Acoustics

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15.4. Continuous delay

Let us now consider the ideal situation whereby we increase the number of rings while reducing their widths until the delay becomes continuously variable along the radius of the membrane. Then we can isolate the effect of the delay profile from the discretization of the rings. If we treat the membrane as a pure pressure source with zero mass and stiffness, the far-field radiated sound pressure at a distance r and angle θ from its center is obtained by inserting Eqs. (13.70) and (13.121) into Eq. (13.124) to yield

\tilde{p} (r, θ) = j k \cos θ \frac{e^{- j k r}}{r} \int_{0}^{a} {\tilde{p}}_{+} (w) J_{0} (k w \sin θ) w d w,

$\tilde{p} (r, θ) = j k \cos θ \frac{e^{- j k r}}{r} \int_{0}^{a} {\tilde{p}}_{+} (w) J_{0} (k w \sin θ) w d w,$

(15.1)

where J ₀ is the zero-order Bessel function, k = ω/c is the wave number, ω = 2πf is the angular frequency, and

{\tilde{p}}_{+} (w)

${\tilde{p}}_{+} (w)$

is the radial distribution of the electrostatic driving pressure. The tilde denotes a harmonically varying quantity where the term e ^jωt has been suppressed.

No Delay. If there is no delay, then the pressure everywhere on the surface of each side is just half the driving pressure

{\tilde{p}}_{+} (w) = \frac{{\tilde{p}}_{0}}{2},

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

(15.2)

where

{\tilde{p}}_{0} = \frac{ε_{0} E_{P}}{d^{2}} {\tilde{e}}_{in}

${\tilde{p}}_{0} = \frac{ε_{0} E_{P}}{d^{2}} {\tilde{e}}_{in}$

(15.3)

and ε ₀ is the permittivity of air. The far-field pressure response then becomes that of Eq. (13.125)

\tilde{p} (r, θ) = j k a^{2} ε_{0} \frac{E_{P}}{d} \cdot \frac{{\tilde{e}}_{in}}{2 d} \cdot \frac{e^{- j k r}}{2 r} D (θ),

$\tilde{p} (r, θ) = j k a^{2} ε_{0} \frac{E_{P}}{d} \cdot \frac{{\tilde{e}}_{in}}{2 d} \cdot \frac{e^{- j k r}}{2 r} D (θ),$

(15.4)

where the directivity function is given by

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

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

(15.5)

The normalized directivity pattern 20 log₁₀|D(θ)| − 20 log₁₀|D(0)| is plotted in Fig. 15.2.

Noting that D(0) = 1, let the normalized on-axis response be

p_{norm} = \frac{\tilde{p} (r, 0)}{a {\tilde{p}}_{0} e^{- j k r} / (4 r)} = j k a D (0) = j k a,

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

(15.6)

which is plotted in Fig. 15.7 (black dashed). The input current

{\tilde{I}}_{in} \approx j ω C_{E} {\tilde{e}}_{in}

${\tilde{I}}_{in} \approx j ω C_{E} {\tilde{e}}_{in}$

is almost entirely due to the static capacitance C _E = ε ₀ πa ²/(2d) so that the on-axis pressure simply becomes Walker's equation [7]

Figure 15.2 Directivity patterns at various frequencies of 280 mm diameter membrane with no delay. Naturally, the high frequencies are extremely directive.

Figure 15.3 Geometry of virtual point source, which behaves like two back-to-back spherical caps with a discontinuity where they join, unlike a smooth oscillating sphere.

\tilde{p} (r, 0) = \frac{E_{P}}{d} \cdot \frac{e^{- j k r}}{2 π r} \cdot \frac{{\tilde{I}}_{in}}{c} .

$\tilde{p} (r, 0) = \frac{E_{P}}{d} \cdot \frac{e^{- j k r}}{2 π r} \cdot \frac{{\tilde{I}}_{in}}{c} .$

(15.7)

Virtual point source. The geometry of a traditional “virtual point source” is shown in Fig. 15.3.

Because of its finite radius a, the membrane can only reproduce the part of the wavefront emanating from the source, which forms a spherical cap with half angle α and radius of curvature R, where

R = a \cot α .

$R = a \cot α .$

(15.8)

To reproduce this, the delay must account for the time taken for the wave to travel the distance ΔR at each point w along the radius according to

Δ R = \sqrt{R^{2} + w^{2}} - R .

