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13.18. Mutual radiation impedance between rigid circular pistons in an infinite baffle [41–43]

Boundary conditions

When sound sources are radiating in close proximity to each other, their radiation characteristics may be affected by their acoustic interaction, depending on their spacing and the frequency. Here we consider two circular pistons of radii a ₁ and a ₁, as shown in Fig. 13.47, which are mounted in an infinite rigid baffle in the xy plane with a separation distance d between their centers and oscillate in phase in the z direction with a harmonically time-dependent velocity

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

Figure 13.43 Normalized specific radiation resistance R _s/ρ ₀ c of the air load on one side of a plane rectangular piston in an infinite flat baffle for five different aspect ratios q = l_y /l_x , where l_x and l_y are the dimensions of the piston. Frequency is plotted on a normalized scale, where kl = 2πl/λ = 2πl/c and l = l_x is the smallest dimension. In the case of q = ∞, the impedance is that of an infinitely long strip.

According to the principle of superposition of fields, we obtain the directivity pattern from the combinations shown in Fig. 13.48.

Figure 13.44 Normalized specific radiation reactance X _s/ρ ₀ c of the air load on one side of a plane rectangular piston in an infinite flat baffle for four different aspect ratios q = l_y /l_x , where l_x and l_y are the dimensions of the piston. Frequency is plotted on a normalized scale, where kl = 2πl/λ = 2πl/c and l = l_x is the smallest dimension. In the case of q = ∞, the impedance is that of an infinitely long strip.

Directivity

According to the product theorem of Section 13.16, we multiply the directivity pattern of a single piston from Eq. (13.102) by that of two point sources in phase from Eq. (4.79) to produce the directivity patterns of Fig. 13.48(b) and (d). Similarly, we multiply the directivity pattern of a single piston by that of two point sources in antiphase from Eq. (4.80) to produce the directivity patterns of Fig. 13.48(c) and (e). Hence, the directivity pattern of Fig. 13.48(a) is given by

$D (θ, ϕ) = \frac{a_{1}^{2} D_{1} (θ)}{a_{1}^{2} + a_{2}^{2}} e^{j \frac{k d}{2} \sin θ \sin ϕ} + \frac{a_{2}^{2} D_{2} (θ)}{a_{1}^{2} + a_{2}^{2}} e^{- j \frac{k d}{2} \sin θ \sin ϕ},$

(13.330)

where

$e^{\pm j \frac{k d}{2} \sin θ \sin ϕ} = \cos (\frac{k d}{2} \sin θ \sin ϕ) \pm j \sin (\frac{k d}{2} \sin θ \sin ϕ)$

(13.331)

and D ₁(θ) is the directivity pattern of the piston of radius a ₁ as given by

$D_{1} (θ) = \frac{2 J_{1} (k a_{1} \sin θ)}{k a_{1} \sin θ}$

(13.332)

and D ₂(θ) is the directivity pattern of the piston of radius a ₂ as given by

Figure 13.45 Normalized specific radiation resistance R _s/ρ ₀ c of the air load on one side of a plane rectangular piston in an infinite flat baffle for five different aspect ratios q = l_y /l_x , where l_x and l_y are the dimensions of the piston. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πa/c and a is a notional radius that gives the same circular area S as the actual area of the rectangular piston, which is given by S = πa ² = l_y l_x .

$D_{2} (θ) = \frac{2 J_{1} (k a_{2} \sin θ)}{k a_{2} \sin θ} .$

(13.333)

Figure 13.46 Normalized specific radiation reactance X _s/ρ ₀ c of the air load on one side of a plane rectangular piston in an infinite flat baffle for five different aspect ratios q = l_y /l_x , where l_x and l_y are the dimensions of the piston. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c and a is a notional radius that gives the same circular area S as the actual area of the rectangular piston, which is given by S = πa ² = l_xl_y .

