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4.21. Plane circular piston radiating from one side only in free space

The specific impedance (N·s/m³) of the air load on a plane piston in free space, which has one surface vibrating sinusoidally while the other remains stationary, is given by Eqs. (13.254) and (13.255) and plotted in Fig. 4.39.

Table 4.5

Radiation impedance and admittance for one side of a plane circular piston in free space ^a .
Impedance	Mechanical	Specific	Acoustic	Analogous circuits
Impedance	f = drop u = flow	p = drop u = flow	p = drop U = flow	Analogous circuits
ka < 0.5: Series resistance, R Series resistance, R Mass, M ₁ Compliance, C ₁ ka > 5: Resistance, R ₂	$R_{M} = \frac{8 a^{6} ρ_{0} ω^{4}}{27 π c^{3}}$ $R_{M} = \frac{8 a^{6} ρ_{0} ω^{4}}{27 π c^{3}}$ $M_{M 1} = \frac{4 a^{3} ρ_{0}}{3}$ $M_{M 1} = \frac{4 a^{3} ρ_{0}}{3}$ $C_{M 1} = \frac{1}{\sqrt{6} π a ρ_{0} c^{2}}$ $C_{M 1} = \frac{1}{\sqrt{6} π a ρ_{0} c^{2}}$ R _M = πa ² ρ ₀ c	$R_{S} = \frac{8 a^{4} ρ_{0} ω^{4}}{27 π^{2} c^{3}}$ $R_{S} = \frac{8 a^{4} ρ_{0} ω^{4}}{27 π^{2} c^{3}}$ $M_{S 1} = \frac{4 a ρ_{0}}{3 π}$ $M_{S 1} = \frac{4 a ρ_{0}}{3 π}$ $C_{S 1} = \frac{a}{\sqrt{6} ρ_{0} c^{2}}$ $C_{S 1} = \frac{a}{\sqrt{6} ρ_{0} c^{2}}$ R _S = ρ ₀ c	$R_{A} = \frac{8 a^{2} ρ_{0} ω^{4}}{27 π^{3} c^{3}}$ $R_{A} = \frac{8 a^{2} ρ_{0} ω^{4}}{27 π^{3} c^{3}}$ $M_{A 1} = \frac{4 ρ_{0}}{3 π^{2} a}$ $M_{A 1} = \frac{4 ρ_{0}}{3 π^{2} a}$ $C_{A 1} = \frac{π a^{3}}{\sqrt{6} ρ_{0} c^{2}}$ $C_{A 1} = \frac{π a^{3}}{\sqrt{6} ρ_{0} c^{2}}$ $R_{A} = \frac{ρ_{0} c}{π a^{2}}$ $R_{A} = \frac{ρ_{0} c}{π a^{2}}$

Admittance	u = drop f = flow	u = drop p = flow	U = drop p = flow
ka < 0.5: Series conductance, G Mass, M ₁ Compliance, C ₁ ka > 5: Conductance, G ₂	$G_{M} = \frac{ω^{2}}{6 π ρ_{0} c^{3}}$ $G_{M} = \frac{ω^{2}}{6 π ρ_{0} c^{3}}$ $M_{M 1} = \frac{4 a^{3} ρ_{0}}{3}$ $M_{M 1} = \frac{4 a^{3} ρ_{0}}{3}$ $C_{M 1} = \frac{1}{\sqrt{6} π a ρ_{0} c^{2}}$ $C_{M 1} = \frac{1}{\sqrt{6} π a ρ_{0} c^{2}}$ $G_{M} = \frac{1}{π a^{2} ρ_{0} c}$ $G_{M} = \frac{1}{π a^{2} ρ_{0} c}$	$G_{S} = \frac{a^{2} ω^{2}}{6 ρ_{0} c^{3}}$ $G_{S} = \frac{a^{2} ω^{2}}{6 ρ_{0} c^{3}}$ $M_{S 1} = \frac{4 a ρ_{0}}{3 π}$ $M_{S 1} = \frac{4 a ρ_{0}}{3 π}$ $C_{M 1} = \frac{π a^{3}}{\sqrt{6} ρ_{0} c^{2}}$ $C_{M 1} = \frac{π a^{3}}{\sqrt{6} ρ_{0} c^{2}}$ $G_{S} = \frac{1}{ρ_{0} c}$ $G_{S} = \frac{1}{ρ_{0} c}$	$G_{A} = \frac{π a^{4} ω^{2}}{6 ρ_{0} c^{3}}$ $G_{A} = \frac{π a^{4} ω^{2}}{6 ρ_{0} c^{3}}$ $M_{A 1} = \frac{4 ρ_{0}}{3 π^{2} a}$ $M_{A 1} = \frac{4 ρ_{0}}{3 π^{2} a}$ $C_{A 1} = \frac{π a^{3}}{\sqrt{6} ρ_{0} c^{2}}$ $C_{A 1} = \frac{π a^{3}}{\sqrt{6} ρ_{0} c^{2}}$ $G_{A} = \frac{π a^{2}}{ρ_{0} c}$ $G_{A} = \frac{π a^{2}}{ρ_{0} c}$

