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13.9. Radiation from a resilient disk in an infinite baffle [21]

A resilient disk in an infinite baffle, like the previous example, represents a source with a uniform pressure distribution over its radiating surface, unlike the rigid piston where the velocity is uniform. This makes the problem slightly harder to solve because we have to include a trial function for the disk velocity distribution in the surface integral. The trial function is in the form of a series expansion, the unknown coefficients of which have to be calculated via a set of simultaneous equations. However, it is worth the effort because, as we shall see, this particular source represents the diffraction pattern due to a plane wave passing through a circular aperture in an infinite screen, which is an important result in optics too. The transmission coefficient, or radiation conductance, was first calculated by Bouwkamp [20] in his PhD dissertation using the boundary value method in the oblate-spheroidal coordinate system. Less than a decade later, Spence [22] calculated the surface velocity distribution and directivity pattern. 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.

Boundary conditions

The configuration is the same as that shown in Fig. 13.3. The infinitesimally thin membrane-like resilient disk is mounted in an infinite baffle in the xy plane with its center at the origin. It 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. As with the rigid piston in an infinite baffle, we will model this as a “breathing” disk in free space. Because of the symmetry of the pressure fields on either side of the baffle

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

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

(13.155)

Consequently, there is a Neumann boundary condition in the plane of the disk where these fields meet:

\frac{\partial}{\partial_{Z}} \tilde{p} {(w, z) |}_{z = 0} = 0, a < w \leq \infty,

$\frac{\partial}{\partial_{Z}} \tilde{p} {(w, z) |}_{z = 0} = 0, a < w \leq \infty,$

(13.156)

which is satisfied automatically. 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 .

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

(13.157)

The pressure gradient is given by

\frac{\partial}{\partial z_{0}} \tilde{p} {(w_{0}, z_{0}) |}_{z_{0} = 0 +} = {\begin{matrix} - j k ρ_{0} c \tilde{u} (w_{0}), & 0 \leq w_{0} \leq a, \\ 0, & w_{0} > a, \end{matrix}

$\frac{\partial}{\partial z_{0}} \tilde{p} {(w_{0}, z_{0}) |}_{z_{0} = 0 +} = {\begin{matrix} - j k ρ_{0} c \tilde{u} (w_{0}), & 0 \leq w_{0} \leq a, \\ 0, & w_{0} > a, \end{matrix}$

(13.158)

where

\tilde{u} (w_{0})

$\tilde{u} (w_{0})$

is the unknown surface velocity distribution 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. We will use the following trial function, which is itself a solution to the free-space Helmholtz wave equation in oblate-spheroidal coordinates [20],

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

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

(13.159)

where A _n are the as-yet unknown power series coefficients that will be evaluated by means of a set of simultaneous equations in matrix form. Note that the n = 0 term is singular when w ₀ = a. This is due to the discontinuity at the perimeter, which is inherent in the problem [2]. Otherwise, if we were modeling a problem with zero velocity at the perimeter, such as a membrane with a clamped perimeter [19], we would replace

(n + \frac{1}{2})

$(n + \frac{1}{2})$

in the index with

(n - \frac{1}{2})

$(n - \frac{1}{2})$

. Using a trial function, any velocity distribution is possible, and this is not the only trial function that may be used [24,25].

Calculation of the velocity series coefficients

The near-field pressure distribution is given by Eq. (13.27) taking into account the double strength source

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

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

(13.160)

where the Green's function in cylindrical coordinates given by Eq. (13.52) is used. In this form Eq. (13.160) is known as the monopole King integral. Inserting Eqs. (13.52), (13.158),(13.159), and (13.99) into Eq. (13.160) and integrating over ϕ ₀ gives

\tilde{p} (w, z) = k \frac{{\tilde{p}}_{0}}{2} \sum_{n = 0}^{\infty} A_{n} (n + \frac{1}{2}) \int_{0}^{a} {(1 - \frac{w_{0}^{2}}{a^{2}})}^{n - \frac{1}{2}} \int_{0}^{\infty} J_{0} (k_{w} w) J_{0} (k_{w} w_{0}) \frac{e^{- j k_{z} z}}{k_{z}} k_{w} d k_{w} w_{0} d w_{0},

