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Piston radiation impedance.

Evaluate the one-sided acoustic radiation impedance of a rigid circular piston in an infinite baffle, including a numerically stable low-frequency limit.

01 / THE INPUTS
Piston geometry
Operating point
Hz
Air & conventions
m/s
kg/m3
02 / THE RESULTS
Acoustic radiation resistance
19.77Pa s/m3
Acoustic radiation reactance
483.82Pa s/m3
Dimensionless size (ka)
0.06933
Low-frequency one-sided air mass
6.425g
Mechanical resistance
0.0494N s/m
Mechanical reactance
1.2096N s/m

One-sided acoustic radiation impedance

LOG FREQUENCY
One-sided acoustic radiation impedance: Acoustic impedance (Pa s/m3) against frequency in hertz.02013.024026.056039.078052.0910k510205010020050010002000Acoustic impedance (Pa s/m3)Frequency (Hz)

Do not add this mass to manufacturer Mms without reconciling its existing air-load convention.

Read this result with its assumptions.

Ideal rigid circular piston in an infinite baffle, one radiating side. Acoustic impedance is pressure divided by volume velocity.

The low-frequency mass approximation is appropriate only when ka is much smaller than one.

Units & assumptions

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03 / UNDER THE HOOD

The math, in plain sight.

The air in front of the cone is part of the load.

One side of a baffled rigid piston

For piston radius a, area S, and wavenumber k, the acoustic impedance is pressure divided by volume velocity. J1 is the Bessel function and H1 is the Struve function.

S=πa2,k=ωcS=\pi a^2,\qquad k=\frac\omega c
ZA,rad=ρcS[1−J1(2ka)ka+jH1(2ka)ka]Z_{A,\mathrm{rad}}=\frac{\rho c}{S}\left[1-\frac{J_1(2ka)}{ka}+j\frac{H_1(2ka)}{ka}\right]
ZM,rad=S2ZA,radZ_{M,\mathrm{rad}}=S^2Z_{A,\mathrm{rad}}

The stable low-frequency approximation

The implementation uses the leading-order form for ka below 0.001 to avoid subtracting nearly equal values. Otherwise it evaluates stable integral representations with composite Simpson quadrature. The numerical operating-point limit is ka = 100.

ZA,rad≈ρcS[(ka)22+j8ka3π]Z_{A,\mathrm{rad}}\approx\frac{\rho c}{S}\left[\frac{(ka)^2}{2}+j\frac{8ka}{3\pi}\right]
Mrad,one side≈83ρa3,Im⁡ZM,rad≈ωMrad,one sideM_{\mathrm{rad,one\ side}}\approx\frac83\rho a^3,\qquad\operatorname{Im}Z_{M,\mathrm{rad}}\approx\omega M_{\mathrm{rad,one\ side}}

Mathematical series and model boundary

J1(z)=∑m=0∞(−1)m(z/2)2m+1m! Γ(m+2)J_1(z)=\sum_{m=0}^{\infty}\frac{(-1)^m(z/2)^{2m+1}}{m!\,\Gamma(m+2)}
H1(z)=∑m=0∞(−1)m(z/2)2m+2Γ(m+32)Γ(m+52)H_1(z)=\sum_{m=0}^{\infty}\frac{(-1)^m(z/2)^{2m+2}}{\Gamma(m+\tfrac32)\Gamma(m+\tfrac52)}

Manufacturer moving mass commonly includes an air-load convention. Do not simply add the calculated radiation mass to published Mms. Cone breakup, finite baffles, enclosure edges, and noncircular diaphragms lie outside this model.