Graphs of the real and imaginary parts of the normalized mechanical impedance Z _M/πa ² ρ ₀ c as a function of ka for a piston so mounted are shown in Fig. 12.36. It is simply the mean of the impedances of a plane circular piston in an infinite baffle and the same in free space.

To a fair approximation, the radiation impedance for a one-sided piston in free space may be represented over the entire frequency range by the same analogous circuits used for the piston in an infinite baffle and shown in Fig. 4.37, where the elements now are

R_{M 2} = π a^{2} ρ_{0} c N \cdot s / m

$R_{M 2} = π a^{2} ρ_{0} c N \cdot s / m$

(4.161)

\begin{matrix} R_{M} = R_{M 2} + R_{M 1} = 16 a^{2} ρ_{0} c / π \\ = 5.09 a^{2} ρ_{0} c N \cdot s / m \end{matrix}

$\begin{matrix} R_{M} = R_{M 2} + R_{M 1} = 16 a^{2} ρ_{0} c / π \\ = 5.09 a^{2} ρ_{0} c N \cdot s / m \end{matrix}$

(4.162)

R_{M 1} = 1.95 a^{2} ρ_{0} c N \cdot s / m

$R_{M 1} = 1.95 a^{2} ρ_{0} c N \cdot s / m$

(4.163)

C_{M 1} = 1 / (π a ρ_{0} c^{2}) = 0.318 / (a ρ_{0} c^{2}) m / N

$C_{M 1} = 1 / (π a ρ_{0} c^{2}) = 0.318 / (a ρ_{0} c^{2}) m / N$

(4.164)

Figure 4.39 Real and imaginary parts of the normalized specific radiation impedance Z _s/ρ ₀ c of the air load on one side of a plane circular piston of radius a radiating from one side only into free space. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

M_{M 1} = 2 a^{3} ρ_{0} kg

$M_{M 1} = 2 a^{3} ρ_{0} kg$

(4.165)

G_{M 2} = 1 / (π a^{2} ρ_{0} c) = 0.318 / (a^{2} ρ_{0} c) m \cdot N^{- 1} \cdot s^{- 1}

$G_{M 2} = 1 / (π a^{2} ρ_{0} c) = 0.318 / (a^{2} ρ_{0} c) m \cdot N^{- 1} \cdot s^{- 1}$

(4.166)

G_{M 1} = 0.513 / (a^{2} ρ_{0} c) m \cdot N^{- 1} \cdot s^{- 1}

$G_{M 1} = 0.513 / (a^{2} ρ_{0} c) m \cdot N^{- 1} \cdot s^{- 1}$

(4.167)

R_{A 2} = ρ_{0} c / (π a^{2}) = 0.318 ρ_{0} c / a^{2} N \cdot s / m^{5}

$R_{A 2} = ρ_{0} c / (π a^{2}) = 0.318 ρ_{0} c / a^{2} N \cdot s / m^{5}$

(4.168)

\begin{matrix} R_{A} = R_{A 2} + R_{A 1} = 16 ρ_{0} c / (π^{3} a^{2}) \\ = 0.516 ρ_{0} c / a^{2} N \cdot s / m^{5} \end{matrix}

$\begin{matrix} R_{A} = R_{A 2} + R_{A 1} = 16 ρ_{0} c / (π^{3} a^{2}) \\ = 0.516 ρ_{0} c / a^{2} N \cdot s / m^{5} \end{matrix}$

(4.169)

R_{A 1} = 0.1977 ρ_{0} c / a^{2} N \cdot s / m^{5}

$R_{A 1} = 0.1977 ρ_{0} c / a^{2} N \cdot s / m^{5}$

(4.170)

C_{A 1} = π a^{3} / (ρ_{0} c^{2}) = 3.142 a^{3} / (ρ_{0} c^{2}) m^{5} / N

$C_{A 1} = π a^{3} / (ρ_{0} c^{2}) = 3.142 a^{3} / (ρ_{0} c^{2}) m^{5} / N$

(4.171)

