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13.10. Radiation from a rigid circular piston in a finite circular open baffle [26,27]

A disk in a circular baffle is a useful model for an open-baffle type loudspeaker and in the limiting case a loudspeaker without a baffle of any sort. Loudspeaker drive units are often measured in a finite baffle such as the rectangular IEC 268-5 baffle. See IEC 60268-5, ed. 3.1, “Sound system equipment - Part 5: Loudspeakers,” available from http://webstore.iec.ch/. For example, for a nominal 8-in (200 mm) diameter loudspeaker, the baffle size would be 1.65 m long by 1.35 m wide, with the loudspeaker offset from the center by 22.5 cm lengthways and 15 cm widthways. If we have a rigorous model of the baffle, we can subtract its diffraction effects from the measurement to reveal the true response of the drive unit. The problem was first solved by Nimura and Watanabe [28] using the boundary value method in the oblate-spheroidal coordinate system. However, oblate-spheroidal functions are rather complicated, so instead we shall use the boundary integral method with the Green's function in cylindrical coordinates and a trial function first used by Streng [23] for a membrane. Previous solutions for the limiting case of a disk in free space have been obtained by Bouwkamp [20] using the boundary value method and Sommerfeld [29] using the boundary integral method in cylindrical coordinates. Meixner and Fritze [30] plotted the near-field pressure, a formidable task without the benefit of modern computing power, and Wiener [31] plotted the far-field directivity pattern.

Boundary conditions

The circular piston of radius a shown in Fig. 13.21 is mounted in a finite circular baffle of radius b in the xy plane with its center at the origin and oscillates in the z direction with a harmonically time-dependent velocity

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

, thus radiating sound from both sides into a homogeneous loss-free medium. The dipole source elements shown in Fig. 13.21 form the piston source. The area of each surface element is given by

$δ S_{0} = w_{0} δ w_{0} δ ϕ_{0} \cdot$

(13.197)

The pressure field on one side of the xy plane is the symmetrical “negative” of that on the other, so that

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

(13.198)

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

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

(13.199)

which is satisfied automatically. On the front and rear surfaces of the baffle, there is a Neumann boundary condition

Figure 13.21 Geometry of rigid circular piston in finite baffle. The point of observation P is located at a distance r and angle θ with respect to the origin at the center of the piston.

$\frac{\partial}{\partial z} \tilde{p} {(w, z) |}_{z = 0 \pm} = 0, a < w \leq b .$

(13.200)

Also, on the front and rear surfaces of the disk, there is the coupling condition

$\frac{\partial}{\partial z} \tilde{p} {(w, z) |}_{z = 0 \pm} = - j k ρ_{0} c {\tilde{u}}_{0}, 0 \leq w \leq a,$

(13.201)

where k is the wave number given by k = ω/c = 2π/λ, 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. To tackle this problem, we shall use the dipole surface integral of Eq. (13.28). However, some prior expression for the frontal surface pressure distribution

${\tilde{p}}_{+} (w_{0})$

is needed. In addition, because the disk can radiate from both sides, the rear surface pressure distribution

${\tilde{p}}_{-} (w_{0})$

must be included too, where

${\tilde{p}}_{+} (w_{0}) = - {\tilde{p}}_{-} (w_{0})$

. Streng [23] showed that the surface pressure distribution for any flat axially symmetric unbaffled source (or sink), based on Bouwkamp's solution [20] to the free-space wave equation in oblate-spheroidal coordinates, could be written as

${\tilde{p}}_{+} (w_{0}) = - {\tilde{p}}_{-} (w_{0}) = k b ρ_{0} c {\tilde{u}}_{0} \frac{a^{2}}{b^{2}} \sum_{n = 0}^{\infty} A_{n} (n + \frac{3}{2}) {(1 - \frac{w_{0}^{2}}{b^{2}})}^{n + \frac{1}{2}}, 0 \leq w_{0} \leq b .$

(13.202)

where A _n are the as-yet unknown power series coefficients, which will be evaluated by means of a set of simultaneous equations in matrix form.

