Let us now consider the ideal situation whereby we increase the number of rings while reducing their widths until the delay becomes continuously variable along the radius of the membrane. Then we can isolate the effect of the delay profile from the discretization of the rings. If we treat the membrane as a pure pressure source with zero mass and stiffness, the far-field radiated sound pressure at a distance r and angle θ from its center is obtained by inserting Eqs. (13.70) and (13.121) into Eq. (13.124) to yield

$\tilde{p}\left(r,\theta \right)=jk\mathrm{cos}\theta \frac{{e}^{-jkr}}{r}{\int}_{0}^{a}{\tilde{p}}_{+}\left(w\right){J}_{0}\left(kw\mathrm{sin}\theta \right)wdw\text{,}$

(15.1)

where J
_{0} is the zero-order Bessel function, k
=
ω/c is the ...

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