$Δ R = \sqrt{R^{2} + w^{2}} - R .$

(15.9)

Hence, the surface pressure distribution is given by

{\tilde{p}}_{+} (w) = \frac{{\tilde{p}}_{0}}{2} e^{j k Δ R} = \frac{{\tilde{p}}_{0}}{2} e^{- j k (\sqrt{a^{2} \cot^{2} α + w^{2}} - a \cot α)},

${\tilde{p}}_{+} (w) = \frac{{\tilde{p}}_{0}}{2} e^{j k Δ R} = \frac{{\tilde{p}}_{0}}{2} e^{- j k (\sqrt{a^{2} \cot^{2} α + w^{2}} - a \cot α)},$

(15.10)

which leads to the directivity pattern

D (θ) = 2 \cos θ \int_{0}^{1} e^{- j k a (\sqrt{\cot^{2} α + s^{2}} - \cot α)} J_{0} (k a s \sin θ) s d s,

$D (θ) = 2 \cos θ \int_{0}^{1} e^{- j k a (\sqrt{\cot^{2} α + s^{2}} - \cot α)} J_{0} (k a s \sin θ) s d s,$

(15.11)

where we have substituted s = w/a. The directivity pattern 20 log₁₀|D(θ)| − 20 log₁₀|D(0)| is plotted in Fig. 15.4. The on-axis response is

Figure 15.4 Directivity patterns at various frequencies of a virtual point source (unshaded) using a 280 mm diameter membrane, where the half angle α = 40 degrees is close to that of the Quad ESL63 [6].

p_{norm} = j k a D (0) = \frac{2 j}{k a} ((1 + \frac{j k a}{\sin α}) e^{\frac{- j k a}{\sin α}} - (1 + \frac{j k a}{\tan α}) e^{\frac{- j k a}{\tan α}}),

$p_{norm} = j k a D (0) = \frac{2 j}{k a} ((1 + \frac{j k a}{\sin α}) e^{\frac{- j k a}{\sin α}} - (1 + \frac{j k a}{\tan α}) e^{\frac{- j k a}{\tan α}}),$

(15.12)

which is plotted in Fig. 15.7 (dark gray).

Virtual oscillating sphere. Naively, we might insert the pressure produced by an oscillating sphere, given by Eq. (4.129), into Eq. (15.1), while setting

{\tilde{P}}_{0} = ρ_{0} c {\tilde{U}}_{0} / (π a^{2})

${\tilde{P}}_{0} = ρ_{0} c {\tilde{U}}_{0} / (π a^{2})$

, to yield

p_{norm} = \frac{- 2 k^{2} a^{2}}{2 - k^{2} a^{2} + j 2 k a} (e^{- j k a} - \frac{e^{- j \sqrt{2} k a}}{\sqrt{2}}),

$p_{norm} = \frac{- 2 k^{2} a^{2}}{2 - k^{2} a^{2} + j 2 k a} (e^{- j k a} - \frac{e^{- j \sqrt{2} k a}}{\sqrt{2}}),$

(15.13)

where we have substituted r ² = w ² + a ² and cos θ = a/r. The first term in parentheses gives the true response of an oscillating sphere that would be obtained if the membrane were infinitely large. However, the second term, which is a “diffraction” term because of the membrane's finite size, interferes with the first term to produce an irregular response as shown in Fig. 15.7 (light gray dashed). Hence, for our virtual oscillating sphere, we shall adopt the geometry shown in Fig. 15.5, where the axial distance between each point on the front surface of the virtual sphere and the membrane is

a - \sqrt{a^{2} - w^{2}}

$a - \sqrt{a^{2} - w^{2}}$

Figure 15.5 Geometry of virtual oscillating sphere.

The amount of delay required at each point along the radius of the membrane is the time taken for sound to travel this distance axially as a plane wave. Hence,

{\tilde{p}}_{+} (w) = \frac{{\tilde{p}}_{0}}{2} e^{- j k (a - \sqrt{a^{2} - w^{2}})},

${\tilde{p}}_{+} (w) = \frac{{\tilde{p}}_{0}}{2} e^{- j k (a - \sqrt{a^{2} - w^{2}})},$

(15.14)

which leads to the directivity pattern

D (θ) = 2 \cos θ \int_{0}^{1} e^{- j k a (1 - \sqrt{1 - s^{2}})} J_{0} (k a s \sin θ) s d s,

$D (θ) = 2 \cos θ \int_{0}^{1} e^{- j k a (1 - \sqrt{1 - s^{2}})} J_{0} (k a s \sin θ) s d s,$

(15.15)

where we have substituted s = w/a. The directivity pattern 20 log₁₀|D(θ)| − 20 log₁₀|D(0)| is plotted in Fig. 15.6.

