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12.10. Radiation from an oscillating concave dome in an infinite baffle

A concave dome [5] of radius a and radius of curvature R in an infinite baffle is shown in Fig. 12.30. In this problem, we shall introduce the concept of coupling whereby the field

{\tilde{p}}_{I} (r, θ)

${\tilde{p}}_{I} (r, θ)$

inside the imaginary sphere, which includes the space inside the dome, is coupled to an external field

\tilde{p} (r, θ)

$\tilde{p} (r, θ)$

. Again, the baffle can be removed so that we have an equivalent field which is symmetrical either side of the plane of the baffle. However, in the external field, we completely ignore the dome as if it inhabited some other universe. What we have in effect is a breathing disk in free space with identical pressure distributions on both faces, which are also the same as that of the mouth of the dome. The velocity distributions are also equal in magnitude to that of the mouth, but have opposite directions. This enables us to apply the same field-matching condition as in the convex dome.

Figure 12.29 Real and imaginary parts of the normalized specific radiation impedance Z _s/ρ ₀ c of the air load on a convex dome of radius a in an infinite baffle, where α is the half angle of the arc formed by the dome. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Figure 12.30 Geometry of concave dome in infinite baffle.

Near-field pressure

We assume that the external pressure field

\tilde{p} (r, θ)

$\tilde{p} (r, θ)$

in the region r ≥ R is a general axisymmetric solution to Eq. (2.180), the Helmholtz wave equation in spherical coordinates:

\tilde{p} (r, θ) = ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} x_{n} h_{n}^{(2)} (k r) P_{n} (\cos θ),

$\tilde{p} (r, θ) = ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} x_{n} h_{n}^{(2)} (k r) P_{n} (\cos θ),$

(12.143)

where x _n are the as-yet unknown expansion coefficients, which will be calculated by means of a set of simultaneous equations in matrix form. The normal particle velocity

{\tilde{u}}_{S} (R, θ)

${\tilde{u}}_{S} (R, θ)$

at the surface of the sphere is given by

\begin{matrix} \tilde{u} (R, θ) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial r} {\tilde{p} (r, θ) |}_{r = R} \\ = \frac{{\tilde{u}}_{0}}{- j k} \sum_{n = 0}^{\infty} x_{n} h_{n}^{' (2)} (k R) P_{n} (\cos θ), \end{matrix}

$\begin{matrix} \tilde{u} (R, θ) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial r} {\tilde{p} (r, θ) |}_{r = R} \\ = \frac{{\tilde{u}}_{0}}{- j k} \sum_{n = 0}^{\infty} x_{n} h_{n}^{' (2)} (k R) P_{n} (\cos θ), \end{matrix}$

(12.144)

where the derivative of the spherical Hankel function

{h^{'}}_{n}^{(2)} (k R)

${h^{'}}_{n}^{(2)} (k R)$

is given by Eq. (12.32). The internal field

{\tilde{p}}_{I} (r, θ)

${\tilde{p}}_{I} (r, θ)$

must be continuous everywhere in the region r ≤ R. Hence, we omit the spherical Bessel function of the second kind, which has a singularity at r = 0, from the spherical Hankel function so that

{\tilde{p}}_{I} (r, θ) = ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} y_{n} j_{n} (k r) P_{n} (\cos θ),

${\tilde{p}}_{I} (r, θ) = ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} y_{n} j_{n} (k r) P_{n} (\cos θ),$

(12.145)

where y _n are the unknown expansion coefficients. The normal particle velocity

{\tilde{u}}_{I} (R, θ)

${\tilde{u}}_{I} (R, θ)$

at the surface of the sphere is given by

\begin{matrix} {\tilde{u}}_{I} (R, θ) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial r} {p_{I} (r, θ) |}_{r = R} \\ = \frac{{\tilde{u}}_{0}}{- j k} \sum_{n = 0}^{\infty} y_{n} j_{n}^{'} (k R) P_{n} (\cos θ) . \end{matrix}

