direction (out of three) at any given point in space. This means that in the direction perpendicular to propagation, the wave is spread out to form a ``wave front''. The wave front can be nearly arbitrary in shape initially; thereafter it evolves according to the mathematics of the wave equation in three dimensions (which is similar to but a bit more complicated than the wave equation in one dimension).

To avoid this complication and focus on general properties that are commonly encountered, we will concentrate on two particular kinds of solutions:

1. Plane Wave solutions. In these solutions, the entire wave moves in one direction (say the $x$ direction) and the wave front is a 2-D plane perpendicular to the direction of propagation. These (displacement) solutions can be written as (e.g.):
   

   $$s(x,t) = s_0 \sin(kx - \omega t)$$ (162)

   
   
   where $s_0$ is the maximum displacement in the travelling wave (which moves in the $x$ direction) and where all molecules in the entire plane at position $x$ are displaced by the same amount.

   Waves far away from the sources that created them are best described as plane waves. So are waves propagating down a constrained environment such as a tube that permits waves to only travel in ``one direction''.

2. Spherical Wave solutions. Sound is often emitted from a source that is highly localized (such as a hammer hitting a nail, or a loudspeaker). If the sound is emitted equally in all directions from the source, a spherical wavefront is formed. Even if it is not emitted equally in all directions, sound from a localized source will generally form a spherically curved wavefront as it travels away from the point with constant speed. The displacement of a spherical wavefront decreases as one moves further away from the source because the energy in the wavefront is spread out on larger and larger surfaces. Its form is given by:
   

   $$s(r,t) = \frac{s_0}{r} \sin(kr - \omega t)$$ (163)

   
   
   where $r$ is the radial distance away from the point-like source.