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13.8. Radiation from a resilient circular disk without a baffle [17]

The resilient circular disk in free space is the simplest dipole planar source and the dipole complement of the rigid circular piston in an infinite baffle. It can be used as an approximate model for unbaffled loudspeakers of the electrostatic or planar magnetic type, in which it is assumed that a perfectly uniform driving pressure is applied to a very light flexible membrane diaphragm in free space. Because of the dipole nature of the source, there is zero pressure in the plane of the disk extending beyond its perimeter. Walker [18] pointed out that such a source is acoustically transparent, in that it does not disturb the field around it, and used this idealized model to derive the far-field on-axis pressure response of an electrostatic loudspeaker, which provides a useful approximation over the loudspeaker's working range. However, it should be noted that the model assumes a freely suspended membrane, whereas in reality it is usually clamped at the perimeter, which effectively removes the singularity from the perimeter of the idealized model [19].

Boundary conditions

The basic configuration is shown in Fig. 13.8. The infinitesimally thin membrane-like resilient disk is assumed to be perfectly flexible, has zero mass, and is free at its perimeter. It is driven by a uniformly distributed harmonically varying pressure

{\tilde{p}}_{0}

${\tilde{p}}_{0}$

and thus radiates sound from both sides into a homogeneous loss-free acoustic medium. In fact, there need not be a disk present at all and instead the driving pressure could be acting on the air particles directly. However, for expedience, the area over which this driving pressure is applied shall be referred to as a disk from here onwards. The pressure field on one side of the xy plane is the symmetrical “negative” of that on the other, so that

Figure 13.8 Geometry of resilient circular disk in free space. The point of observation P is located at a distance r and angle θ with respect to the origin at the center of the disk.

\tilde{p} (w, z) = - \tilde{p} (w, - z) .

$\tilde{p} (w, z) = - \tilde{p} (w, - z) .$

(13.121)

Consequently, there is a Dirichlet boundary condition in the plane of the disk where these equal and opposite fields meet.

\tilde{p} (w, 0) = 0, a < w \leq \infty .

$\tilde{p} (w, 0) = 0, a < w \leq \infty .$

(13.122)

On the front and rear surfaces of the disk, the pressures are

{\tilde{p}}_{+}

${\tilde{p}}_{+}$

and

{\tilde{p}}_{_}

${\tilde{p}}_{_}$

respectively, which are given by

\tilde{p} + (w_{0}) = - {\tilde{p}}_{-} (w_{0}) = {\tilde{p}}_{0} / 2, 0 \leq w_{0} \leq a

$\tilde{p} + (w_{0}) = - {\tilde{p}}_{-} (w_{0}) = {\tilde{p}}_{0} / 2, 0 \leq w_{0} \leq a$

(13.123)

and k is the wave number given by k = ω/c = 2π/λ, where ω is the angular frequency of excitation, ρ ₀ is the density of the surrounding medium, c is the speed of sound in that medium, and λ is the wavelength.

Far-field pressure

The far-field pressure distribution is given by the dipole boundary integral of Eq. (13.28), taking into account the surface pressure on both sides:

\tilde{p} (r, θ) = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - {\tilde{p}}_{-} (w_{0})) \frac{\partial}{\partial z_{0}} g {(r, θ | w_{0}, ϕ_{0}) |}_{z_{0} = 0 +} w_{0} d w_{0} d ϕ_{0},

$\tilde{p} (r, θ) = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - {\tilde{p}}_{-} (w_{0})) \frac{\partial}{\partial z_{0}} g {(r, θ | w_{0}, ϕ_{0}) |}_{z_{0} = 0 +} w_{0} d w_{0} d ϕ_{0},$

(13.124)

where the far-field Green's function in spherical-cylindrical coordinates given by Eq. (13.70) is used. Inserting Eqs. (13.70), (13.121) and (13.123) into Eq. (13.124) and integrating over the surface, using Eqs. (76) and (95) from Appendix II [with z = kw ₀ sin θ, b = k sin θ, and letting ϕ = π/2 so that cos(ϕ − ϕ ₀) = sin ϕ ₀], gives