Formulation of the coupled equation

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

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

(13.203)

where the Green's function in cylindrical coordinates given by Eq. (13.52) is used. In this form Eq. (13.203) is known as the dipole King integral. Inserting Eqs. (13.52), (13.198) and (13.202) into Eq. (13.203) and integrating over ϕ ₀ gives

$\begin{matrix} \tilde{p} (w, z) = k b ρ_{0} c {\tilde{u}}_{0} \frac{a^{2}}{b^{2}} \sum_{n = 0}^{\infty} A_{n} (n + \frac{3}{2}) \int_{0}^{b} {(1 - \frac{w_{0}^{2}}{b^{2}})}^{n + \frac{1}{2}} \\ \times \int_{0}^{\infty} J_{0} (k_{w} w) J_{0} (k_{w} w_{0}) e^{- j k_{z} | z |} k_{w} d k_{w} w_{0} d w_{0}, \end{matrix}$

(13.204)

where k _z is given by Eq. (13.51). At the surface of the disk, we have the coupling condition

$\frac{\partial}{\partial z} {\tilde{p} (w, z) |}_{z = 0} = - j k ρ_{0} c {\tilde{u}}_{0} Φ (w),$

(13.205)

where Φ(w) is a dimensionless function of the surface velocity distribution. We will use different expressions for Φ(w) when considering a piston in free space and a piston or point source in a circular baffle. This leads to the following coupled equation

$\sum_{n = 0}^{\infty} A_{n} I_{n} (w) = Φ (w),$

(13.206)

where

$I_{n} (w) = \frac{a^{2}}{b} (n + \frac{3}{2}) \int_{0}^{b} {(1 - \frac{w_{0}^{2}}{b^{2}})}^{n + \frac{1}{2}} \int_{0}^{\infty} J_{0} (k_{w} w) J_{0} (k_{w} w_{0}) \sqrt{k^{2} - k_{w}^{2}} k_{w} d k_{w} w_{0} d w_{0} .$

(13.207)

The infinite integral [46–48] is given by Eq. (A2.106b), together with Eqs. (A2.11a) and (A2.150), from Appendix II with m = n = 0 and γ = 1

$\begin{matrix} \int_{0}^{\infty} J_{0} (k_{w} w) J_{0} (k_{w} w_{0}) \sqrt{k^{2} - k_{w}^{2}} k_{w} d k_{w} \\ = - \frac{2 j \sqrt{π}}{w_{0}^{3}} \sum_{m = 0}^{\infty} \frac{1}{{(m!)}^{2}} {(\frac{w}{w_{0}})}^{2 m} \sum_{r = 0}^{\infty} \frac{Γ (\frac{r}{2} - \frac{1}{2} + δ_{r, 1})}{Γ (\frac{r}{2} + 1) Γ^{2} (\frac{r}{2} - m - \frac{1}{2})} {(\frac{- j k w_{0}}{2})}^{r}, \end{matrix}$

(13.208)

where δ _r,1 is the Kronecker delta function that is included to ensure that the r = 1 term goes to zero in an orderly way without having a singularity due to Γ(0) in the numerator. Hence

$\begin{matrix} I_{n} (w) = - 2 j \sqrt{π} \frac{a^{2}}{b} (n + \frac{3}{2}) \sum_{m = 0}^{\infty} \frac{w^{2 m}}{{(m!)}^{2}} \\ \times \sum_{r = 0}^{\infty} \frac{Γ (\frac{r}{2} - \frac{1}{2} + δ_{r, 1})}{Γ (\frac{r}{2} + 1) Γ^{2} (\frac{r}{2} - m - \frac{1}{2})} {(\frac{- j k}{2})}^{r} \int_{0}^{b} {(1 - \frac{w_{0}^{2}}{b^{2}})}^{n + \frac{1}{2}} w_{0}^{r - 2 m - 2} d w_{0}, \end{matrix}$

