Acoustics

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12.2. Radiation from an infinite line source

In the limit as the radius of the cylinder shrinks to zero, we have an infinite line source. In Section 13.14, we will use this as a building block for an infinite ribbon, which can be treated as an array of line sources using the Huygens–Fresnel principle. When the radius is very small, we find that

H_{1}^{(2)} {(k a) |}_{a \to 0} = \frac{2 j}{π k a},

$H_{1}^{(2)} {(k a) |}_{a \to 0} = \frac{2 j}{π k a},$

(12.3)

which, after inserting into Eq. (12.2), gives the pressure field of an infinite line source:

\tilde{p} (w) = \frac{k ρ_{0} c ({\tilde{U}}_{0} / l)}{4} H_{1}^{(2)} (k w) .

$\tilde{p} (w) = \frac{k ρ_{0} c ({\tilde{U}}_{0} / l)}{4} H_{1}^{(2)} (k w) .$

(12.4)

In the far field, we find that

H_{1}^{(2)} {(k w) |}_{w \to \infty} = \sqrt{\frac{2}{π k w}} e^{- j (k w - \frac{π}{4})},

$H_{1}^{(2)} {(k w) |}_{w \to \infty} = \sqrt{\frac{2}{π k w}} e^{- j (k w - \frac{π}{4})},$

(12.5)

so that the far-field pressure for a line source is given by

\tilde{p} (w) = \frac{ρ_{0} c ({\tilde{U}}_{0} / l)}{2} \sqrt{\frac{k}{2 π w}} e^{- j (k w - \frac{π}{4})} .

$\tilde{p} (w) = \frac{ρ_{0} c ({\tilde{U}}_{0} / l)}{2} \sqrt{\frac{k}{2 π w}} e^{- j (k w - \frac{π}{4})} .$

(12.6)

Interestingly, the far-field pressure given by Eq. (12.6) varies with the inverse square root of the radial distance w from the source so that the sound pressure falls by 3 dB for every doubling of distance, which is a characteristic of cylindrically diverging waves. This is in contrast to a spherically diverging wave, where the pressure given by Eq. (2.142) varies with the inverse square of the radial distance r from the source so that the SPL falls by 6 dB for every doubling of distance. Hence, line sources, in the form of vertical stacks of loudspeakers, are popular in auditoriums because they give a more uniform sound pressure distribution.

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12.2. Radiation from an infinite line source

Table of Contents for
Acoustics