\tilde{p} (r, θ) = j k a^{2} {\tilde{p}}_{0} \frac{e^{- j k r}}{4 r} D (θ),

$\tilde{p} (r, θ) = j k a^{2} {\tilde{p}}_{0} \frac{e^{- j k r}}{4 r} D (θ),$

(13.125)

where the directivity function D(θ) is given by

D (θ) = \frac{2 J_{1} (k a \sin θ)}{k a \sin θ} \cos θ .

$D (θ) = \frac{2 J_{1} (k a \sin θ)}{k a \sin θ} \cos θ .$

(13.126)

The on-axis pressure is evaluated by setting θ = 0 in Eq. (13.70) before inserting it in Eq. (13.124) and integrating over the surface to give

D (0) = 1

$D (0) = 1$

(13.127)

so that the on-axis response can be written as

\tilde{p} (r, 0) = j f {\tilde{F}}_{0} \frac{e^{- j k r}}{2 r c},

$\tilde{p} (r, 0) = j f {\tilde{F}}_{0} \frac{e^{- j k r}}{2 r c},$

(13.128)

which just gives a constant 6 dB/octave rising response at all frequencies for a given driving force

{\tilde{F}}_{0} = S {\tilde{p}}_{0}

${\tilde{F}}_{0} = S {\tilde{p}}_{0}$

, where S = πa ² is the area. Eq. (13.128) is true for a planar resilient radiator of any shape. The normalized directivity function 20 log₁₀|D(θ)/D(0)| is plotted in Fig. 13.9 for four values of ka = 2πa/λ, that is, for four values of the ratio of the circumference of the disk to the wavelength. The directivity pattern is that of a rigid piston in an infinite baffle multiplied by cos θ. When the circumference of the disk (2πa) is less than one-half wavelength, that is, ka < 0.5, the resilient disk behaves essentially like a dipole point source. When ka becomes greater than 3, the resilient disk is highly directional, like the piston in an infinite baffle. In fact, at very high frequencies, they both radiate sound as a narrow central lobe (Airy disk) accompanied by a number of very small side lobes, in which case the factor of cos θ makes relatively little difference. In the case of a push-pull electrostatic loudspeaker,

{\tilde{p}}_{0} = \frac{E_{P}}{d} \cdot \frac{2 {\tilde{I}}_{i n}}{j ωπ a^{2}},

${\tilde{p}}_{0} = \frac{E_{P}}{d} \cdot \frac{2 {\tilde{I}}_{i n}}{j ωπ a^{2}},$

(13.129)

where E _P is the polarizing voltage, d is the membrane-electrode separation, and

{\tilde{I}}_{i n}

${\tilde{I}}_{i n}$

is the static input current to each electrode, assuming that the motional current is negligible in comparison. Substituting this in Eq. (13.128) yields

Figure 13.9 Far-field directivity patterns for a resilient circular disk in free space as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the disk.

\tilde{p} (r, 0) = \frac{E_{P}}{d} \cdot \frac{{\tilde{I}}_{i n} e^{- j k r}}{2 π r c},

$\tilde{p} (r, 0) = \frac{E_{P}}{d} \cdot \frac{{\tilde{I}}_{i n} e^{- j k r}}{2 π r c},$

(13.130)

which is Walker's equation [18], albeit obtained by a slightly different method.

