Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition

15.15. Radiation impedance

To derive the radiation impedance Z _AR , we modify Eq. (13.212) from Section 13.10 for the uniform velocity distribution of a rigid piston to obtain that for one with a parabolic velocity distribution

Φ ( w ) = 2 ( 1 − w 2 a 2 ) = 2 ∑ m = 0 ∞ ( δ m 0 − δ m 1 ) ( w a ) 2 m , 0 ≤ w ≤ a

(15.88)

and replace Eq. (13.215) with b(m   +   1)   =   2(− δ _m0   +   δ _m1). By analyzing the low frequency asymptotic behavior, we obtain the radiation impedance elements

M A R = 8 ρ 0 5 π 2 a ,

(15.89)

C A R = 5 π a 3