$\begin{matrix} {\tilde{u}}_{I} (R, θ) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial r} {p_{I} (r, θ) |}_{r = R} \\ = \frac{{\tilde{u}}_{0}}{- j k} \sum_{n = 0}^{\infty} y_{n} j_{n}^{'} (k R) P_{n} (\cos θ) . \end{matrix}$

(12.146)

At the surface of the dome, the normal particle velocity has to match the axial velocity

{\tilde{u}}_{0}

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

of the dome. Hence,

{\tilde{u}}_{I} (R, θ) = - {\tilde{u}}_{0} \cos θ, 0 \leq θ \leq α .

${\tilde{u}}_{I} (R, θ) = - {\tilde{u}}_{0} \cos θ, 0 \leq θ \leq α .$

(12.147)

In addition, we have the coupling condition whereby the normal particle velocity on the inner surface of the imaginary sphere has to match that on its outer surface

{\tilde{u}}_{I} (R, θ) = \tilde{u} (R, θ), α \leq θ \leq π .

${\tilde{u}}_{I} (R, θ) = \tilde{u} (R, θ), α \leq θ \leq π .$

(12.148)

Likewise, the pressure on the inner surface of the sphere has to match that on its outer surface:

\tilde{p} (R, θ) = {\tilde{p}}_{I} (R, θ), α < θ \leq π .

$\tilde{p} (R, θ) = {\tilde{p}}_{I} (R, θ), α < θ \leq π .$

(12.149)

Finally, we apply the field matching whereby the pressure on the outer surface of the imaginary sphere is equal to that of its mirror image, which lies inside the image sphere:

\tilde{p} (R, θ) = {\tilde{p}}_{I} (r_{1}, θ_{1}), 0 < θ \leq α .

$\tilde{p} (R, θ) = {\tilde{p}}_{I} (r_{1}, θ_{1}), 0 < θ \leq α .$

(12.150)

From the geometry of the problem, we can write

r_{1} = R \sqrt{1 + 4 \cos α (\cos α - \cos θ)},

$r_{1} = R \sqrt{1 + 4 \cos α (\cos α - \cos θ)},$

(12.151)

\cos θ_{1} = \frac{R}{r_{1}} (2 \cos α - \cos θ) .

$\cos θ_{1} = \frac{R}{r_{1}} (2 \cos α - \cos θ) .$

(12.152)

Now, we have all of the boundary conditions in place, we can create the following pair of infinite simultaneous equations in the unknown coefficients x _n and y _n in the usual manner by multiplying through P _m(cos θ) and integrating over the surface of the imaginary sphere together with its image and the surface of the dome:

\int_{0}^{π} \tilde{p} (R, θ) P_{m} (\cos θ) \sin θ d θ = \int_{0}^{α} {\tilde{p}}_{I} (r_{1}, θ_{1}) P_{m} (\cos θ) \sin θ d θ + \int_{α}^{π} {\tilde{p}}_{I} (R, θ) P_{m} (\cos θ) \sin θ d θ, m = 0,1,2, \dots,

$\int_{0}^{π} \tilde{p} (R, θ) P_{m} (\cos θ) \sin θ d θ = \int_{0}^{α} {\tilde{p}}_{I} (r_{1}, θ_{1}) P_{m} (\cos θ) \sin θ d θ + \int_{α}^{π} {\tilde{p}}_{I} (R, θ) P_{m} (\cos θ) \sin θ d θ, m = 0,1,2, \dots,$

(12.153)

\int_{0}^{π} {\tilde{u}}_{I} (R, θ) P_{m} (\cos θ) \sin θ d θ = - {\tilde{u}}_{0} \int_{0}^{a} P_{m} (\cos θ) \cos θ \sin θ d θ + \int_{a}^{π} \tilde{u} (R, θ) P_{m} (\cos θ) \sin θ d θ, m = 0,1,2, \dots .