The on-axis response is

p_{norm} = k a D (0) = \frac{2 j}{k a} (1 - e^{- j k a} - j k a),

$p_{norm} = k a D (0) = \frac{2 j}{k a} (1 - e^{- j k a} - j k a),$

(15.16)

which is plotted in Fig. 15.7 (black) along with the following first-order high-pass filter approximation (light gray)

Figure 15.6 Directivity patterns at various frequencies of a virtual oscillating sphere using a 280 mm diameter membrane. The broad figure-of-eight pattern is almost constant at all frequencies.

p_{norm} \approx \frac{2 j k a}{2 + j k a} .

$p_{norm} \approx \frac{2 j k a}{2 + j k a} .$

(15.17)

This approximation can be included as part of a crossover filter response, for example. The cut-off frequency is given by f _C = c/(πa). Above f _C, we have

p_{norm} \approx 2, f > \frac{c}{π a} .

$p_{norm} \approx 2, f > \frac{c}{π a} .$

(15.18)

Then from Eqs. (15.4), (15.6), and (15.18), the voltage sensitivity is

\tilde{p} (r, 0) = ε_{0} a \frac{E_{P}}{d} \cdot \frac{{\tilde{e}}_{in}}{2 d} \cdot \frac{e^{- j k r}}{r}, f > \frac{c}{π a}

$\tilde{p} (r, 0) = ε_{0} a \frac{E_{P}}{d} \cdot \frac{{\tilde{e}}_{in}}{2 d} \cdot \frac{e^{- j k r}}{r}, f > \frac{c}{π a}$

(15.19)

and at lower frequencies we have

\tilde{p} (r, 0) = j ε_{0} k a^{2} \frac{E_{P}}{d} \cdot \frac{{\tilde{e}}_{in}}{2 d} \cdot \frac{e^{- j k r}}{2 r}, f < \frac{c}{π a},

$\tilde{p} (r, 0) = j ε_{0} k a^{2} \frac{E_{P}}{d} \cdot \frac{{\tilde{e}}_{in}}{2 d} \cdot \frac{e^{- j k r}}{2 r}, f < \frac{c}{π a},$

(15.20)

which is the same as Eq. (15.4) (with D(0) = 1) for the voltage sensitivity of a massless flexible membrane at all frequencies. It is equivalent to Walker's Eq. (15.7) but given in terms of input voltage instead of current, so we shall call it Walker's voltage equation. The maximum field strength that we can realistically expect without breakdown is E _P/d = 2000 V/mm. Similarly, the input voltage should not exceed 4000 V peak across 2 mm. If the radius a is 14 cm and the permittivity of free space ε ₀ is 8.85 pF/m, the maximum RMS sound pressure from Eq. (15.19) is 105 dB SPL at 1 m re 20 μPa. This pressure increases by 6 dB for every doubling of the diameter.

Figure 15.7 On-axis responses 20 log₁₀|p _norm/2| of a 280 mm diameter membrane with a continuous and unattenuated delay line configured to simulate a virtual oscillating sphere (black), a naïve oscillating sphere (dark gray dashed), and a point source (dark gray) where the half angle of the arc is 40 degrees. Also shown is a first-order filter response that approximates the oscillating sphere (light gray with f _C = 784 Hz) and the on-axis response of the membrane with no delay (black dashed), which keeps rising as the energy is focused on-axis. The −3 dB frequency of the virtual oscillating sphere is 0.75 f _C = 588 Hz.

The on-axis plot of a virtual oscillating sphere shown in Fig. 15.7 tells us that, in theory, a continuously increasing delay in the driving pressure along the radius of the membrane produces a very smooth response, with just some very small ripples, and an almost constant figure-of-eight directivity pattern at all frequencies, as shown in Fig. 15.6. Although perfectly constant directivity is not achieved, the result is remarkably good considering the finite size of the membrane. Next, we will look at the discretization of the delay into rings of various widths.

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15.4. Continuous delay

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Acoustics