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−w2a2)=2∑m=0∞(δm0−δm1)(wa)2m,0≤w≤a
image (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
MAR=8ρ05π2a,
image (15.89)
CAR=5πa36ρ0c2,
image (15.90)
RAR=ρ0cπa2.
image (15.91)
From Eq. (13.125), the far-field on-axis pressure with no delay is given by
p˜(r)=jka2p˜0e−jkr4r
image (15.92)
Notice that the far-field pressure is the first-order derivative of the driving pressure. At high frequencies, where the wavelength is small compared to the membrane; this risng response is caused by the decreasing beam width that concentrates the radiated energy on axis. At low frequencies, the antiphase rear radiation partially cancels that from the front. Only the phase shift due to the path difference prevents the cancellation from being complete. However, as the frequency decreases, the cancellation becomes more complete as the path difference is equal to an ever-smaller portion of the wavelength.
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