Near-field pressure

The near-field pressure distribution is given by Eq. (13.28) taking into account the surface pressure on both sides:

\tilde{p} (r, θ) = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - {\tilde{p}}_{_} (w_{0})) \frac{\partial}{\partial z_{0}} g {(r, θ | w_{0}, ϕ_{0}) |}_{z_{0} = 0 +} w_{0} d w_{0} d ϕ_{0},

$\tilde{p} (r, θ) = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - {\tilde{p}}_{_} (w_{0})) \frac{\partial}{\partial z_{0}} g {(r, θ | w_{0}, ϕ_{0}) |}_{z_{0} = 0 +} w_{0} d w_{0} d ϕ_{0},$

(13.131)

where the Green's function in spherical-cylindrical coordinates given by Eq. (13.68) is used. It has been shown [17] that inserting Eqs. (13.68) and (13.123) into Eq. (13.131) and integrating over the surface gives

\begin{matrix} \tilde{p} (r, θ) = - j {\tilde{p}}_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} Γ (n + \frac{3}{2})}{Γ (n + 2) Γ (2 n + \frac{3}{2})} {(\frac{k a}{2})}^{2 n + 2} \\ \times_{1} F_{2} (n + 1; n + 2,2 n + \frac{5}{2}; - \frac{k^{2} a^{2}}{4}) h_{2 n + 1}^{(2)} (k r) P_{2 n + 1} (\cos θ), \end{matrix}

$\begin{matrix} \tilde{p} (r, θ) = - j {\tilde{p}}_{0} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} Γ (n + \frac{3}{2})}{Γ (n + 2) Γ (2 n + \frac{3}{2})} {(\frac{k a}{2})}^{2 n + 2} \\ \times_{1} F_{2} (n + 1; n + 2,2 n + \frac{5}{2}; - \frac{k^{2} a^{2}}{4}) h_{2 n + 1}^{(2)} (k r) P_{2 n + 1} (\cos θ), \end{matrix}$

(13.132)

which converges for r > a but is generally used for w ≥ a. The other part of the Green's function of Eq. (13.68) could be used to derive an expression for r < a. However, a better expression is provided by moving the origin of the coordinate system to a point on the z axis that lies on the same plane as the observation point to give

\tilde{p} (w, z) = \frac{j {\tilde{p}}_{0}}{\sqrt{π} k w} \sum_{n = 0}^{\infty} {(- 1)}^{n} (4 n + 3) \frac{Γ (n + \frac{3}{2})}{Γ (n + 1)} j_{2 n + 1} (k w) f_{2 n + 1},

$\tilde{p} (w, z) = \frac{j {\tilde{p}}_{0}}{\sqrt{π} k w} \sum_{n = 0}^{\infty} {(- 1)}^{n} (4 n + 3) \frac{Γ (n + \frac{3}{2})}{Γ (n + 1)} j_{2 n + 1} (k w) f_{2 n + 1},$

(13.133)

where f _2n+1 is given by the following recursion formulas:

f_{1} = j (\frac{z}{r_{a}} e^{- j k r_{a}} - e^{- j k z}),

$f_{1} = j (\frac{z}{r_{a}} e^{- j k r_{a}} - e^{- j k z}),$

(13.134)

f_{2 n + 1} = - f_{2 n - 1} + k r_{a} h_{2 n}^{(2)} (k r_{a}) (P_{2 n + 1} (z / r_{a}) - P_{2 n - 1} (z / r_{a})),

$f_{2 n + 1} = - f_{2 n - 1} + k r_{a} h_{2 n}^{(2)} (k r_{a}) (P_{2 n + 1} (z / r_{a}) - P_{2 n - 1} (z / r_{a})),$

(13.135)

and

r_{a} = \sqrt{z^{2} + a^{2}},

$r_{a} = \sqrt{z^{2} + a^{2}},$

(13.136)

which converges for w ² < a ² + z ² but is generally used for w < a and is thus termed the paraxial solution. The number of terms in the expansion needed for convergence decreases linearly toward the z-axis until just a single term is needed. This is the closed-form axial solution:

\tilde{p} (0, z) = \frac{{\tilde{p}}_{0}}{2} (e^{- j k z} - \frac{z}{\sqrt{z^{2} + a^{2}}} e^{- j k \sqrt{z^{2} + a^{2}}}) .

