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4.19. Plane circular piston in infinite baffle

The specific impedance in N·s/m³ of the air load on one side of a plane piston mounted in an infinite baffle (see Fig. 13.3) and vibrating sinusoidally is given by Eqs. (13.116)–(13.118). Plots of the real and imaginary parts of

\frac{Z_{s}}{ρ_{0} c} = \frac{R_{s} + j X_{s}}{ρ_{0} c}

$\frac{Z_{s}}{ρ_{0} c} = \frac{R_{s} + j X_{s}}{ρ_{0} c}$

(4.145)

are shown in Fig. 4.35 as a function of ka. Similar graphs of the real and imaginary parts of the specific admittance

Y_{s} ρ_{0} c = ρ_{0} c (G_{s} + j B_{s}) = ρ_{0} c (\frac{R_{s}}{R_{s}^{2} + X_{X}^{s}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}})

$Y_{s} ρ_{0} c = ρ_{0} c (G_{s} + j B_{s}) = ρ_{0} c (\frac{R_{s}}{R_{s}^{2} + X_{X}^{s}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}})$

(4.146)

are shown in Fig. 4.36. The specific admittance is in m³·N⁻¹·s⁻¹ (rayls⁻¹).

Figure 4.35 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 in an infinite flat baffle. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Figure 4.36 Real and imaginary parts of the normalized specific radiation admittance ρ ₀ cY _s of the air load on one side of a plane circular piston of radius a in an infinite flat baffle. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

The data of Fig. 4.35 are used in dealing with impedance analogies and the data of Fig. 4.36 in dealing with admittance analogies.

We see from Fig. 4.35 that, for ka < 0.5, the reactance varies as the first power of frequency while the resistance varies as the second power of frequency. At high frequencies, for ka > 5, the reactance becomes small compared with the resistance, and the resistance approaches a constant value.

The admittance, on the other hand, is better behaved. The conductance is constant for ka < 0.5, and it is also constant for ka > 5 although its value is larger.

Approximate analogous circuits

The behavior just noted suggests that, except for the ripples in the curves for ka between 1 and 5, the impedance and the admittance for a piston in an infinite baffle can be approximated over the whole frequency range by the analogous circuits of Fig. 4.37. Those circuits give the mechanical and acoustic impedances and admittances, where

Figure 4.37 Approximate radiation impedances and admittances for a piston in an infinite baffle or for a closed-back piston for all values of ka.(a) Mechanical-impedance analogy; (b) acoustic-impedance analogy; (c) mechanical-admittance analogy; (d) acoustic-admittance analogy.

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

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

(4.147)

\begin{matrix} R_{M} = R_{M 2} + R_{M 1} = 128 a^{2} ρ_{0} c / (9 π) \\ = 4.53 a^{2} ρ_{0} c N \cdot s / m \end{matrix}

$\begin{matrix} R_{M} = R_{M 2} + R_{M 1} = 128 a^{2} ρ_{0} c / (9 π) \\ = 4.53 a^{2} ρ_{0} c N \cdot s / m \end{matrix}$

(4.148)

R_{M 1} = 1.386 a^{2} ρ_{0} c N \cdot s / m

$R_{M 1} = 1.386 a^{2} ρ_{0} c N \cdot s / m$

(4.149)

C_{M 1} = 1.89 / (π a ρ_{0} c^{2}) = 0.6 / (a ρ_{0} c^{2}) m / N

$C_{M 1} = 1.89 / (π a ρ_{0} c^{2}) = 0.6 / (a ρ_{0} c^{2}) m / N$

(4.150)

M_{M 1} = 8 a^{3} ρ_{0} / 3 = 2.67 a^{3} ρ_{0} kg

$M_{M 1} = 8 a^{3} ρ_{0} / 3 = 2.67 a^{3} ρ_{0} kg$

(4.151)

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

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

(4.152)

G_{M 1} = 0.722 / (a^{2} ρ_{0} c) m \cdot N^{- 1} \cdot s^{- 1}

$G_{M 1} = 0.722 / (a^{2} ρ_{0} c) m \cdot N^{- 1} \cdot s^{- 1}$

(4.153)

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

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

(4.154)

\begin{matrix} R_{A} = R_{A 2} + R_{A 1} = 128 ρ_{0} c / (9 π^{3} a^{2}) \\ = 0.459 ρ_{0} c / a^{2} N \cdot s / m^{5} \end{matrix}

$\begin{matrix} R_{A} = R_{A 2} + R_{A 1} = 128 ρ_{0} c / (9 π^{3} a^{2}) \\ = 0.459 ρ_{0} c / a^{2} N \cdot s / m^{5} \end{matrix}$

(4.155)

R_{A 1} = 0.1404 ρ_{0} c / a^{2} N \cdot s / m^{5}

$R_{A 1} = 0.1404 ρ_{0} c / a^{2} N \cdot s / m^{5}$

(4.156)

C_{A 1} = 1.89 π a^{3} / (ρ_{0} c^{2}) = 5.94 a^{3} / (ρ_{0} c^{2}) m^{5} / N

$C_{A 1} = 1.89 π a^{3} / (ρ_{0} c^{2}) = 5.94 a^{3} / (ρ_{0} c^{2}) m^{5} / N$

(4.157)

M_{A 1} = 8 ρ_{0} / (3 π^{2} a) = 0.27 ρ_{0} / a kg / m^{4}

$M_{A 1} = 8 ρ_{0} / (3 π^{2} a) = 0.27 ρ_{0} / a kg / m^{4}$

(4.158)

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

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

(4.159)

G_{A 1} = 7.12 a^{2} / (ρ_{0} c) m^{5} \cdot N^{- 1} \cdot s^{- 1}

$G_{A 1} = 7.12 a^{2} / (ρ_{0} c) m^{5} \cdot N^{- 1} \cdot s^{- 1}$

(4.160)

All constants are dimensionless and were chosen to give the best average fit to the functions of Figs. 4.35 and 4.36.