Note that Eqs. (4.79) and (4.80) have been modified to include the ϕ dependency. Then

${| D (θ, ϕ) |}^{2} = \frac{a_{1}^{4} D_{1}^{2} (θ) + a_{2}^{4} D_{2}^{2} (θ) + 2 a_{1}^{2} a_{2}^{2} D_{1} (θ) D_{2} (θ) \cos (k d \sin θ \sin ϕ)}{{(a_{1}^{2} + a_{2}^{2})}^{2}},$

(13.334)

where we have noted that sin²(kd sin θ sin ϕ) = 1 − cos²(kd sin θ sin ϕ).

Impedance

We now obtain the total radiation impedance of the two pistons using Bouwkamp's impedance theorem of Eqs. (13.271) and (13.272)

$Z_{S} = R_{S} + j X_{S} = \frac{k^{2} ρ_{0} c S}{4 π^{2}} \int_{0}^{2 π} (\int_{0}^{\frac{π}{2}} {| D (θ, ϕ) |}^{2} + \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} {| D (θ, ϕ) |}^{2}) \sin θ d θ d ϕ,$

(13.335)

Figure 13.47 Geometry of two circular pistons in an infinite baffle. The point of observation P is located at a distance r from the origin at an inclination angle θ with respect to the z axis.

Figure 13.48 Combination of pistons of radii a ₁ and a ₂.

where

$S = π (a_{1}^{2} + a_{2}^{2})$

is the total radiating area. The total specific radiation resistance is given by

$\begin{matrix} R_{S} = \frac{a_{1}^{2}}{a_{1}^{2} + a_{2}^{2}} R_{11} + \frac{a_{2}^{2}}{a_{1}^{2} + a_{2}^{2}} R_{22} + \frac{2 a_{1} a_{2}}{a_{1}^{2} + a_{2}^{2}} R_{12} \\ = \frac{4 k^{2} ρ_{0} c}{π (a_{1}^{2} + a_{2}^{2})} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} (a_{1}^{4} D_{1}^{2} (θ) + a_{2}^{4} D_{2}^{2} (θ) \\ + a_{1}^{2} a_{2}^{2} D_{1} (θ) D_{2} (θ) \cos (k d \sin θ \sin ϕ)) \sin θ d θ d ϕ, \end{matrix}$

(13.336)

where R ₁₁ and R ₂₂ are the self-resistances of each piston and R ₁₂ is the mutual resistance between the two pistons. The specific radiation reactance is given by

$\begin{matrix} X_{S} = \frac{a_{1}^{2}}{a_{1}^{2} + a_{2}^{2}} X_{11} + \frac{a_{2}^{2}}{a_{1}^{2} + a_{2}^{2}} X_{22} + \frac{2 a_{1} a_{2}}{a_{1}^{2} + a_{2}^{2}} X_{12} \\ = - j \frac{4 k^{2} ρ_{0} c}{π (a_{1}^{2} + a_{2}^{2})} \int_{0}^{2 π} \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} (a_{1}^{4} D_{1}^{2} (θ) + a_{2}^{4} D_{2}^{2} (θ) \\ + a_{1}^{2} a_{2}^{2} D_{1} (θ) D_{2} (θ) \cos (k d \sin θ \sin ϕ)) \sin θ d θ d ϕ, \end{matrix}$

(13.337)

where X ₁₁ and X ₂₂ are the self-reactances of each piston and X ₁₂ is the mutual reactance between the two pistons. The first term in each integral, which is independent of the spacing d, may be identified as the self-impedance Z ₁₁ of the first piston

$Z_{11} = 2 ρ_{0} c (\int_{0}^{\frac{π}{2}} \frac{J_{1}^{2} (k a_{1} \sin θ)}{\sin θ} d θ + \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} \frac{J_{1}^{2} (k a_{1} \sin θ)}{\sin θ} d θ),$

