Acoustics

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13.11. Radiation from a rigid circular piston in a finite circular closed baffle [32] (one-sided radiator)

The configuration is the same as that shown in Fig. 13.21 for a piston in an open baffle except that the velocity on the rear of the piston is zero. We can achieve this boundary condition by superposition of fields (or Gutin concept) whereby we combine the field of a piston in an open finite baffle with that of a piston in an infinite baffle. The former has negative velocity

- {\tilde{u}}_{0}

$- {\tilde{u}}_{0}$

on its rear surface, whereas the latter, if treated as a “breathing” disk in free space, has positive velocity

{\tilde{u}}_{0}

${\tilde{u}}_{0}$

on its rear surface. Hence, when the two fields are combined, their rear surface velocities cancel to produce a zero velocity boundary condition as illustrated in Fig. 13.29. However, if we wish the front velocity to be

{\tilde{u}}_{0}

${\tilde{u}}_{0}$

and not

2 {\tilde{u}}_{0}

$2 {\tilde{u}}_{0}$

, we must halve the result.

Although the piston and baffle are both infinitesimally thin in this model, it can be used to model a finite cylindrical enclosure with reasonable accuracy. In fact, the radiation characteristics of the single-sided piston without a baffle (b = a) are remarkably similar to those of a piston at the end of an infinite tube [33]. In the case of a finite cylinder, there will be secondary reflections from the far end, but they will be considerably weaker than the primary ones from the perimeter of the baffle.

Figure 13.29 Gutin concept: By superposition of fields, a one-sided piston in a finite closed baffle is equivalent to the sum of a double-sided dipole piston in a finite open baffle and a monopole piston in an infinite baffle. Also see Fig.13.4 for the monopole piston model.

Far-field pressure

The directivity function D(θ) is half the sum of that from Eq. (13.102) for a piston in an infinite baffle and that from Eq. (13.235) for a piston in a finite open baffle:

D (θ) = \frac{J_{1} (k a \sin θ)}{k a \sin θ} + \frac{k b}{2} \cos θ \sum_{n = 0}^{N} A_{n} Γ (n + \frac{5}{2}) {(\frac{2}{k b \sin θ})}^{n + \frac{3}{2}} J_{n + \frac{3}{2}} (k b \sin θ) .

$D (θ) = \frac{J_{1} (k a \sin θ)}{k a \sin θ} + \frac{k b}{2} \cos θ \sum_{n = 0}^{N} A_{n} Γ (n + \frac{5}{2}) {(\frac{2}{k b \sin θ})}^{n + \frac{3}{2}} J_{n + \frac{3}{2}} (k b \sin θ) .$

(13.252)

Similarly, the on-axis pressure is obtained from Eqs. (13.103) and (13.236) to give

D (0) = \frac{1}{2} (1 + k b \sum_{n = 0}^{N} A_{n}) .

$D (0) = \frac{1}{2} (1 + k b \sum_{n = 0}^{N} A_{n}) .$

(13.253)

The on-axis response for five values of b is shown in Fig. 13.30, calculated from the magnitude of D(0). The normalized directivity function 20 log₁₀|D(θ)/D(0)| for a one-sided piston in free space is plotted in Fig. 13.31 for four values of ka = 2πa/λ, that is, for four values of the ratio of the circumference of the disk to the wavelength. Also, the directivity function for a piston in a finite closed baffle is plotted in Fig. 13.32 for four values of ka with b = 2a and in Fig. 13.33 for four values of b.

Figure 13.30 Plot of 20 log₁₀(D(0)) where D(θ) is the directivity function of a plane circular piston of radius a in a plane closed circular baffle of radius b. When b = a (solid black curve), there is no baffle and the piston is radiating from one side only in free space. When b = ∞ (dotted black curve), the piston is in an infinite baffle. The axial acceleration of the piston is constant. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Figure 13.31 Far-field directivity patterns for a plane circular piston radiating from one side only into free space as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston.

Fig. 13.32 Far-field directivity patterns for a circular piston in a plane closed circular baffle as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston and b = 2a is the radius of the baffle.

Fig. 13.33 Far-field directivity patterns for a circular piston in a plane closed circular baffle as a function of b at ka = π/2 or λ = 4a, where a is the radius of the piston and b is the radius of the baffle.

Near-field pressure

The near-field pressure is simply the average of the pressures from Eqs. (13.106) and (13.240) for r > a or Eqs. (13.107) and (13.241) for r ≤ a. The pressure field for a one-sided piston in free space is plotted in Fig. 13.34 for three values of ka. The pressure response on the shadow side of the one-sided radiator is interesting not only for what it reveals about the diffraction effects around an infinitesimally thin edge but also for the fact that this pressure field is actually the difference between the baffled (monopole) and unbaffled (dipole) responses of the rigid piston. In particular, the differences persist into the high-frequency range. The pressure field for a rigid circular piston in a closed circular baffle of radius b = 2a is plotted in Fig. 13.35 for two values of ka.