(13.209)

which is simplified with help of the integral

$\int_{0}^{b} {(1 - \frac{w_{0}^{2}}{b^{2}})}^{n + \frac{1}{2}} w_{0}^{μ} d w_{0} = \frac{b^{μ + 1} Γ (n + \frac{3}{2}) Γ (\frac{μ + 1}{2})}{2 Γ (n + \frac{μ}{2} + 2)}$

(13.210)

to give

$\begin{matrix} I_{n} (w) = - j \sqrt{π} \frac{a^{2}}{b^{2}} Γ (n + \frac{5}{2}) \sum_{m = 0}^{\infty} \frac{1}{{(m!)}^{2}} {(\frac{w}{b})}^{2 m} \\ \times \sum_{r = 0}^{\infty} \frac{Γ (\frac{r}{2} - \frac{1}{2} + δ_{r, 1})}{Γ (\frac{r}{2} + 1) Γ (\frac{r}{2} - m - \frac{1}{2}) Γ (\frac{r}{2} + n - m + 1)} {(\frac{- j k b}{2})}^{r} . \end{matrix}$

(13.211)

Solution of the power series coefficients for a piston in free space

Eq. (13.211) is an expansion in (w/b)^2m. Hence, to solve for the expansion coefficients, it is useful to also express the disk and baffle velocity distribution Φ(w) as a function of (w/b)^2m. In the case of a disk in free space where b = a, we have

${Φ (w) |}_{b = a} = 1 = \sum_{m = 0}^{\infty} δ_{m, 0} {(\frac{w}{a})}^{2 m}, 0 \leq w \leq a$

(13.212)

where δ _m0 is the Kronecker delta function. Inserting Eqs. (13.211) and (13.212) in Eq. (13.206) and equating the coefficients of (w/a)^2m yields the (N + 1) × (N + 1) matrix equation

$M \cdot a = b,$

(13.213)

where the matrix M and vectors a and b are given by

$M (m + 1, n + 1) = {{}_{n}B}_{m} (k a), {\begin{matrix} m = 0, 1, \dots, N \\ n = 0, 1, \dots, N \end{matrix},$

(13.214)

$b (m + 1) = δ_{m, 0}, m = 0, 1, \dots, N,$

(13.215)

$a (n + 1) = A_{n}, n = 0, 1, \dots, N,$

(13.216)

and the infinite power series limits have been truncated to order N. The dipole cylindrical wave function _n B _m is named the Bouwkamp–Streng function in tribute to their pioneering work and is defined by

${{}_{n}B}_{m} (k a) = - j \sqrt{π} Γ (n + \frac{5}{2}) \frac{1}{{(m!)}^{2}} \sum_{r = 0}^{\infty} {{}_{n}S}_{m} (r) {(\frac{- j k a}{2})}^{r},$

(13.217)

where

${{}_{n}S}_{m} (r) = \frac{Γ (\frac{r}{2} - \frac{1}{2} + δ_{r, 1})}{Γ (\frac{r}{2} + 1) Γ (\frac{r}{2} - m - \frac{1}{2}) Γ (\frac{r}{2} + n - m + 1)} .$

(13.218)

Solution of the power series coefficients for a piston in a circular baffle

For a finite baffle, where b ≠ a, we can employ the following least-mean-squares (LMS) algorithm. From Eq. (13.206), let an error function be defined by

$E (A_{n}) = \int_{0}^{b} {| \sum_{n = 0}^{N} A_{n} I_{n} (w) - Φ (w) |}^{2} w d w .$

(13.219)

where

$Φ (w) = {\begin{matrix} 1, 0 \leq w \leq a \\ 0, a < w \leq b, \end{matrix}$

(13.220)

To find the values of A _n that minimize the error, we take the derivative of E with respect to A _n and equate the result to zero

$\frac{\partial}{\partial A_{n}} E (A_{n}) = 2 \int_{0}^{b} I_{m}^{*} (w) (\sum_{n = 0}^{N} A_{n} I_{n} (w) - Φ (w)) w d w = 0,$

(13.221)

which, after truncating the infinite series limit to order N, yields the following set of N + 1 simultaneous equations