$\tilde{p} (0, z) = \frac{{\tilde{p}}_{0}}{2} (e^{- j k z} - \frac{z}{\sqrt{z^{2} + a^{2}}} e^{- j k \sqrt{z^{2} + a^{2}}}) .$

(13.137)

The pressure field for three values of ka is plotted in Fig. 13.10. We can see that the plane wave region near the surface forms more readily than in the case of the rigid piston (see Fig. 13.6), no doubt aided by the uniform pressure distribution at the surface of the resilient disk. At ka = 6π, the pressure field fluctuations in the vicinity of the resilient disk are smaller than for the rigid piston. Furthermore, the axial pressure response of a rigid disk given by Eq. (13.111) has nulls, whereas the resilient disk axial response given by Eq. (13.137) is oscillatory but with decreasing magnitude toward the face of the disk. An alternative expression to Eq. (13.133) is given in Reference [48].

Surface velocity

Using the solutions for the near-field pressure from Eqs. (13.133)–(13.135), and taking the normal pressure gradient at the surface of the disk, the surface velocity is given by

\begin{matrix} {\tilde{u}}_{0} (w) = \frac{j}{k ρ c} \frac{d}{d z} \tilde{p} {(w, z) |}_{z = 0 +} \\ = - \frac{{\tilde{p}}_{0}}{ρ c \sqrt{π}} \sum_{n = 0}^{\infty} {(- 1)}^{n} (4 n + 3) \frac{Γ (n + \frac{3}{2})}{Γ (n + 1)} f_{2 n}^{'} \frac{j_{2 n + 1} (k w)}{k w}, \end{matrix}

$\begin{matrix} {\tilde{u}}_{0} (w) = \frac{j}{k ρ c} \frac{d}{d z} \tilde{p} {(w, z) |}_{z = 0 +} \\ = - \frac{{\tilde{p}}_{0}}{ρ c \sqrt{π}} \sum_{n = 0}^{\infty} {(- 1)}^{n} (4 n + 3) \frac{Γ (n + \frac{3}{2})}{Γ (n + 1)} f_{2 n}^{'} \frac{j_{2 n + 1} (k w)}{k w}, \end{matrix}$

(13.138)

where

f_{0}^{'} = 1 - j \frac{e^{- j k a}}{k a},

$f_{0}^{'} = 1 - j \frac{e^{- j k a}}{k a},$

(13.139)

f_{2 n}^{'} = - f_{2 n - 2}^{'} - h_{2 n}^{(2)} (k a) ((2 n + 1) P_{2 n} (0) - (2 n - 1) P_{2 n - 2} (0)) .

$f_{2 n}^{'} = - f_{2 n - 2}^{'} - h_{2 n}^{(2)} (k a) ((2 n + 1) P_{2 n} (0) - (2 n - 1) P_{2 n - 2} (0)) .$

(13.140)

The magnitude and phase of the normalized velocity are shown in Figs. 13.11 and 13.12, respectively, for four values of ka. For small k, it can be shown to agree well with the asymptotic expression given by Eq. (13.144). We see that the velocity increases rapidly toward the perimeter, where it is singular. This is a feature of uniform pressure sources in general due to the discontinuity at the perimeter. However, it is exacerbated in this case by the acoustic short circuit between the front and rear surfaces of the dipole source.

Figure 13.10 Normalized near-field pressure plots for a resilient circular disk in free space as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the disk. $| \tilde{p} |$ $| \tilde{p} |$ is the pressure magnitude, ${\tilde{p}}_{0}$ ${\tilde{p}}_{0}$ is the driving pressure.

Figure 13.11 Normalized surface velocity magnitude for a resilient circular disk in free space as a function of w/a, where w is the radial ordinate and ka = 2πa/λ = ω/c, where a is the radius of the disk.

Figure 13.12 Surface velocity phase for a resilient circular disk in free space as a function of w/a, where w is the radial ordinate and ka = 2πa/λ = ω/c, where a is the radius of the disk.