Low- and high-frequency approximations

At low and high frequencies, these circuits may be approximated by the simpler circuits given in the last column of Table 4.4.

It is apparent that when ka < 0.5, that is, when the circumference of the piston 2πa is less than one-half wavelength λ/2, the impedance load presented by the air on the vibrating piston is that of a mass shunted by a very large resistance. In other words, R ² = (R ₁ + R ₂)² is large compared with ω ² M ₁ ². In fact, this loading mass may be imagined to be a layer of air equal in area to the area of the piston and equal in thickness to about 0.85 times the radius because

(π a^{2}) (0.85 a) ρ_{0} \approx 2.67 a^{3} ρ_{0} = M_{M 1}

$(π a^{2}) (0.85 a) ρ_{0} \approx 2.67 a^{3} ρ_{0} = M_{M 1}$

At high frequencies, ka > 5, the air load behaves exactly as though it were connected to one end of a tube of the same diameter as the piston, with the other end of the tube perfectly absorbing. As we saw in Eq. (2.124), the input mechanical resistance for such a tube is πa ² ρ ₀ c. Hence, intuitively one might expect that at high frequencies the vibrating rigid piston beams the sound outward in lines perpendicular to the face of the piston. This is actually the case for the immediate near field close to the piston. At a distance, however, the far-field radiation spreads, as we learned earlier in this chapter.

Table 4.4

Radiation impedance and admittance for one side of a plane circular piston in an infinite baffle ^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	$R_{M} = \frac{π a^{4} ρ_{0} ω^{2}}{2 c}$ $R_{M} = \frac{π a^{4} ρ_{0} ω^{2}}{2 c}$	$R_{S} = \frac{a^{2} ρ_{0} ω^{2}}{2 c}$ $R_{S} = \frac{a^{2} ρ_{0} ω^{2}}{2 c}$	$R_{A} = \frac{ρ_{0} ω^{2}}{2 π c}$ $R_{A} = \frac{ρ_{0} ω^{2}}{2 π c}$
Shunt resistance, R	$R_{M} = \frac{128 a^{2} ρ_{0} c}{9 π}$ $R_{M} = \frac{128 a^{2} ρ_{0} c}{9 π}$	$R_{S} = \frac{128 ρ_{0} c}{9 π^{2}}$ $R_{S} = \frac{128 ρ_{0} c}{9 π^{2}}$	$R_{A} = \frac{128 ρ_{0} c}{9 π^{3} a^{2}}$ $R_{A} = \frac{128 ρ_{0} c}{9 π^{3} a^{2}}$
Mass, M ₁ ka > 5:	$M_{M 1} = \frac{8 a^{3} ρ_{0}}{3}$ $M_{M 1} = \frac{8 a^{3} ρ_{0}}{3}$	$M_{S 1} = \frac{8 a ρ_{0}}{3 π}$ $M_{S 1} = \frac{8 a ρ_{0}}{3 π}$	$M_{A 1} = \frac{8 ρ_{0}}{3 π^{2} a}$ $M_{A 1} = \frac{8 ρ_{0}}{3 π^{2} a}$
Resistance, R ₂	R _M2 = πa ² ρ ₀ c	R _S2 = ρ ₀ c	$R_{A 2} = \frac{ρ_{0} c}{π a^{2}}$ $R_{A 2} = \frac{ρ_{0} c}{π a^{2}}$

Admittance	u = drop f = flow	u = drop p = flow	U = drop p = flow
ka < 0.5:
Series conductance, G	$G_{M} = \frac{9 π}{128 a^{2} ρ_{0} c}$ $G_{M} = \frac{9 π}{128 a^{2} ρ_{0} c}$	$G_{S} = \frac{9 π^{2}}{128 ρ_{0} c}$ $G_{S} = \frac{9 π^{2}}{128 ρ_{0} c}$	$G_{A} = \frac{9 π^{3} a^{2}}{128 ρ_{0} c}$ $G_{A} = \frac{9 π^{3} a^{2}}{128 ρ_{0} c}$
Mass, M ₁ ka > 5:	$M_{M 1} = \frac{8 a^{3} ρ_{0}}{3}$ $M_{M 1} = \frac{8 a^{3} ρ_{0}}{3}$	$M_{S 1} = \frac{8 a ρ_{0}}{3 π}$ $M_{S 1} = \frac{8 a ρ_{0}}{3 π}$	$M_{A 1} = \frac{8 ρ_{0}}{3 π^{2} a}$ $M_{A 1} = \frac{8 ρ_{0}}{3 π^{2} a}$
Conductance, G ₂	$G_{M 2} = \frac{1}{π a^{2} ρ_{0} c}$ $G_{M 2} = \frac{1}{π a^{2} ρ_{0} c}$	$G_{S 2} = \frac{1}{ρ_{0} c}$ $G_{S 2} = \frac{1}{ρ_{0} c}$	$G_{A 2} = \frac{π a^{2}}{ρ_{0} c}$ $G_{A 2} = \frac{π a^{2}}{ρ_{0} c}$