Acoustics

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15.7. Far-field sound pressure

The far-field pressure is derived in the same way as that for a resilient disk in free space in Section 13.8 and is the sum of the pressures radiated from each individual ring

\tilde{p} (r, θ) = - j a {\tilde{p}}_{0} \frac{e^{- j k r}}{4 r} D (θ)

$\tilde{p} (r, θ) = - j a {\tilde{p}}_{0} \frac{e^{- j k r}}{4 r} D (θ)$

(15.52)

where D(θ) is the directivity function given by

D (θ) = \frac{2}{a} \cot θ (a_{0} J_{1} (k a_{0} \sin θ) + \sum_{n = 1}^{N} \frac{{\tilde{p}}_{n}}{{\tilde{p}}_{0}} (a_{n} J_{1} (k a_{n} \sin θ) - a_{n - 1} J_{1} (k a_{n - 1} \sin θ))) .

$D (θ) = \frac{2}{a} \cot θ (a_{0} J_{1} (k a_{0} \sin θ) + \sum_{n = 1}^{N} \frac{{\tilde{p}}_{n}}{{\tilde{p}}_{0}} (a_{n} J_{1} (k a_{n} \sin θ) - a_{n - 1} J_{1} (k a_{n - 1} \sin θ))) .$

(15.53)

The normalized directivity pattern 20 log₁₀|D(θ)| − 20 log₁₀|D(0)| is plotted in Fig. 15.13. The on-axis response is given by

p_{norm} = k a D (0) = \frac{k a_{0}^{2}}{a} + \sum_{n = 1}^{N} \frac{{\tilde{p}}_{n} k (a_{n}^{2} - a_{n - 1}^{2})}{{\tilde{p}}_{0} a},

$p_{norm} = k a D (0) = \frac{k a_{0}^{2}}{a} + \sum_{n = 1}^{N} \frac{{\tilde{p}}_{n} k (a_{n}^{2} - a_{n - 1}^{2})}{{\tilde{p}}_{0} a},$

(15.54)

which is plotted in Fig. 15.14 using the quantities shown in Table 15.2, where the total delay is 255 μs and the width of each ring is 2 cm.

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Table of Contents for Acoustics

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15.7. Far-field sound pressure

Table of Contents for
Acoustics