$\int_{0}^{π} {\tilde{u}}_{I} (R, θ) P_{m} (\cos θ) \sin θ d θ = - {\tilde{u}}_{0} \int_{0}^{a} P_{m} (\cos θ) \cos θ \sin θ d θ + \int_{a}^{π} \tilde{u} (R, θ) P_{m} (\cos θ) \sin θ d θ, m = 0,1,2, \dots .$

(12.154)

In matrix form, this becomes

x = A \cdot y,

$x = A \cdot y,$

(12.155)

y = B \cdot x + c,

$y = B \cdot x + c,$

(12.156)

where

x = [x_{0}, x_{1}, \dots, x_{m}], y = [y_{0}, y_{1}, \dots, y_{m}],

$x = [x_{0}, x_{1}, \dots, x_{m}], y = [y_{0}, y_{1}, \dots, y_{m}],$

where the matrices A and B and vector c are given by

A (m + 1, n + 1) = \frac{m + 1 / 2}{h_{m}^{(2)} (k R)} (I_{m n} + j_{n} (k R) K_{m n}), {\begin{matrix} m = 0,1, \dots, N \\ n = 0,1, \dots, N \end{matrix},

$A (m + 1, n + 1) = \frac{m + 1 / 2}{h_{m}^{(2)} (k R)} (I_{m n} + j_{n} (k R) K_{m n}), {\begin{matrix} m = 0,1, \dots, N \\ n = 0,1, \dots, N \end{matrix},$

(12.157)

B (m + 1, n + 1) = \frac{m + 1 / 2}{j_{m}^{'} (k R)} h_{n}^{' (2)} (k R) K_{m n,} {\begin{matrix} m = 0,1, \dots, N \\ n = 0,1, \dots, N \end{matrix},

$B (m + 1, n + 1) = \frac{m + 1 / 2}{j_{m}^{'} (k R)} h_{n}^{' (2)} (k R) K_{m n,} {\begin{matrix} m = 0,1, \dots, N \\ n = 0,1, \dots, N \end{matrix},$

(12.158)

c (m + 1) = j k \frac{m + 1 / 2}{j_{m}^{'} (k R)} L_{m}, m = 0,1, \dots, N,

$c (m + 1) = j k \frac{m + 1 / 2}{j_{m}^{'} (k R)} L_{m}, m = 0,1, \dots, N,$

(12.159)

where

I_{m n} = \int_{0}^{a} j_{n} (k r_{1}) P_{n} (\cos θ_{1}) P_{m} (\cos θ) \sin θ d θ,

$I_{m n} = \int_{0}^{a} j_{n} (k r_{1}) P_{n} (\cos θ_{1}) P_{m} (\cos θ) \sin θ d θ,$

(12.160)

\begin{matrix} K_{m n} = \int_{α}^{π} P_{n} (\cos θ) P_{m} (\cos θ) \sin θ d θ \\ = {\begin{matrix} \frac{\sin α (P_{m} (\cos α) P_{n}^{'} (\cos α) - P_{n} (\cos α) P_{m}^{'} (\cos α))}{m (m + 1) - n (n + 1)}, & m \neq n \\ \frac{1 + {(P_{m} (\cos α))}^{2} \cos α + 2 \sum_{j = 1}^{m - 1} P_{j} (\cos α) (P_{j} (\cos α) \cos α - P_{j + 1} (\cos α))}{2 m + 1}, & m = n \end{matrix} \end{matrix}

$\begin{matrix} K_{m n} = \int_{α}^{π} P_{n} (\cos θ) P_{m} (\cos θ) \sin θ d θ \\ = {\begin{matrix} \frac{\sin α (P_{m} (\cos α) P_{n}^{'} (\cos α) - P_{n} (\cos α) P_{m}^{'} (\cos α))}{m (m + 1) - n (n + 1)}, & m \neq n \\ \frac{1 + {(P_{m} (\cos α))}^{2} \cos α + 2 \sum_{j = 1}^{m - 1} P_{j} (\cos α) (P_{j} (\cos α) \cos α - P_{j + 1} (\cos α))}{2 m + 1}, & m = n \end{matrix} \end{matrix}$