$\tilde{p} (w, z) = k \frac{{\tilde{p}}_{0}}{2} \sum_{n = 0}^{\infty} A_{n} (n + \frac{1}{2}) \int_{0}^{a} {(1 - \frac{w_{0}^{2}}{a^{2}})}^{n - \frac{1}{2}} \int_{0}^{\infty} J_{0} (k_{w} w) J_{0} (k_{w} w_{0}) \frac{e^{- j k_{z} z}}{k_{z}} k_{w} d k_{w} w_{0} d w_{0},$

(13.161)

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

{\tilde{p} (w, z) |}_{z = 0 +} = \frac{{\tilde{p}}_{0}}{2}, 0 \leq w \leq a

${\tilde{p} (w, z) |}_{z = 0 +} = \frac{{\tilde{p}}_{0}}{2}, 0 \leq w \leq a$

(13.162)

which leads to the following coupled equation

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

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

(13.163)

which is to be solved for the power series coefficients A _n, where

Φ (w) = 1, 0 \leq w \leq a

$Φ (w) = 1, 0 \leq w \leq a$

(13.164)

and

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

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

(13.165)

The infinite integral [49-51] is given by Eq. (A2.106a), 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}) \frac{k_{w}}{\sqrt{k^{2} - k_{w}^{2}}} d k_{w} = \frac{\sqrt{π} k}{2} \sum_{m = 0}^{\infty} \frac{1}{{(m!)}^{2}} {(\frac{w}{w_{0}})}^{2 m} \\ \times \sum_{r = 0}^{\infty} \frac{Γ (\frac{r}{2} + \frac{1}{2})}{Γ (\frac{r}{2} + 1) Γ^{2} (\frac{r}{2} - m + \frac{1}{2})} {(\frac{- j k w_{0}}{2})}^{r - 1}, \end{matrix}

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

(13.166)

so that

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

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

(13.167)

which is simplified with help of the integral

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

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

(13.168)

to give

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

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

(13.169)

which is an expansion in (w/a)^2m. We also note that

Φ (w) = \sum_{m = 0}^{\infty} δ_{m 0} {(\frac{w}{a})}^{2 m}, 0 \leq w \leq a

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

(13.170)

where δ _m0 is the Kronecker delta function. Inserting Eqs. (13.169) and (13.170) into Eq. (13.163) and equating the coefficients of (w/a)^2m yields the following (N + 1) × (N + 1) matrix equation

M \cdot a = b \Rightarrow a = M^{- 1} \cdot b,

$M \cdot a = b \Rightarrow a = M^{- 1} \cdot b,$

(13.171)

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

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

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

(13.172)

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

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

(13.173)

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

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

(13.174)

and the infinite power series limits have been truncated to order N. The monopole cylindrical wave function _n T _m is named the Stenzel–Spence function in tribute to their pioneering work and is defined by

{{}_{n}T}_{m} (k a) = - \sqrt{π} \frac{Γ (n + \frac{3}{2})}{{(m!)}^{2}} \sum_{r = 0}^{N} {{}_{n}P}_{m} (r) {(\frac{- j k a}{2})}^{r + 1}

${{}_{n}T}_{m} (k a) = - \sqrt{π} \frac{Γ (n + \frac{3}{2})}{{(m!)}^{2}} \sum_{r = 0}^{N} {{}_{n}P}_{m} (r) {(\frac{- j k a}{2})}^{r + 1}$

(13.175)

{{}_{n}P}_{m} (r) = \frac{Γ (\frac{r}{2} + \frac{1}{2})}{Γ (\frac{r}{2} + 1) Γ (\frac{r}{2} - m + \frac{1}{2}) Γ (\frac{r}{2} - m + n + 1)}

${{}_{n}P}_{m} (r) = \frac{Γ (\frac{r}{2} + \frac{1}{2})}{Γ (\frac{r}{2} + 1) Γ (\frac{r}{2} - m + \frac{1}{2}) Γ (\frac{r}{2} - m + n + 1)}$

(13.176)

Now that we have the surface velocity series coefficients A _n, we can derive some radiation characteristics for the resilient disk.