$\sum_{n = 0}^{\infty} A_{n} \int_{0}^{b} I_{m}^{*} (w) I_{n} (w) w d w = \int_{0}^{a} I_{m}^{*} (w) w d w, m = 0, 1, \dots N,$

(13.222)

where

$I_{m}^{*} (w) = \frac{a^{2}}{b^{2}} \sum_{p = 0}^{P} {{}_{m}B}_{p}^{*} (k b) {(\frac{w}{b})}^{2 p},$

(13.223)

$I_{n} (w) = \frac{a^{2}}{b^{2}} \sum_{q = 0}^{Q} {{}_{n}B}_{q} (k b) {(\frac{w}{b})}^{2 q} .$

(13.224)

Integrating over w yields the following (N + 1) × (N + 1) matrix equation

$M \cdot a = b,$

(13.225)

where the matrix M and vectors a and b are given by

$M (m + 1, n + 1) = \sum_{p = 0}^{P} \sum_{q = 0}^{Q} \frac{{{}_{m}B}_{q}^{*} (k b) {{}_{n}B}_{q} (k b)}{p + q + 1}, {\begin{matrix} m = 0, 1, \dots, N \\ n = 0, 1, \dots, N \end{matrix},$

(13.226)

$b (m + 1) = - \sum_{p = 0}^{P} \frac{{{}_{m}B}_{p}^{*} (k b)}{p + 1} {(\frac{a}{b})}^{2 p}, m = 0, 1, \dots, N,$

(13.227)

$a (n + 1) = A_{n}, n = 0, 1, \dots, N$

(13.228)

Solution of the power series coefficients for a point or ring source in a circular baffle

In the case of a ring source of radius a in a circular baffle, we have

$Φ (w) = \frac{a}{2} δ (w - a),$

(13.229)

where δ is the Dirac delta function. Inserting this into Eq. (13.221) and truncating the infinite series limit to order N, yields the following set of N + 1 simultaneous equations

$\sum_{n = 0}^{N} A_{n} \int_{0}^{b} I_{m}^{*} (w) I_{n} (w) w d w = \frac{a}{2} \int_{0}^{b} δ (w - a) I_{m}^{*} (w) w d w, m = 0, 1, \dots N,$

(13.230)

where

$I_{m}^{*}$

(w) and I _n(w) are given by Eqs. (13.223) and (13.224), respectively. Integrating over w and using the property of the Dirac delta function yields the same matrix equations as Eqs. (13.225)–(13.228) except that

$b (m + 1) = \sum_{p = 0}^{P} {{}_{m}B}_{p}^{*} (k b) {(\frac{a}{b})}^{2 p} .$

(13.231)

In the limiting case of a point source at the center of a circular baffle, we let a → 0 so that

$b (m + 1) = {{}_{m}B}_{0}^{*} (k b) .$

(13.232)

Now that we have the surface pressure series coefficients A _n, we can derive some radiation characteristics for the disk in free space or open circular baffle or a point source in a circular baffle.

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:

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

(13.233)

where the far-field Green's function in spherical-cylindrical coordinates given by Eq. (13.70) is used. Inserting Eqs. (13.70), (13.198), and (13.202) into Eq. (13.233) and integrating over the surface, using Eqs. (76) and (96) 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

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

(13.234)

where the directivity function D(θ) is given by

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

(13.235)

The on-axis pressure is evaluated by setting θ = 0 in Eq. (13.70) before inserting it into Eq. (13.233) and integrating over the surface to give

$D (0) = k b \sum_{n = 0}^{N} A_{n},$

(13.236)

so that the on-axis response can be written as

$\tilde{p} (r, 0) = j ρ_{0} f {\tilde{U}}_{0} \frac{e^{- j k r}}{r} k b \sum_{n = 0}^{N} A_{n} .$

(13.237)

where

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

is the total volume velocity. It is worth noting that in the unbaffled case, where b = a, D(0) is simply the normalized radiation impedance, that is D(0) = ( R _s + jX _s)/(ρ ₀ c) where R _s and X _s are given by Eqs. (13.249) and (13.250), respectively. Using standard curve-fitting methods, the following asymptotic expression can be written as

$D (0) \approx j 0.66 (\frac{b}{a} - 0.3) k a, k b < 0.5.$

(13.238)

The on-axis response for five values of b is shown in Fig. 13.22, calculated from the magnitude of D(0).

