Compressible and incompressible flow: retarded time

The second concept we saw in §1 was that of the time delay between cause and effect. This is due to the compressibility of a fluid. So far, you have mainly studied incompressible fluid dynamics, in which disturbances propagate through a fluid instantaneously. Acoustics, however, depends on a fluid being compressible and on disturbances (sound) propagating with some time delay.

Figure 2.2: Compressible and incompressible flow models

a	b

The reason for this time delay is shown in figure 2.2. If we think of a fluid as being like a set of spheres (think of one of those `executive toys'), an incompressible fluid is made up of spheres in contact. If we apply a force at one end of the line of spheres, they all move immediately and together: because the volume $V$ (the number of spheres per unit length in this case) is constant a disturbance must propagate instantaneously. On the other hand, a disturbance in a compressible fluid, figure 2.2b, propagates with some delay. Because the fluid is compressible, it can accomodate variations in density and a disturbance at one point does not make itself felt immediately at another.

Returning to Figure 2.1, we can see that the time delay depends on distance and the speed of sound $c$. We also know that both of these are fixed, the first by position and the second because the speed of sound is constant. The only thing that matters then is when the sound is generated. We can combine the first and second principles of §1 into a single statement and say that $\vert x-y\vert=c(t-\tau)$. Two new pieces of information have been added: the first is $y$, the position of the source, so that $\vert x-y\vert$ is the distance the sound travels; the second is $\tau$, the time when the sound left the point $y$, and is called the retarded time or emission time.

We now know what a realistic theory of sound has to look like. In the next section, we will start from the fundamental theory of fluid motion and derive the wave equation for acoustics.