Far-field pressure

The far-field pressure distribution is given by Eq. (13.27) taking into account the double-strength source:

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

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

(13.177)

where the far-field Green's function in spherical-cylindrical coordinates given by Eq. (13.70) is used. Inserting Eqs. (13.70), (13.158) and (13.159) into Eq. (13.177) 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} {\tilde{p}}_{0} \frac{e^{- j k r}}{4 r} D (θ),

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

(13.178)

where the directivity function D(θ) is given by

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

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

(13.179)

The on-axis pressure is evaluated by setting θ = 0 in Eq. (13.70) before inserting it in Eq. (13.177) and integrating over the surface to give

D (0) = \sum_{n = 0}^{N} A_{n} \approx {\begin{matrix} 4 j / (π k a), k a < 0.5 \\ 1, k a > 2. \end{matrix}

$D (0) = \sum_{n = 0}^{N} A_{n} \approx {\begin{matrix} 4 j / (π k a), k a < 0.5 \\ 1, k a > 2. \end{matrix}$

(13.180)

It is worth noting that D(0) is simply the normalized radiation admittance, that is

D (0) = ρ_{0} c (G_{S} + j B_{S}),

$D (0) = ρ_{0} c (G_{S} + j B_{S}),$

where G _s and B _s are given by Eqs. (13.193) and (13.194) respectively. The asymptotic expression for low-frequency on-axis pressure is then simply

\tilde{p} (r, 0) \approx - \frac{a}{π r} {\tilde{p}}_{0} e^{- j k r}, k a < 0.5

$\tilde{p} (r, 0) \approx - \frac{a}{π r} {\tilde{p}}_{0} e^{- j k r}, k a < 0.5$

(13.181)

and at high frequencies

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

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

(13.182)

which is the same as for a resilient disk in free space at all frequencies. The on-axis response is shown in Fig. 13.15, calculated from the magnitude of D(0). The normalized directivity function 20 log₁₀|D(θ)/D(0)| is plotted in Fig. 13.16 for four values of ka = 2πa/λ, that is, for four values of the ratio of the circumference of the piston to the wavelength. When the circumference of the piston (2πa) is less than one-half wavelength, that is, ka < 0.5, the disk behaves essentially like a point source. When ka becomes greater than 3, the resilient disk is highly directional, rather like the rigid piston in an infinite baffle except without the nulls.

Figure 13.15 Plot of 20 log₁₀ (π ka | D(0)/4) where D(θ) is the directivity function of a resilient disk of radius a in an infinite baffle. The uniform driving pressure is constant. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Figure 13.16 Far-field directivity patterns for a resilient disk in an infinite baffle as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the disk.

Near-field pressure

The near-field pressure distribution is given by Eq. (13.27) taking into account the surface pressure on both sides:

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

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

(13.183)

where the Green's function in spherical-cylindrical coordinates given by Eq. (13.68) is used. It has been shown [21] that inserting Eqs. (13.68), (13.158) and (13.159) into Eq. (13.183) and integrating over the surface gives

p (r, θ) = {\tilde{p}}_{0} \sum_{n = 0}^{N} A_{n} \sum_{p = 0}^{P} \frac{{(- 1)}^{p} Γ (p + \frac{1}{2}) Γ (n + \frac{3}{2}) h_{2 p}^{(2)} (k r) P_{2 p} (\cos θ)}{Γ (2 p + \frac{1}{2}) Γ (p + n + \frac{3}{2})} \times {(\frac{k a}{2})}^{2 p + 2} {}_{1}{F_{2}} (p + 1; p + n + \frac{3}{2}, 2 p + \frac{3}{2}; - \frac{k^{2} a^{2}}{4}),

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

(13.184)

which converges for r > a. For r ≤ a, we derive a suitable expression from Eq. (13.161), which is weakly singular at k _w = k. However, we can remove this singularity as follows: First, we substitute

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

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

and

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

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

in Eq. (13.161) to obtain

\tilde{p} (w, z) = {\tilde{p}}_{0} \sum_{n = 0}^{\infty} A_{n} Γ (n + \frac{3}{2}) (I_{F i n} + j I_{I n f}),