Radiation impedance

The same principle can also be applied to the radiation impedance, which is proportional to the sum of the surface pressures of a piston in a finite baffle and an infinite baffle. Hence the real part can be obtained from Eqs. (13.117) and (13.249) as follows:

\begin{matrix} {R_{s} |}_{b \neq a} = \frac{ρ_{0} c}{2} {1 - \frac{J_{1} (2 k a)}{k a} + k b ℜ (\sum_{n = 0}^{N} A_{n} {1 - {(1 - \frac{a^{2}}{b^{2}})}^{n + \frac{3}{2}}})} \\ {R_{s} |}_{b = a} = \frac{ρ_{0} c}{2} {1 - \frac{J_{1} (2 k a)}{k a} + k a ℜ (\sum_{n = 0}^{N} A_{n})} \approx ρ_{0} c \frac{k^{2} a^{2}}{4}, k a < 0.5, \end{matrix}

$\begin{matrix} {R_{s} |}_{b \neq a} = \frac{ρ_{0} c}{2} {1 - \frac{J_{1} (2 k a)}{k a} + k b ℜ (\sum_{n = 0}^{N} A_{n} {1 - {(1 - \frac{a^{2}}{b^{2}})}^{n + \frac{3}{2}}})} \\ {R_{s} |}_{b = a} = \frac{ρ_{0} c}{2} {1 - \frac{J_{1} (2 k a)}{k a} + k a ℜ (\sum_{n = 0}^{N} A_{n})} \approx ρ_{0} c \frac{k^{2} a^{2}}{4}, k a < 0.5, \end{matrix}$

(13.254)

Figure 13.34 Normalized near-field pressure plots for a rigid circular piston radiating from one side only into free space as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston. $| \tilde{p} |$ $| \tilde{p} |$ is the pressure magnitude, ${\tilde{u}}_{0}$ ${\tilde{u}}_{0}$ is the piston velocity, ρ ₀ is the density of the acoustic medium, and c is the speed of sound in that medium.

Figure 13.35 Normalized near-field pressure plots for a rigid circular piston in a closed circular baffle of radius b = 2a as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston. $| \tilde{p} |$ $| \tilde{p} |$ is the pressure magnitude, ${\tilde{u}}_{0}$ ${\tilde{u}}_{0}$ is the piston velocity, ρ ₀ is the density of the acoustic medium, and c is the speed of sound in that medium.

Likewise, the imaginary part can be obtained from Eqs. (13.118) and (13.250) as follows:

\begin{matrix} {X_{s} |}_{b \neq a} = \frac{ρ_{0} c}{2} {\frac{H_{1} (2 k a)}{k a} + k b ℑ (\sum_{n = 0}^{N} A_{n} {1 - {(1 - \frac{a^{2}}{b^{2}})}^{n + \frac{3}{2}}})} \\ {X_{s} |}_{b = a} = \frac{ρ_{0} c}{2} {\frac{H_{1} (2 k a)}{k a} + k a ℑ (\sum_{n = 0}^{N} A_{n})} \approx ρ_{0} c \frac{2 k a}{π}, k a < 0.5. \end{matrix}

$\begin{matrix} {X_{s} |}_{b \neq a} = \frac{ρ_{0} c}{2} {\frac{H_{1} (2 k a)}{k a} + k b ℑ (\sum_{n = 0}^{N} A_{n} {1 - {(1 - \frac{a^{2}}{b^{2}})}^{n + \frac{3}{2}}})} \\ {X_{s} |}_{b = a} = \frac{ρ_{0} c}{2} {\frac{H_{1} (2 k a)}{k a} + k a ℑ (\sum_{n = 0}^{N} A_{n})} \approx ρ_{0} c \frac{2 k a}{π}, k a < 0.5. \end{matrix}$

(13.255)

Plots of the real and imaginary parts of

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

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

(13.256)

are shown in Fig. 13.36 as a function of ka.

Figure 13.36 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 a flat circular closed baffle of radius b. When b = a (solid black curve), there is no baffle and the piston is radiating from one side only into free space. When b = ∞ (dotted black curve), the piston is in an infinite baffle. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

The specific admittance is given by

\begin{matrix} Y_{s} = G_{s} + j B_{s} = \frac{R_{s}}{R_{s}^{2} + X_{s}^{2}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}} \\ {Y_{s} |}_{b = a} = {(G_{s} + j B_{s}) |}_{b = a} \approx \frac{1}{ρ_{0} c} (\frac{π^{2}}{16} - j \frac{π}{2 k a}), k a < 0.5. \end{matrix}

$\begin{matrix} Y_{s} = G_{s} + j B_{s} = \frac{R_{s}}{R_{s}^{2} + X_{s}^{2}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}} \\ {Y_{s} |}_{b = a} = {(G_{s} + j B_{s}) |}_{b = a} \approx \frac{1}{ρ_{0} c} (\frac{π^{2}}{16} - j \frac{π}{2 k a}), k a < 0.5. \end{matrix}$

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Table of Contents for Acoustics

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13.11. Radiation from a rigid circular piston in a finite circular closed baffle [32] (one-sided radiator)

Far-field pressure

Near-field pressure

Radiation impedance

Table of Contents for
Acoustics