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A cylindrical loudspeaker in a wall can be modelled by a piston of radius R vibrating with velocity vz located in a rigid wall. For convenience the surface integral will be expressed in cylindrical coordinates r0 and φ0. The receiver coordinates are given as spherical coordinates r and ϑ(Figure 2.11). Without loss of generality the azimuthal angle φ is set to zero.

(2.142)

Figure 2.11 Coordinate definitions for the piston in the wall. Source: Alexander Peiffer.

In the far field approximation we assume l≈r and get

(2.143)

So, the approximate result is

(2.144)

The integral is the Bessel function of first order

(2.145)

The results for some values of kR are shown in Figure 2.12 over the angular range of ±π/2. For a piston size small compared to the wavelength (kR≤1) the radiation pattern is similar to a point source. The smaller the wavelength gets in relation to the piston radius R the more a specific radiation pattern develops.

Figure 2.12 Angular distribution (radiation pattern) of the pressure field of the piston. Source: Alexander Peiffer.