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Part XXXVI: Radiation and scattering in cylindrical-spherical coordinates

13.7. Radiation from a rigid circular piston in an infinite baffle

The simplest monopole planar source is the oscillating circular piston (or rigid disk) in an infinite baffle. The piston is assumed to be rigid so that all parts of its surface vibrate in phase, and its velocity amplitude is independent of the mechanical or acoustic loading on its radiating surface. Remarkably, its radiation impedance was first derived by Rayleigh [10] before the direct radiator loudspeaker had even been invented [11], yet it has been widely accepted as an idealized model for such when mounted in an enclosure situated near a wall or, even better, mounted directly in a wall as commonly found in recording studios. The model is useful in the frequency range up to the first diaphragm break-up mode. It should be noted that here the term “infinite baffle” refers to an infinitely large plane rigid wall that surrounds the piston and not a finite sealed enclosure, which is often referred to as an infinite baffle enclosure. The only thing they have in common is that they both block the transmission path between the back and the front of the radiating surface. However, the infinitely large wall model does not take into account reflections from the edges of a real finite enclosure. Also both sides of the radiating surface are open to half space so that the loading effects of a real finite enclosure such as compliance, standing waves, absorption, and wall vibration, etc. are ignored. The original derivation of the radiation impedance by Rayleigh over 100 years ago used the nonintegral Green's function of Eq. (13.16) with an ingenious coordinate system. Here we shall follow the approach of King [12] using the integral Green's function in cylindrical coordinates given by Eq. (13.52).

Boundary conditions

The circular piston of radius a shown in Fig. 13.3 is mounted in an infinite baffle 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}

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

, thus radiating sound into a homogeneous loss-free medium. The area of each surface element is given by

δ S_{0} = w_{0} δ w_{0} δ ϕ_{0} .

$δ S_{0} = w_{0} δ w_{0} δ ϕ_{0} .$

(13.96)

The monopole source elements shown in Fig. 13.4, together with their images, form the piston source. As they are coincident in the plane of the baffle, they coalesce to form elements of double strength. Hence the piston in an infinite baffle can be modeled as a “breathing” disk in free space. It may also be considered as a pulsating sphere of the same radius compressed into the plane of the disk. Because of the symmetry of the pressure fields on either side of the baffle,

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

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

(13.97)

Consequently, there is the following Neumann boundary condition on its surface:

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

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

(13.98)

which is satisfied automatically. On the surface of the disk there is the coupling condition

Figure 13.3 Geometry of rigid circular piston in infinite 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.

Figure 13.4 Equivalence between circular piston in infinite baffle and double-sided monopole piston in free space or “breathing” disk in free space.

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

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

(13.99)

and 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.

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},

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

(13.100)

where the far-field Green's function in spherical-cylindrical coordinates given by Eq. (13.70) is used. Inserting Eqs. (13.70) and (13.99) into Eq. (13.100) 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} ρ_{0} c {\tilde{u}}_{0} \frac{e^{- j k r}}{2 r} D (θ),

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

(13.101)

where the directivity function D(θ) is given by

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

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

(13.102)

which is often referred to as the Fraunhofer or Airy diffraction pattern. The normalized directivity function 20 log₁₀|D(θ)| is plotted in Fig. 13.5 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 piston behaves essentially like a point source. When ka becomes greater than 3, the piston is highly directional.

The on-axis pressure is evaluated by setting θ = 0 in Eq. (13.70) before inserting it in Eq. (13.100) and integrating over the surface to give

D (0) = 1,

$D (0) = 1,$

(13.103)

which means that the on-axis far-field pressure is proportional to the piston acceleration at all frequencies and is often written as

\tilde{p} (r, 0) = j ρ_{0} f {\tilde{U}}_{0} \frac{e^{- j k r}}{r},

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

(13.104)

where

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

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

is the total volume velocity. This is a general expression for a planar source in an infinite baffle and also applies to nonuniform velocity distributions where the volume velocity is the product of the average velocity and the radiating area, which can be of arbitrary shape.