We can see from Fig. 13.22 that, in the case of an unbaffled piston (b = a) radiating from both sides, the on-axis sound pressure falls at 6 dB/octave for small values of ka owing to the decreasing path difference (as a proportion of wavelength λ) between the antiphase rear radiation and the front radiation, which it partially cancels. This is also true of the oscillating sphere (see Fig. 4.26), but the attenuation is not as great because of the longer path difference around the sphere.

Figure 13.22 Plot of 20 log₁₀(D(0)) where D(θ) is the directivity function of a plane circular piston of radius a in a flat open circular baffle of radius b. When b = a (solid black curve), there is no baffle and the piston is radiating from both sides 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.

At larger values of ka, the rear radiation moves in and out of phase with that from the front. However, the comb-filter effect is fairly “smeared,” the largest peak being 3 dB at ka = π/√2 (or λ = 2√2 a), the reason being that rear radiation is due to the sum of many ring sources spread over the radius of the piston, each with a different path length to the front, so that at no particular frequency do they combine to produce a source that is either directly in phase or out of phase with that from the front. Unlike the oscillating sphere, the on-axis response does not roll-off at high frequencies, which is a property of planar sources, as already discussed in Section 12.8.

By contrast, when we include a circular baffle and increase its size, the actual radiating area decreases in proportion to the total so that it behaves more like a coherent point source at the center. Hence, when b = 4a, a deep null can be seen at ka = π/2 or λ = 4a, which is the distance from the center to the edge. Of course, a piston at the center of a circular baffle is the “worst case,” and it would be interesting to compare these results with those of an offset piston in a circular, rectangular, or elliptical baffle, for example, to “smear” the path difference effect.

Figure 13.23 Far-field directivity patterns for a plane circular piston radiating from both sides into free space as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston.

The normalized directivity function 20 log₁₀|D(θ)/D(0)| for a piston in free space is plotted in Fig. 13.23 for four values of ka = 2πa/λ, that is, for four values of the ratio of the circumference of the disk to the wavelength. When the circumference of the piston (2πa) is less than one-half wavelength, that is, ka < 0.5, it behaves essentially like a dipole point source. In fact, to a first approximation, an unbaffled thin piston is simply a doublet because an axial movement in one direction compresses the air on one side of it and causes a rarefaction of the air on the other side. When ka becomes greater than 3, the piston is highly directional, like the piston in an infinite baffle. Also, the directivity function for a piston in a finite open baffle is plotted in Fig. 13.24 for four values of ka with b = 2a and in Fig. 13.25 for four values of b.

Near-field pressure

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

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

(13.239)

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), (13.198) and (13.202) into Eq. (13.239) and integrating over the surface gives

Figure 13.24 Far-field directivity patterns for a circular piston in a plane open 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.

Figure 13.25 Far-field directivity patterns for a circular piston in a plane open 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.

$\begin{matrix} \tilde{p} (r, θ) = 2 j k b ρ_{0} c {\tilde{u}}_{0} \frac{a^{2}}{b^{2}} \sum_{n = 0}^{N} A_{n} \sum_{p = 0}^{P} \frac{{(- 1)}^{p} Γ (p + \frac{3}{2}) Γ (n + \frac{5}{2}) h_{2 p + 1}^{(2)} (k r) P_{2 p + 1} (\cos θ)}{Γ (2 p + \frac{3}{2}) Γ (p + n + \frac{5}{2})} \\ \times {(\frac{k b}{2})}^{2 p + 2} {}_{1}{F_{2}} (p + 1; p + n + \frac{5}{2}, 2 p + \frac{5}{2}; - \frac{k^{2} b^{2}}{4}), \end{matrix}$

