15.5. Effect of discretization into rings of finite width

Rings of Equal Delay. One option might be to vary the widths of the rings so that there are equal delay sections between them, in which case the on-axis pressure is
pnorm=kaD(0)=ka∑n=0Ne−jkan+1/2N+1(an2−an−12a2)
image (15.21)
where the radius of the nth ring is
an=a1−(1−n+1N+1)2,
image (15.22)
a 0 is the radius of the center disk and a −1   =   0. In Eq. (15.21), the term in parentheses is proportional to the area of the nth ring while the exponent term represents the delay applied to that ring. The cross section of a stator with rings of equal delay is shown in Fig. 15.8.
Rings of Equal Area. Another option might be to vary the widths of the rings so that they all have the same area and capacitance, in which case the on-axis pressure is
pnorm=kaD(0)=ka∑n=0Ne−jka(1−1−an−1/22a2)(an2−an−12a2),
image (15.23)
where the radius of the nth ring is
an=an+1N+1.
image (15.24)
The cross section of a stator with rings of equal delay is shown in Fig. 15.9.
Rings of Equal Width. The last option we shall consider is one in which the rings are of equal width, in which case the on-axis pressure is again given by Eq. (15.23), but with the radius of the nth ring given by
an=an+1N+1.
image (15.25)
The cross section of a stator with rings of equal delay is shown in Fig. 15.10, while the on-axis responses with equal delay, equal area, and equal width are plotted in Fig. 15.11. The ring widths are listed in Table 15.1. Arguably, the rings of equal delay produce the smoothest response at higher frequencies because of their finer resolution of the rapid increase in delay near the rim, as shown in Fig. 15.8.
image
Figure 15.8 Cross section of a stator divided into concentric rings of equal delay, where purely for illustration each ring has been shifted to the left by the distance that a wave would have traveled during the time delay applied to that ring.
However, the outer rings are so thin that stray capacitances will dominate the ring capacitances, whereas a stator with rings of equal width largely avoids this problem. Also, the wide center disk (144   mm diameter) will produce high-frequency beaming. If the rings have equal width, they can all be narrow compared to the wavelength over most of the audio spectrum.
image
Figure 15.9 Cross section of a stator divided into concentric rings of equal area, where purely for illustration each ring has been shifted to the left by the distance that a wave would have traveled during the time delay applied to that ring.
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