Radiation admittance and low-frequency asymptotic surface velocity

The near-field pressure distribution is given by Eq. (13.28) taking into account the surface pressure on both sides:

\tilde{p} (w, z) = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - {\tilde{p}}_{-} (w_{0})) \frac{\partial}{\partial z_{0}} g {(w, z | w_{0}, z_{0}) |}_{z_{0} = 0 +} w_{0} d w_{0} d ϕ_{0},

$\tilde{p} (w, z) = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - {\tilde{p}}_{-} (w_{0})) \frac{\partial}{\partial z_{0}} g {(w, z | w_{0}, z_{0}) |}_{z_{0} = 0 +} w_{0} d w_{0} d ϕ_{0},$

(13.141)

where the integral Green's function in cylindrical coordinates given by Eq. (13.52) is used. In this form Eq. (13.141) is known as the dipole King integral. Inserting Eqs. (13.52) and (13.123) into Eq. (13.141) and integrating over the surface gives

\tilde{p} (w, z) = - k a ρ_{0} c {\tilde{u}}_{0} \int_{0}^{\infty} J_{0} (k_{w} w) J_{1} (k_{w} a) \frac{1}{k_{z}} e^{- j k_{z} z} d k_{w},

$\tilde{p} (w, z) = - k a ρ_{0} c {\tilde{u}}_{0} \int_{0}^{\infty} J_{0} (k_{w} w) J_{1} (k_{w} a) \frac{1}{k_{z}} e^{- j k_{z} z} d k_{w},$

(13.142)

where we have again used the integral of Eq. (95) from Appendix II and k _z is given by Eq. (13.51). The disk velocity

{\tilde{u}}_{0} (w)

${\tilde{u}}_{0} (w)$

can be derived using the following relationship for the normal pressure gradient:

\begin{matrix} {\tilde{u}}_{0} (w) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial z} \tilde{p} {(w, z) |}_{z = 0 +} \\ = \frac{a {\tilde{p}}_{0}}{2 k ρ_{0} c} \int_{0}^{\infty} J_{1} (k_{w} a) J_{0} (k_{w} w) k_{z} d k_{w} . \end{matrix}

$\begin{matrix} {\tilde{u}}_{0} (w) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial z} \tilde{p} {(w, z) |}_{z = 0 +} \\ = \frac{a {\tilde{p}}_{0}}{2 k ρ_{0} c} \int_{0}^{\infty} J_{1} (k_{w} a) J_{0} (k_{w} w) k_{z} d k_{w} . \end{matrix}$

(13.143)

For small k, we obtain

\begin{matrix} {\tilde{u}}_{0} {(w) |}_{k \to 0} = \frac{j a {\tilde{p}}_{0}}{2 k ρ_{0} c} \int_{0}^{\infty} J_{1} (k_{w} a) J_{0} (k_{w} w) k_{w} d k_{w} \\ = \frac{j {\tilde{p}}_{0} E (w^{2} / a^{2})}{π k a ρ_{0} c} {(1 - \frac{w^{2}}{a^{2}})}^{- 1}, \end{matrix}

$\begin{matrix} {\tilde{u}}_{0} {(w) |}_{k \to 0} = \frac{j a {\tilde{p}}_{0}}{2 k ρ_{0} c} \int_{0}^{\infty} J_{1} (k_{w} a) J_{0} (k_{w} w) k_{w} d k_{w} \\ = \frac{j {\tilde{p}}_{0} E (w^{2} / a^{2})}{π k a ρ_{0} c} {(1 - \frac{w^{2}}{a^{2}})}^{- 1}, \end{matrix}$

(13.144)

where E is the complete elliptic integral of the second kind. Hence there is a singularity at the perimeter. The total volume velocity