$\tilde{p} (w, z) = {\tilde{p}}_{0} \sum_{n = 0}^{\infty} A_{n} Γ (n + \frac{3}{2}) (I_{F i n} + j I_{I n f}),$

(13.185)

where

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

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

(13.186)

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

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

(13.187)

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

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

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

(13.188)

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{3}{2})} {(\frac{k a}{2})}^{2 p + 2} {}_{2}{F_{1}} (- p, - n - p - \frac{1}{2}; 1; \frac{w^{2}}{a^{2}}) \\ \times {(\frac{2}{k z})}^{p + \frac{1}{2}} (J_{p + \frac{1}{2}} (k z) - j H_{p + \frac{1}{2}} (k z)), \end{matrix}

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

(13.189)

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

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

(13.190)

Eq. (13.189) converges everywhere and is therefore suitable for r < a. Unfortunately, Eq. (13.190) only converges for z ² > w ² + a ² and is therefore not suitable. However, Eq. (13.187) converges everywhere and can be calculated numerically without problem and is therefore suitable for r < a. Using the Babinet–Bouwkamp principle, this represents the field scattered by a hole in an infinite screen in the presence of an incident plane wave, as plotted in Fig. 13.39 for three values of ka.

Surface velocity

The magnitude and phase of the normalized velocity from Eq. (13.159) are shown in Figs. 13.17 and 13.18, respectively, for four values of ka. We see that the velocity increases rapidly toward the perimeter where it is singular. This is a feature of uniform pressure sources in general because of the discontinuity at the perimeter.

Figure 13.18 Surface velocity phase for a resilient circular disk in an infinite baffle 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

The total volume velocity

{\tilde{U}}_{0}

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

is found by integrating the velocity from Eq. (13.159) over the surface of the disk to give

{\tilde{U}}_{0} = \int_{0}^{2 π} \int_{0}^{a} {\tilde{u}}_{0} (w_{0}) w_{0} d w_{0} d ϕ_{0} = \frac{π a^{2} {\tilde{p}}_{0}}{2 ρ_{0} c} \sum_{n = 0}^{N} A_{n} .

${\tilde{U}}_{0} = \int_{0}^{2 π} \int_{0}^{a} {\tilde{u}}_{0} (w_{0}) w_{0} d w_{0} d ϕ_{0} = \frac{π a^{2} {\tilde{p}}_{0}}{2 ρ_{0} c} \sum_{n = 0}^{N} A_{n} .$

(13.191)

The specific radiation admittance is then given by

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

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

(13.192)

where

{\tilde{U}}_{0}

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

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} ℜ (\sum_{n = 0}^{N} A_{n}) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{8}{π^{2}}, k a < 0.5, \end{matrix}

$\begin{matrix} G_{s} = \frac{1}{ρ_{0} c} ℜ (\sum_{n = 0}^{N} A_{n}) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{8}{π^{2}}, k a < 0.5, \end{matrix}$

(13.193)

where the bold G indicates that the quantity varies with frequency. 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} ℑ (\sum_{n = 0}^{N} A_{n}) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{4}{π k a}, k a < 0.5. \end{matrix}

$\begin{matrix} B_{s} = \frac{1}{ρ_{0} c} ℑ (\sum_{n = 0}^{N} A_{n}) \\ \approx \frac{1}{ρ_{0} c} \cdot \frac{4}{π k a}, k a < 0.5. \end{matrix}$

(13.194)

Plots of the real and imaginary parts of

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

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

(13.195)

are shown in Fig. 13.19 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{k^{2} a^{2}}{2} - j \frac{π k a}{4}, k a < 0.5 \end{matrix}

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

(13.196)

are shown in Fig. 13.20. 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.

Figure 13.19 Real and imaginary parts of the normalized specific radiation admittance Y _s/ρ ₀ c of the air load on one side of a plane circular resilient disk of radius a in an infinite baffle. Frequency is plotted on a normalized scale, where ka = 2πa/λ = 2πfa/c.

Figure 13.20 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 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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13.9. Radiation from a resilient disk in an infinite baffle [21]

Boundary conditions

Calculation of the velocity series coefficients

Far-field pressure

Near-field pressure

Surface velocity

Radiation admittance

Table of Contents for
Acoustics