(13.338)

which after substituting μ = k sin θ, as discussed in Section 13.13, gives

$Z_{11} = R_{11} + j X_{11} = ρ_{0} c {(1 - \frac{J_{1}^{2} (k a_{1})}{k a_{1}}) + j \frac{H_{1} (k a_{1})}{k a_{1}}} .$

(13.339)

Similarly, the second term in each integral may be identified as the self-impedance Z ₂₂ of the second piston

$Z_{22} = R_{22} + j X_{22} = ρ_{0} c {(1 - \frac{J_{1}^{2} (k a_{2})}{k a_{2}}) + j \frac{H_{1} (k a_{2})}{k a_{2}}} .$

(13.340)

The third term in each integral then gives the mutual impedance Z ₁₂ so that, after integrating over ϕ using the integral solution of Eq. (A2.77) from Appendix II (with z = kd sin θ), we have

$\begin{matrix} Z_{12} = 2 ρ_{0} c (\int_{0}^{\frac{π}{2}} \frac{J_{1} (k a_{1} \sin θ) J_{1} (k a_{2} \sin θ)}{\sin θ} J_{0} (k d \sin θ) d θ \\ + \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} \frac{J_{1} (k a_{1} \sin θ) J_{1} (k a_{2} \sin θ)}{\sin θ} J_{0} (k d \sin θ) d θ), \end{matrix}$

(13.341)

which after substituting s = sin θ becomes

$Z_{12} = 2 ρ_{0} c (\int_{0}^{1} \frac{J_{1} (k a_{1} s) J_{1} (k a_{2} s)}{s \sqrt{1 - s^{2}}} J_{0} (k d s) d s + \int_{1}^{\infty} \frac{J_{1} (k a_{1} s) J_{1} (k a_{2} s)}{s \sqrt{1 - s^{2}}} J_{0} (k d s) d s) .$

(13.342)

We then expand the J ₁ functions using the following Lommel expansion

$J_{ν} (k a s) = s^{ν} \sum_{m = 0}^{\infty} {(\frac{k a}{2})}^{m} \frac{{(1 - s^{2})}^{m}}{m!} J_{ν + m} (k a)$

(13.343)

to give

$\begin{matrix} Z_{12} = 2 ρ_{0} c \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} {(\frac{k a_{1}}{2})}^{m} {(\frac{k a_{2}}{2})}^{n} \frac{J_{m + 1} (k a_{1}) J_{n + 1} (k a_{2})}{m! n!} \\ \times (\int_{0}^{1} J_{0} (k d s) {(1 - s^{2})}^{m + n - 1 / 2} s d s + j {(- 1)}^{m + n} \int_{1}^{\infty} J_{0} (k d s) {(s^{2} - 1)}^{m + n - 1 / 2} s d s), \end{matrix}$

(13.344)

which can be solved with help from Eqs. (A2.96), (A2.97), and (A2.98) of Appendix II to yield

$Z_{12} = \frac{2 ρ_{0} c}{\sqrt{π}} \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} {(\frac{k a_{1}}{k d})}^{m} {(\frac{k a_{2}}{k d})}^{n} \frac{Γ (m + n + 1 / 2) J_{m + 1} (k a_{1}) J_{n + 1} (k a_{2})}{m! n!} h_{m + n}^{(2)} (k d),$

(13.345)

where the total specific radiation impedance is given by

$Z_{S} = \frac{a_{1}^{2}}{a_{1}^{2} + a_{2}^{2}} Z_{11} + \frac{a_{2}^{2}}{a_{1}^{2} + a_{2}^{2}} Z_{22} + \frac{2 a_{1} a_{2}}{a_{1}^{2} + a_{2}^{2}} Z_{12}$

(13.346)

and h _m+n ⁽²⁾ is the spherical Hankel function defined in Eq. (A2.133) of Appendix II. For very large wavelengths and separations, where ka << 1 and d >> a, we have

$Z_{12} \approx ρ_{0} c \frac{(k a_{1}) (k a_{2})}{2} (\frac{\sin (k d)}{k d} + j \frac{\cos (k d)}{k d}), (k a_{1}) (k a_{2}) < < 1, \frac{\sqrt{a_{1} a_{2}}}{d} < < 1,$

(13.347)

where R ₁₁ = (ka)²/2. Plots of the real normalized mutual radiation resistance ( R ₁₂/ R ₁₁) and reactance (X ₁₂/ R ₁₁) for ka = 1 are shown in Fig. 13.49 as a function of kd.