{\tilde{U}}_{0}

${\tilde{U}}_{0}$

is found by integrating the velocity from Eq. (13.143) over the surface of the disk and again using the integral of Eq. (95) from Appendix II to give

\begin{matrix} {\tilde{U}}_{0} = \int_{0}^{2 π} \int_{0}^{a} {\tilde{u}}_{0} (w) w d w d ϕ \\ = \frac{π a^{2} {\tilde{p}}_{0}}{k ρ_{0} c} (\int_{0}^{k} J_{1}^{2} (k_{w} a) \frac{\sqrt{k^{2} - k_{w}^{2}}}{k_{w}} d k_{w} - j \int_{k}^{\infty} J_{1}^{2} (k_{w} a) \frac{\sqrt{k_{w}^{2} - k^{2}}}{k_{w}} d k_{w}), \end{matrix}

$\begin{matrix} {\tilde{U}}_{0} = \int_{0}^{2 π} \int_{0}^{a} {\tilde{u}}_{0} (w) w d w d ϕ \\ = \frac{π a^{2} {\tilde{p}}_{0}}{k ρ_{0} c} (\int_{0}^{k} J_{1}^{2} (k_{w} a) \frac{\sqrt{k^{2} - k_{w}^{2}}}{k_{w}} d k_{w} - j \int_{k}^{\infty} J_{1}^{2} (k_{w} a) \frac{\sqrt{k_{w}^{2} - k^{2}}}{k_{w}} d k_{w}), \end{matrix}$

(13.145)

The solution [17,20] has been shown to be

Y_{s} = \frac{{\tilde{U}}_{0}}{S {\tilde{p}}_{0}} = G_{s} + j B_{s},

$Y_{s} = \frac{{\tilde{U}}_{0}}{S {\tilde{p}}_{0}} = G_{s} + j B_{s},$

(13.146)

where

{\tilde{U}}_{0} = π a^{2} {\tilde{u}}_{0}

${\tilde{U}}_{0} = π a^{2} {\tilde{u}}_{0}$

is the total volume velocity and G _s is the specific radiation conductance in m³·N⁻¹ s⁻¹ (rayl⁻¹) given by

\begin{matrix} G_{s} = \frac{1}{ρ_{0} c} (1 + \frac{J_{1} (2 k a)}{k a} - 2 J_{0} (2 k a) - π (J_{1} (2 k a) H_{0} (2 k a) - J_{0} (2 k a) H_{1} (2 k a))) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{k^{2} a^{2}}{6}, k a < 0.5, \end{matrix}

$\begin{matrix} G_{s} = \frac{1}{ρ_{0} c} (1 + \frac{J_{1} (2 k a)}{k a} - 2 J_{0} (2 k a) - π (J_{1} (2 k a) H_{0} (2 k a) - J_{0} (2 k a) H_{1} (2 k a))) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{k^{2} a^{2}}{6}, k a < 0.5, \end{matrix}$

(13.147)

where the bold G indicates that the quantity varies with frequency and B _s is the specific radiation susceptance in m³·N⁻¹ s⁻¹ (rayl⁻¹) given by

\begin{matrix} B_{s} = - \frac{1}{ρ_{0} c} (\frac{4}{π k a} - \frac{H_{1} (2 k a)}{k a} + \frac{4 k a}{π} {}_{2}{F_{3}} (1,1; \frac{3}{2}, \frac{3}{2}, 2; - k^{2} a^{2})) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{4}{π k a}, k a < 0.5, \end{matrix}

$\begin{matrix} B_{s} = - \frac{1}{ρ_{0} c} (\frac{4}{π k a} - \frac{H_{1} (2 k a)}{k a} + \frac{4 k a}{π} {}_{2}{F_{3}} (1,1; \frac{3}{2}, \frac{3}{2}, 2; - k^{2} a^{2})) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{4}{π k a}, k a < 0.5, \end{matrix}$

(13.148)

where J _n and H _n are Bessel and Struve functions respectively and ₂ F ₃ is a hypergeometric function. Plots of the real and imaginary parts of