(13.240)

which converges for r > b. For r ≤ b, we derive a suitable expression from Eq. (13.204), which is weakly singular at k _w = k. However, we can remove this singularity as follows: Firstly, we substitute

$k_{w} = k \sqrt{1 - t^{2}} for k_{w} \leq k$

and

$k_{w} = k \sqrt{1 + t^{2}} for k_{w} > k$

in Eq. (13.204) to obtain

$\tilde{p} (w, z) = k^{2} a^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} A_{n} Γ (n + \frac{5}{2}) (I_{F i n} + I_{I n f}),$

(13.241)

where

$I_{F i n} = - {(\frac{2}{k b})}^{n + \frac{1}{2}} \int_{0}^{1} {(\frac{1}{1 - t^{2}})}^{\frac{n}{2} + \frac{3}{4}} J_{n + \frac{3}{2}} (k b \sqrt{1 - t^{2}}) J_{0} (k w \sqrt{1 - t^{2}}) e^{- j k z t} t d t,$

(13.242)

$I_{I n f} = - {(\frac{2}{k b})}^{n + \frac{1}{2}} \int_{0}^{\infty} {(\frac{1}{1 + t^{2}})}^{\frac{n}{2} + \frac{3}{4}} J_{n + \frac{3}{2}} (k b \sqrt{1 + t^{2}}) J_{0} (k w \sqrt{1 + t^{2}}) e^{- k z t} t d t .$

(13.243)

We then apply the expansion of Eq. (109) from Appendix II to give

$\begin{matrix} I_{F i n} + I_{I n f} = - \sum_{p = 0}^{\infty} \frac{{(- 1)}^{p}}{p! Γ (n + p + \frac{5}{2})} {(\frac{k b}{2})}^{2 p + 1} {}_{2}{F_{1}} (- p, - n - p - \frac{3}{2}; 1; \frac{w^{2}}{b^{2}}) \\ \times (\int_{0}^{1} {(1 - t^{2})}^{p} e^{- j k z t} t d t + \int_{0}^{\infty} {(1 + t^{2})}^{p} e^{- k z t} t d t), \end{matrix}$

(13.244)

which after integrating yields

$\begin{matrix} I_{F i n} = \frac{\sqrt{π}}{2} \sum_{p = 0}^{\infty} \frac{{(- 1)}^{p}}{Γ (n + p + \frac{5}{2})} {(\frac{k b}{2})}^{2 p + 1} {}_{2}{F_{1}} (- p, - n - p - \frac{3}{2}; 1; \frac{w^{2}}{b^{2}}) \\ \times {{(\frac{2}{k z})}^{p + \frac{1}{2}} ({j^{J}}_{p + \frac{3}{2}} (k z) + H_{p + \frac{3}{2}} (k z)) - \frac{1}{\sqrt{π} (p + 1)!}}, \end{matrix}$

(13.245)

$\begin{matrix} I_{I n f} = \frac{\sqrt{π}}{2} \sum_{p = 0}^{\infty} \frac{{(- 1)}^{p}}{p! Γ (n + p + \frac{5}{2})} {(\frac{k b}{2})}^{2 p + 1} {}_{2}{F_{1}} (- p, - n - p - \frac{3}{2}; 1; \frac{w^{2}}{b^{2}}) \\ \times {{(\frac{2}{k z})}^{p + \frac{1}{2}} (Y_{p + \frac{3}{2}} (k z) - H_{p + \frac{3}{2}} (k z)) + \frac{1}{\sqrt{π} (p + 1)!}} . \end{matrix}$

(13.246)

Eq. (13.245) converges everywhere and is therefore suitable for r < b. Unfortunately, Eq. (13.246) only converges for z ² > w ² + b ² and is therefore not suitable. However, Eq. (13.243) converges everywhere and can be calculated numerically without problem and is therefore suitable for r < b. The pressure field of a rigid piston in free space is plotted in Fig. 13.26 for three values of ka. At ka = 6π, the sound field of the unbaffled piston shows similar characteristics to the baffled one, except that the radial pressure beyond its perimeter is zero, as with any planar dipole source. This suggests that, at high frequencies, objects on either side of the source have less effect on the sound field, except that the axial nulls are not as deep and the peaks are slightly higher. It can be shown that at low frequencies, where ka < 1, the on-axis pressure converges to the far-field approximation at increasingly greater distances due to the proximity effect (bass tip-up). The pressure field of a rigid piston in a circular baffle of radius b = 2a is plotted in Fig. 13.27 for two values of ka.