(12.161)

\begin{matrix} L_{m} = \int_{0}^{α} P_{m} (\cos θ) \cos θ \sin θ d θ \\ = {\begin{matrix} (1 - \cos^{3} α) / 3, & m = 1 \\ - \sin α \frac{\sin α P_{m} (\cos α) + \cos α P_{m}^{1} (\cos α)}{(m - 1) (m + 2)}, & m \neq 1, \end{matrix} \end{matrix}

$\begin{matrix} L_{m} = \int_{0}^{α} P_{m} (\cos θ) \cos θ \sin θ d θ \\ = {\begin{matrix} (1 - \cos^{3} α) / 3, & m = 1 \\ - \sin α \frac{\sin α P_{m} (\cos α) + \cos α P_{m}^{1} (\cos α)}{(m - 1) (m + 2)}, & m \neq 1, \end{matrix} \end{matrix}$

(12.162)

where the identities of Eqs. (A2.66), (A2.68), and (A2.70) from Appendix II have been applied. Unfortunately, the integral I _mn has no analytical solution and has to be evaluated numerically using, for example,

\int_{0}^{a} f (θ) d θ = \frac{α}{P} \sum_{p = 1}^{P} f (θ_{p}), where θ_{p} = \frac{p - 1 / 2}{P} α .

$\int_{0}^{a} f (θ) d θ = \frac{α}{P} \sum_{p = 1}^{P} f (θ_{p}), where θ_{p} = \frac{p - 1 / 2}{P} α .$

(12.163)

Solving Eqs. (12.155) and (12.156) for x and y gives

y = {[I - B \cdot A]}^{- 1} \cdot c,

$y = {[I - B \cdot A]}^{- 1} \cdot c,$

(12.164)

x = A \cdot y .

$x = A \cdot y .$

(12.165)

Far-field pressure

In the far field, we can use the asymptotic expression for the spherical Hankel function from Eq. (12.18), which when inserted into Eq. (12.143) gives

{\tilde{p} (r, θ) |}_{r \to \infty} = - j k ρ_{0} c S \frac{{\tilde{u}}_{0}}{2 π r} e^{- j k r} D (θ),

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

(12.166)

where S is the dome effective area given by S = πa ² and

D (θ) = - \frac{2}{k^{2} R^{2} \sin^{2} α} \sum_{n = 0}^{N} x_{n} j^{n} P_{n} (\cos θ) .

$D (θ) = - \frac{2}{k^{2} R^{2} \sin^{2} α} \sum_{n = 0}^{N} x_{n} j^{n} P_{n} (\cos θ) .$

(12.167)

The directivity pattern 20 log₁₀(|D(θ)|/|D(0)|) for α = 60 degrees is plotted in Fig. 12.31 for various values of ka. Not surprisingly, the directivity is similar to that of the convex dome. The far-field on-axis response is given by

D (0) = - \frac{2}{k^{2} R^{2} \sin^{2} α} \sum_{n = 0}^{N} x_{n} j^{n} .

$D (0) = - \frac{2}{k^{2} R^{2} \sin^{2} α} \sum_{n = 0}^{N} x_{n} j^{n} .$

(12.168)

The on-axis response 20 log₁₀|D(0)| is plotted against ka in Fig. 12.32. This shows some interesting features. The dips in the responses of the convex dome shown in Fig. 12.28 for various values α are now replaced with resonant peaks. In each case, the resonant frequency is determined by the compliance of the dome cavity and the radiation mass. The peak is fairly broad due to the damping effect of the radiation resistance. At ka = 4.1, we see a sharp dip due to a radial standing wave across the mouth of the dome, where the air just circulates back and forth between points of maximum and minimum pressure. Above this frequency, the response is fairly uneven due to standing wave harmonics.

Figure 12.31 Far-field directivity pattern 20 log₁₀(|D(θ)|/|D(0)|) of the far-field pressure due to a concave dome of radius a in an infinite baffle for α = 60 degrees, where α is the half angle of the arc formed by the dome.