ρ_{0} c Y_{s} = ρ_{0} c (G_{s} + j B_{s})

$ρ_{0} c Y_{s} = ρ_{0} c (G_{s} + j B_{s})$

(13.149)

are shown in Fig. 13.13 as a function of ka. Similar graphs of the real and imaginary parts of the specific impedance

\begin{matrix} \frac{Z_{s}}{ρ_{0} c} = \frac{R_{s} + j X_{s}}{ρ_{0} c} = \frac{1}{ρ_{0} c} (\frac{G_{s}}{G_{s}^{2} + B_{s}^{2}} - j \frac{B_{s}}{G_{s}^{2} + B_{s}^{2}}) \\ \approx \frac{π k^{4} a^{4}}{96} - j \frac{π k a}{4}, k a < 0.5 \end{matrix}

$\begin{matrix} \frac{Z_{s}}{ρ_{0} c} = \frac{R_{s} + j X_{s}}{ρ_{0} c} = \frac{1}{ρ_{0} c} (\frac{G_{s}}{G_{s}^{2} + B_{s}^{2}} - j \frac{B_{s}}{G_{s}^{2} + B_{s}^{2}}) \\ \approx \frac{π k^{4} a^{4}}{96} - j \frac{π k a}{4}, k a < 0.5 \end{matrix}$

(13.150)

are shown in Fig. 13.14. The specific admittance is in m³·N⁻¹ s⁻¹ (rayl⁻¹). Although the impedance and admittance functions of the rigid disk in an infinite baffle show ripples (see Figs. 13.35 and 13.36, respectively), those of the resilient disk are smooth almost monotonic functions. We can see that at low frequencies the impedance and admittance curves are more reactive than those of a piston in an infinite baffle, so that less power is radiated. This is due to the cancellation of the acoustic output by the rear wave or acoustic “short circuit,” which is generally the case with all dipole sources.

Figure 13.13 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 resilient disk of radius a in free space. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Figure 13.14 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 resilient disk of radius a in free space. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Relationship between a resilient disk in free space and a rigid piston in an infinite baffle

Suppose that the radiation resistance and reactance of a rigid disk in an infinite baffle are denoted by R _s and X _s respectively and G _s and B _s are the radiation conductance and susceptance respectively of a resilient disk in free space as defined in Eqs. (13.147) and (13.148), then

{(ρ_{0} c)}^{2} \frac{d}{d (k a)} k a G_{s} (k a) = R_{s} (k a) = ρ_{0} c (1 - \frac{J_{1} (2 k a)}{k a}),

${(ρ_{0} c)}^{2} \frac{d}{d (k a)} k a G_{s} (k a) = R_{s} (k a) = ρ_{0} c (1 - \frac{J_{1} (2 k a)}{k a}),$

(13.151)

G_{s} (k a) = \frac{1}{k a {(ρ_{0} c)}^{2}} \int R_{s} (k a) d (k a),

$G_{s} (k a) = \frac{1}{k a {(ρ_{0} c)}^{2}} \int R_{s} (k a) d (k a),$

(13.152)

and

{(ρ_{0} c)}^{2} \frac{d}{d (k a)} k a B_{s} (k a) = X_{s} (k a) = ρ_{0} c (\frac{H_{1} (2 k a)}{k a}),

${(ρ_{0} c)}^{2} \frac{d}{d (k a)} k a B_{s} (k a) = X_{s} (k a) = ρ_{0} c (\frac{H_{1} (2 k a)}{k a}),$

(13.153)

B_{s} (k a) = \frac{1}{k a {(ρ_{0} c)}^{2}} (\int X_{s} (k a) d (k a) + \frac{4}{π}) .

$B_{s} (k a) = \frac{1}{k a {(ρ_{0} c)}^{2}} (\int X_{s} (k a) d (k a) + \frac{4}{π}) .$

(13.154)

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Table of Contents for Acoustics

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