Compactness and interference

Figure 5.4: Acoustic field (absolute value of $p$) along the axis of a vibrating piston. The dashed line shows the $1/z$ fit.

a: $k=0.1$	b: $k=1.0$	c: $k=10.0$

If we examine the acoustic field defined by equation 5.6 as a function of frequency, we can see that it changes quite rapidly as $k$ is increased. Figure 5.4 shows the absolute value of the non-dimensional pressure $\vert p/\rho_{0}cv\vert$ for different values of $k$. For comparison, the curve $1/R_{0}=1/z$ is also shown. The results for $k=0.1$ and $k=1$ are similar with a smooth $1/R_{0}$ decay but the $k=10$ curve is quite different, having a sharp drop before it begins to follow a $1/R_{0}$ curve. This is a result of interference between sound from different parts of the piston. When a body is large compared to the wavelength of the sound it generates, interference between different parts of the body gives rise to a complicated sound pattern, especially in the region near the body. When the body is small on a wavelength scale (or, equivalently, vibrates at low frequency), the phase difference between different parts of the source is not enough to give rise to much interference and the body radiates like a point source. The `size' of the body at a given frequency is called its compactness and is characterized by the parameter $ka$ where $a$ is a characteristic dimension, or by the ratio of characteristic dimension to wavelength $a/\lambda$. A compact source, one with $ka\ll1$, radiates like a point source, while non-compact bodies must be treated in more detail.