Figure 13.26 Normalized near-field pressure plots for a rigid circular piston in free space as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston. $| \tilde{p} |$ is the pressure magnitude, ${\tilde{u}}_{0}$ is the piston velocity, ρ ₀ is the density of the acoustic medium, and c is the speed of sound in that medium.

Figure. 13.27 Normalized near-field pressure plots for a rigid circular piston in a finite open circular baffle of radius b = 2a as a function ka = 2πa/λ = 2πfa/c, where a is the radius of the piston. $| \tilde{p} |$ is the pressure magnitude, ${\tilde{u}}_{0}$ is the piston velocity, ρ ₀ is the density of the acoustic medium, and c is the speed of sound in that medium.

Radiation impedance of a piston in a circular baffle

The total radiation force is found by integrating the pressure from Eq. (13.202) over the surface of the disk on both sides to give

$\begin{matrix} \tilde{F} = \int_{0}^{2 π} \int_{0}^{a} ({\tilde{p}}_{+} (w_{0}) - \tilde{p}_(w_{0})) w d w d ϕ \\ = 2 π a^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{N} A_{n} {1 - {(1 - \frac{a^{2}}{b^{2}})}^{n + \frac{3}{2}}} . \end{matrix}$

(13.247)

The specific radiation impedance Z _s is then given by

$Z_{s} = \frac{\tilde{F}}{{\tilde{U}}_{0}} = R_{s} + j X_{s},$

(13.248)

where

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

is the total volume velocity and R _s is the specific radiation resistance in N·s/m³ (rayl) given by

$\begin{matrix} {R_{s} |}_{b \neq a} = k b ρ_{0} c ℜ (\sum_{n = 0}^{N} A_{n} {1 - {(1 - \frac{a^{2}}{b^{2}})}^{n + \frac{3}{2}}}) \\ {R_{s} |}_{b = a} = k a ρ_{0} c ℜ (\sum_{n = 0}^{N} A_{n}) \approx ρ_{0} c \frac{8 k^{4} a^{4}}{27 π^{2}}, k a < 0.5, \end{matrix}$

(13.249)

where the bold R indicates that the quantity varies with frequency and X _s is the specific radiation reactance in N·s/m³ (rayl) given by

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

(13.250)

Plots of the real and imaginary parts of

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

(13.251)

are shown in Fig. 13.28 as a function of ka.

The data of Fig. 13.28 are used in dealing with impedance analogies. The complex admittance can be obtained by taking the reciprocal of the complex impedance.

The specific admittance is given by

$\begin{matrix} Y_{s} = G_{s} + j B_{s} = \frac{R_{s}}{R_{s}^{2} + X_{s}^{2}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}} \\ {Y_{s} |}_{b = a} = {(G_{s} + j B_{s}) |}_{b = a} \approx \frac{1}{ρ_{0} c} (\frac{k^{2} a^{2}}{6} - j \frac{3 π}{4 k a}), k a < 0.5. \end{matrix}$

Figure 13.28 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 open baffle of radius b. When b = a (solid black curve), there is no baffle and the piston is radiating from both sides 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.

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Table of Contents for Acoustics

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13.10. Radiation from a rigid circular piston in a finite circular open baffle [26,27]

Boundary conditions

Formulation of the coupled equation

Solution of the power series coefficients for a piston in free space

Solution of the power series coefficients for a piston in a circular baffle

Solution of the power series coefficients for a point or ring source in a circular baffle

Far-field pressure

Near-field pressure

Radiation impedance of a piston in a circular baffle

Table of Contents for
Acoustics