Figure 13.5 Far-field directivity patterns for a rigid circular piston in an infinite baffle as a function of ka = 2πa/λ = 2πfa/c, where a is the radius of the piston. The directivity index never becomes less than 3 dB because the piston radiates only into half space.

Although the piston behaves as a more or less omnidirectional source for ka ≤ 1, similar to a pulsating sphere, the output of the piston is 6 dB less than that of the pulsating sphere at very low frequencies. Because the piston is radiating into half space, its output per unit surface area is double that of the pulsating sphere, which is radiating into whole space. However, the sphere has four times the surface area of a piston of the same radius. Therefore it produces twice the output. Unlike the pulsating sphere, the on-axis response of the piston does not roll-off at high frequencies, which is a property of planar sources in general, as already discussed in Section 12.8 regarding a piston in a sphere. Unlike the piston in a sphere, there is no 6 dB level shift between low and high frequencies because the baffled piston effectively radiates into half space at all frequencies. As we shall see, its radiation impedance, like that of a pulsating sphere, is dominated by mass reactance at low frequencies and resistance at high frequencies.

In the low-frequency region, the radiated sound pressure and hence also intensity are held constant under constant piston acceleration. This is because the decreasing velocity is compensated for by the rising radiation resistance, as discussed in greater detail in Section 4.10.

At higher frequencies, where the impedance starts to become more resistive, the beam pattern, coincidentally, becomes increasingly narrow. This phenomenon compensates for the fall in on-axis output that would otherwise occur. Indeed, in the case of the pulsating sphere, the radiated sound pressure is proportional to the surface velocity in the region where the load is resistive and therefore falls under constant acceleration and falling velocity. It seems a remarkable coincidence of nature that this transition occurs so smoothly as to produce a completely flat on-axis response, although it does not seem so surprising when we consider that the on-axis response results from the sum of an array of point sources that are all in phase, where the field of each point source is frequency invariant under constant volume acceleration.

Near-field pressure

The near-field pressure distribution is given by the boundary integral of 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},

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

(13.105)

where the Green's function in spherical-cylindrical coordinates given by Eq. (13.68) is used. Mast and Yu [13] show that inserting Eqs. (13.68) and (13.99) into Eq. (13.105) and integrating over the surface gives [46]

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

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

(13.106)

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 as was done previously by Stenzel [14]. However, a better expression is provided by Mast and Yu [13], which is derived by moving the origin of the coordinate system to a point on the z axis that lies in the same plane as the observation point to give

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

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

(13.107)

where f _2n is given by the following recursion formulas:

f_{0} = e^{- j k z} - e^{- j k r_{a}},

$f_{0} = e^{- j k z} - e^{- j k r_{a}},$

(13.108)

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

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

(13.109)

and

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

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

(13.110)

which converges for w ² < a ² + z ² but is generally used for w < a and is thus termed the paraxial solution. These equations are an elegant and important result for ultrasound because they eliminate the need for inefficient numerical integration at high frequencies. In particular, the number of terms needed for convergence in the paraxial expansion decreases linearly toward the z-axis until just a single term remains. This is the closed-form Backhaus axial solution [47]:

\tilde{p} (0, z) = ρ_{0} c {\tilde{u}}_{0} (e^{- j k z} - e^{- j k r_{a}}) .

$\tilde{p} (0, z) = ρ_{0} c {\tilde{u}}_{0} (e^{- j k z} - e^{- j k r_{a}}) .$

(13.111)

The first term represents a point source at the center of the piston and the second term radiation from the perimeter. The magnitude of the axial pressure is

| \tilde{p} (0, z) | = 2 ρ_{0} c | {\tilde{u}}_{0} \sin k (r_{a} - z) / 2 | .