Radiation impedance

The total radiation force

\tilde{F}

$\tilde{F}$

is given by

\tilde{F} = R^{2} \int_{0}^{2 π} \int_{0}^{a} \tilde{p} (R, θ) \cos θ \sin θ d θ d ϕ

$\tilde{F} = R^{2} \int_{0}^{2 π} \int_{0}^{a} \tilde{p} (R, θ) \cos θ \sin θ d θ d ϕ$

(12.169)

using the identity of equation (A2.68) from Appendix II. The specific impedance Z _s is then given by

Z_{s} = \frac{\tilde{F}}{{\tilde{U}}_{0}} = \frac{2 ρ_{0} c}{\sin^{2} α} \sum_{n = 0}^{N} y_{n} j_{n} (k R) L_{n},

$Z_{s} = \frac{\tilde{F}}{{\tilde{U}}_{0}} = \frac{2 ρ_{0} c}{\sin^{2} α} \sum_{n = 0}^{N} y_{n} j_{n} (k R) L_{n},$

(12.170)

where we have used the expression for

{\tilde{U}}_{o}

${\tilde{U}}_{o}$

from Eq. (12.49). The real and imaginary parts, R _s and X _s, are plotted in Fig. 12.33, where

Figure 12.32 Plot of 20 log₁₀|D(0)| where D(θ) is the directivity function of a concave dome of radius a in an infinite baffle, where α is the half angle of the arc formed by the dome. The axial acceleration of the dome is constant. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Z_{s} = R_{s} + j X_{s} = ℜ (Z_{s}) + j ℑ (Z_{s}) .

$Z_{s} = R_{s} + j X_{s} = ℜ (Z_{s}) + j ℑ (Z_{s}) .$

(12.171)

We can see from these curves that at the first peak in the radiation resistance, which more or less corresponds with the first peak in the on-axis response, the radiation reactance is at a minimum, so that the radiation efficiency is enhanced. Below this resonance, the reactance is positive due to the radiation mass. Immediately above it, the reactance is negative due to the compliance of the dome cavity. However, due to standing wave modes, the reactance is alternately positive and negative as the frequency increases above ka = 4.

Problem 12.1. In Section 12.1, we derived the sound pressure at a radial distance w from the axis of an infinitely long pulsating cylinder of radius a. Divide the pressure at the surface (where w = a) by the surface velocity

{\tilde{u}}_{0}

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

to obtain the specific radiation impedance, after substituting

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

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

. Separate the impedance into real and imaginary parts by multiplying the numerator and denominator by the complex conjugate of the denominator

H_{1}^{(1)} (k a)

$H_{1}^{(1)} (k a)$

and applying the Hankel function identities of Eqs. (A2.74) and (A2.75) of Appendix II. Use the Wronskian of Eq. (A2.111) to simplify the real part and thus show that it is the throat impedance of an infinite hyperbolic horn given by Eq. (9.30) if we set a = x_T .

Figure 12.33 Real and imaginary parts of the normalized specific radiation impedance Z _s/ρ ₀ c of the air load on a concave dome of radius a in an infinite baffle, where α is the half angle of the arc formed by the dome. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Problem 12.2. In Section 12.6, we derived the nearfield pressure, far-field pressure and radiation impedance of an oscillating spherical cap in a sphere. Derive the same for a pulsating spherical cap in a sphere using the same methods. In other words, instead of the cap oscillating axially, it moves radially with a velocity u ₀%, which removes the cosθ terms from Eqs. (12.48), (12.49), (12.52), (12.53), and (12.62). Instead of Eq. (A2.68), use Eq. (A2.69) from Appendix II to evaluate the integral on the right-hand-side of Eq. (12.53) and the integral of Eq. (12.62). Show that the results are the same as those obtained in Section 13.5 using the boundary integral method.

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12.10. Radiation from an oscillating concave dome in an infinite baffle

Near-field pressure

Far-field pressure

Radiation impedance

Table of Contents for
Acoustics