Array of pistons

We now extend the model to an array of N pistons, each of radius a _p, where the pth and qth pistons are separated by a distance d _pq. Multiplying Eq. (13.346) through by the total area

$π (a_{1}^{2} + a_{2}^{2})$

yields the total mechanical impedance

$Z_{M} = π (a_{1}^{2} + a_{2}^{2}) Z_{S} = π a_{1}^{2} Z_{11} + π a_{2}^{2} Z_{22} + 2 π a_{1} a_{2} Z_{12} .$

(13.348)

We may represent this with an N-port model in the following matrix form

$F = Z \cdot u,$

(13.349)

where u is the velocity vector

Figure 13.49 Real and imaginary parts of the normalized mutual radiation impedance Z ₁₂ of the air load on one side of two plane circular pistons of radius a and separation d in an infinite baffle.

$u = {[\begin{matrix} {\tilde{u}}_{1} & {\tilde{u}}_{2} & \dots & {\tilde{u}}_{p} & \dots & {\tilde{u}}_{N} \end{matrix}]}^{T} .$

(13.350)

Because we are extending this model to an unlimited number of ports, we define all the velocities as positive flowing into the ports. Then Z is the mechanical impedance matrix

$Z = [\begin{matrix} z_{11} & z_{12} & \dots & z_{1 q} & \dots & z_{1 N} \\ z_{21} & z_{22} & \dots & z_{2 q} & \dots & z_{2 N} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ z_{p 1} & z_{p 2} & \dots & z_{p q} & \dots & z_{p N} \\ ⋮ & ⋮ & ⋮ \\ z_{N 1} & z_{N 2} & \dots & z_{N q} & \dots & z_{N N} \end{matrix}],$

(13.351)

and F is the force vector

$F = {[\begin{matrix} {\tilde{F}}_{1} & {\tilde{F}}_{2} & \dots & {\tilde{F}}_{q} & \dots & {\tilde{F}}_{N} \end{matrix}]}^{T} .$

(13.352)

The mechanical self-impedance elements on the diagonal are given by

$z_{p p} = π a_{p}^{2} (R_{p p} + j X_{p p}) = π a_{p}^{2} ρ_{0} c {(1 - \frac{J_{1}^{2} (k a_{p})}{k a_{p}}) + j \frac{H_{1} (k a_{p})}{k a_{p}}}$

(13.353)

and the symmetrical mutual-impedance elements are given by

$\begin{matrix} z_{p q} = z_{q p} = π a_{p} a_{q} (R_{p q} + j X_{p q}) \\ = 2 \sqrt{π} a_{p} a_{q} ρ_{0} c \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} {(\frac{k a_{p}}{k d_{p q}})}^{m} {(\frac{k a_{q}}{k d_{p q}})}^{n} \\ \times \frac{Γ (m + n + 1 / 2) J_{m + 1} (k a_{p}) J_{n + 1} (k a_{q})}{m! n!} h_{m + n}^{(2)} (k d_{p q}) . \end{matrix}$

(13.354)

The self- and mutual acoustic impedances are given by

$Z_{p q} = \frac{z_{p q}}{π^{2} a_{p}^{2} a_{q}^{2}} .$

(13.355)

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13.18. Mutual radiation impedance between rigid circular pistons in an infinite baffle [41–43]

Boundary conditions

Directivity

Impedance

Array of pistons

Table of Contents for
Acoustics