$| \tilde{p} (0, z) | = 2 ρ_{0} c | {\tilde{u}}_{0} \sin k (r_{a} - z) / 2 | .$

Near the surface of the piston, it is approximately

| \tilde{p} (0, z) | \approx ρ_{0} cka | {\tilde{u}}_{0} | / (1 + z / a)

$| \tilde{p} (0, z) | \approx ρ_{0} cka | {\tilde{u}}_{0} | / (1 + z / a)$

for ka < 0.5 and z < 0.5a. Hence, at low frequencies, the radiated sound pressure of a loudspeaker may be calculated from the diaphragm velocity [see Eq. 13.101], which in turn may be measured using a probe microphone close to the center. The pressure field for three values of ka is plotted in Fig. 13.6 and for ka = 12π in Fig. 13.7. From these figures, we can see the formation of the central and side lobes of the directivity patterns at the start of the far-field or Fraunhofer diffraction zone, where the waves are spherically diverging. The near-field or Fresnel region is dominated by nonpropagating interference patterns due to the differences in path lengths from different parts of the radiating surface. However, in the immediate near field of Fig. 13.7, the pressure fluctuations are relatively small and we see here the formation of a plane wave, which extends outwards with increasing frequency. The furthest axial peak is a focal point, which is useful for ultrasound applications. Also, we can make the following observations:

Figure 13.6 Normalized near-field pressure plots for a rigid circular piston in an infinite baffle as a function of ka = 2πa/λ = 2πfa/c. Where a is the radius of the piston, $| \tilde{p} |$ $| \tilde{p} |$ is the pressure magnitude, ${\tilde{u}}_{0}$ ${\tilde{u}}_{0}$ is the piston velocity, ρ ₀ is the density of the acoustic medium, and c is the speed of sound in that medium.

Figure 13.7 Normalized near-field pressure plots for a rigid circular piston in an infinite baffle as a function of ka = 2πa/λ = 2πfa/c. Where a is the radius of the piston, $| \tilde{p} |$ $| \tilde{p} |$ is the pressure magnitude, ${\tilde{u}}_{0}$ ${\tilde{u}}_{0}$ is the piston velocity, ρ ₀ is the density of the acoustic medium, and c is the speed of sound in that medium.

1. At low frequencies, where ka < 3, the on-axis pressure of Eq. (13.111) converges to the far-field approximation of Eq. (13.104) at around z = πa/2.
2. At high frequencies, where ka > 3, the on-axis pressure converges to the far-field approximation at around z = ka ²/2, which is known as the Rayleigh distance [15,16]. The on-axis near-field pressure is oscillatory and there are ka/(2π) or a/λ cycles before it converges to the far-field response, where one cycle spans two magnitude peaks or two nulls. The pressure on the face of the piston also oscillates radially with a total of ka/2π or a/λ cycles between the center and perimeter. Furthermore, if ka = nπ or nλ = 2a, where n is an integer, the pressure at the center of the piston is at a null for even n and at a peak for odd n.
3. The number of lobes in the directivity pattern corresponds to the number of axial peaks plus the number of peaks along the radius of the piston. An alternative expression to Eq. (13.107) is given in Reference [48].

Radiation impedance and high-frequency asymptotic expression

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 (r, θ | 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},

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

(13.112)

where the Green's function in cylindrical coordinates given by Eq. (13.52) is used. In this form Eq. (13.112) is known as the monopole King integral [12]. Inserting Eqs. (13.52) and (13.99) into Eq. (13.112) 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},

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

(13.113)

where we have used the integral solution of Eq. (95) from Appendix II and k _z is given by Eq. (13.51). To investigate the asymptotic high-frequency behavior, we let k → ∞ in Eq. (13.113) to give

\begin{matrix} \tilde{p} {(w, z) |}_{k \to \infty} = - ρ_{0} c {\tilde{u}}_{0} e^{- j k z} a \int_{0}^{\infty} J_{1} (k_{w} a) J_{0} (k_{w} w) d k_{w} \\ = {\begin{matrix} - ρ_{0} c {\tilde{u}}_{0} e^{- j k z}, & 0 \leq w \leq a \\ 0, & w > a . \end{matrix} \end{matrix}