M_{A 1} = 2 ρ_{0} / (π^{2} a) = 0.2026 ρ_{0} / a kg / m^{4}

$M_{A 1} = 2 ρ_{0} / (π^{2} a) = 0.2026 ρ_{0} / a kg / m^{4}$

(4.172)

G_{A 2} = π a^{2} / (ρ_{0} c) m^{5} \cdot N^{- 1} \cdot s^{- 1}

$G_{A 2} = π a^{2} / (ρ_{0} c) m^{5} \cdot N^{- 1} \cdot s^{- 1}$

(4.173)

G_{A 1} = 5.06 a^{2} / (ρ_{0} c) m^{5} \cdot N^{- 1} \cdot s^{- 1}

$G_{A 1} = 5.06 a^{2} / (ρ_{0} c) m^{5} \cdot N^{- 1} \cdot s^{- 1}$

(4.174)

In the frequency ranges where ka < 0.5 and ka > 5, analogous circuits of the type shown in Table 4.6 may be used.

Table 4.6

Radiation impedance and admittance for a plane circular piston radiating from one side only in free space ^a
Impedance	Mechanical	Specific	Acoustic	Analogous circuits
Impedance	f = drop u = flow	p = drop u = flow	p = drop U = flow	Analogous circuits
ka < 0.5: Series resistance, R Series resistance, R Mass, M ₁ ka > 5: Resistance, R ₂	$R_{M} = \frac{π a^{4} ρ_{0} ω^{2}}{4 c}$ $R_{M} = \frac{π a^{4} ρ_{0} ω^{2}}{4 c}$ $R_{M} = \frac{16 a^{2} ρ_{0} c}{π}$ $R_{M} = \frac{16 a^{2} ρ_{0} c}{π}$ M _M1 = 2a ³ ρ ₀ R _M2 = πa ² ρ ₀ c	$R_{S} = \frac{a^{2} ρ_{0} ω^{2}}{4 c}$ $R_{S} = \frac{a^{2} ρ_{0} ω^{2}}{4 c}$ $R_{S} = \frac{16 ρ_{0} c}{π^{2}}$ $R_{S} = \frac{16 ρ_{0} c}{π^{2}}$ $M_{S 1} = \frac{2 a ρ_{0}}{π}$ $M_{S 1} = \frac{2 a ρ_{0}}{π}$ R _S2 = ρ ₀ c	$R_{A} = \frac{ρ_{0} ω^{2}}{4 π c}$ $R_{A} = \frac{ρ_{0} ω^{2}}{4 π c}$ $R_{A} = \frac{16 ρ_{0} c}{π^{3} a^{2}}$ $R_{A} = \frac{16 ρ_{0} c}{π^{3} a^{2}}$ $M_{A 1} = \frac{2 ρ_{0}}{π^{2} a}$ $M_{A 1} = \frac{2 ρ_{0}}{π^{2} a}$ $R_{A 2} = \frac{ρ_{0} c}{π a^{2}}$ $R_{A 2} = \frac{ρ_{0} c}{π a^{2}}$

Admittance	u = drop f = flow	u = drop p = flow	U = drop p = flow
ka < 0.5: Series conductance, G Mass, M ₁ ka > 5: Conductance, G ₂	$G_{M} = \frac{π}{16 a^{2} ρ_{0} c}$ $G_{M} = \frac{π}{16 a^{2} ρ_{0} c}$ M _M1 = 2a ³ ρ ₀ $G_{M 2} = \frac{1}{π a^{2} ρ_{0} c}$ $G_{M 2} = \frac{1}{π a^{2} ρ_{0} c}$	$G_{S} = \frac{π^{2}}{16 ρ_{0} c}$ $G_{S} = \frac{π^{2}}{16 ρ_{0} c}$ $M_{S 1} = \frac{2 a ρ_{0}}{π}$ $M_{S 1} = \frac{2 a ρ_{0}}{π}$ $G_{S 2} = \frac{1}{ρ_{0} c}$ $G_{S 2} = \frac{1}{ρ_{0} c}$	$G_{A} = \frac{π^{3} a^{2}}{16 ρ_{0} c}$ $G_{A} = \frac{π^{3} a^{2}}{16 ρ_{0} c}$ $M_{A 1} = \frac{2 ρ_{0}}{π^{2} a}$ $M_{A 1} = \frac{2 ρ_{0}}{π^{2} a}$ $G_{A 2} = \frac{π a^{2}}{ρ_{0} c}$ $G_{A 2} = \frac{π a^{2}}{ρ_{0} c}$

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4.21. Plane circular piston radiating from one side only in free space

Table of Contents for
Acoustics