$\begin{matrix} \tilde{p} {(w, z) |}_{k \to \infty} = - ρ_{0} c {\tilde{u}}_{0} e^{- j k z} a \int_{0}^{\infty} J_{1} (k_{w} a) J_{0} (k_{w} w) d k_{w} \\ = {\begin{matrix} - ρ_{0} c {\tilde{u}}_{0} e^{- j k z}, & 0 \leq w \leq a \\ 0, & w > a . \end{matrix} \end{matrix}$

(13.114)

This slightly trivial solution describes the sound being radiated as a laser beam confined within the diameter of the piston. It can also be regarded as a virtual infinite tube or transmission line in space starting from the perimeter of the piston. At first sight, this may appear to contradict Eq. (13.111), because the axial nulls and peaks never actually disappear. On the contrary, they become more numerous and travel out further with increasing frequency. However, in the high frequency limit, the radial width of this range of hills and dales shrinks so much that they become insignificant.

The total radiation force is found by integrating the pressure from Eq. (13.113) over the surface of the piston and again using the integral of Eq. (95) from Appendix II to give

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

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

(13.115)

King [12] shows the solution to be

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

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

(13.116)

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

R_{s} = ρ_{0} c (1 - \frac{J_{1} (2 k a)}{k a}) \approx ρ_{0} c \frac{k^{2} a^{2}}{2}, k a < 0.5,

$R_{s} = ρ_{0} c (1 - \frac{J_{1} (2 k a)}{k a}) \approx ρ_{0} c \frac{k^{2} a^{2}}{2}, k a < 0.5,$

(13.117)

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

X_{s} = ρ_{0} c \frac{H_{1} (2 k a)}{k a} \approx ρ_{0} c \frac{8 k a}{3 π}, k a < 0.5,

$X_{s} = ρ_{0} c \frac{H_{1} (2 k a)}{k a} \approx ρ_{0} c \frac{8 k a}{3 π}, k a < 0.5,$

(13.118)

where J ₁ and H ₁ are Bessel and Struve functions respectively as defined by Eqs. (71) and (125) in Appendix II. Plots of the real and imaginary parts of

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

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

(13.119)

are shown in Fig. 4.35 as a function of ka. Similar graphs of the real and imaginary parts of the specific admittance

\begin{matrix} Y_{s} ρ_{0} c = ρ_{0} c (G_{s} + j B_{s}) = ρ_{0} c (\frac{R_{s}}{R_{s}^{2} + X_{s}^{2}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}}) \\ \approx \frac{9 π^{2}}{128} - j \frac{3 π}{8 k a}, k a < 0.5 \end{matrix}

$\begin{matrix} Y_{s} ρ_{0} c = ρ_{0} c (G_{s} + j B_{s}) = ρ_{0} c (\frac{R_{s}}{R_{s}^{2} + X_{s}^{2}} - j \frac{X_{s}}{R_{s}^{2} + X_{s}^{2}}) \\ \approx \frac{9 π^{2}}{128} - j \frac{3 π}{8 k a}, k a < 0.5 \end{matrix}$

(13.120)

are shown in Fig 4.36. The specific admittance is in m³·N⁻¹ s⁻¹ (rayl⁻¹).

We see from Fig. 4.35 that, for ka < 0.5, the reactance varies as the first power of frequency while the resistance varies as the second power of frequency. At high frequencies, for ka > 5, the reactance becomes small compared with the resistance, and the resistance approaches a constant value.

The admittance, on the other hand, is better behaved. The conductance is constant for ka < 0.5, and it is also constant for ka > 5 although its value is larger.

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