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6.6. Sound pressure produced at distance r

We show in Eq. (13.104) that the far-field on-axis pressure produced by a plane circular piston in an infinite baffle is given by

\begin{matrix} \tilde{p} (r) = j f ρ_{0} {\tilde{U}}_{c} \frac{e^{- j k r}}{r} \\ = ρ_{0} S_{D} {\tilde{a}}_{c} \frac{e^{- j k r}}{2 π r} \end{matrix}

$\begin{matrix} \tilde{p} (r) = j f ρ_{0} {\tilde{U}}_{c} \frac{e^{- j k r}}{r} \\ = ρ_{0} S_{D} {\tilde{a}}_{c} \frac{e^{- j k r}}{2 π r} \end{matrix}$

(6.31)

where

{\tilde{a}}_{c}

${\tilde{a}}_{c}$

is given by Eqs. (6.8), (6.9), and (6.13) so that

\tilde{p} (r) = \frac{{\tilde{e}}_{g} B l S_{D} ρ_{0}}{(R_{g} + R_{E}) M_{M S}} \cdot \frac{e^{- j k r}}{2 π r} α_{c}

$\tilde{p} (r) = \frac{{\tilde{e}}_{g} B l S_{D} ρ_{0}}{(R_{g} + R_{E}) M_{M S}} \cdot \frac{e^{- j k r}}{2 π r} α_{c}$

(6.32)

The frequency response is then given by α _c in Eq. (6.14). In other words, the frequency response is proportional to the cone acceleration and remains flat above the suspension resonance.

The fact that the on-axis frequency response remains flat, even though the radiated power decreases when the wavelength is small in comparison with the circumference of the piston, may seem slightly surprising. However, what we have not taken into account here is the spatial distribution of the radiated sound pressure which becomes increasingly narrow at high frequencies. Although we are not dealing with an ideal flat piston and have not included the effect of the coil inductance, Eq. (6.32) is useful for defining the voltage sensitivity of a loudspeaker within its working frequency range between the suspension resonance and cone break-up (which we will discuss later in this chapter). It shows that for a given coil resistance R _E, the sensitivity is increased by maximizing the Bl factor and diaphragm area S _D while minimizing the total moving mass M _MS, which includes the radiation mass M _MR, although it is usually very small in comparison with M _MD. These requirements are usually in conflict with each other, so it is not possible to optimize all of them in a practical design. Because the most common nominal impedance of a loudspeaker is 8 Ω, the rms generator voltage e _g(rms) is usually taken as

\sqrt{8}

$\sqrt{8}$

or 2.83 V_rms to deliver 1 W of power into an 8 Ω load. Hence, Eq. (6.32) can then be used to give the power sensitivity which is usually expressed in dB SPL (sound pressure level) (relative to 20 μPa, see Eq. 1.18) for W _E = 1 W at r = 1 m, so that

Sensitivity = 20 \log_{10} (\frac{\sqrt{Z_{nom} W_{E}} B l S_{D} ρ_{0}}{2 π r (R_{g} + R_{E}) M_{M S} \times 20 \times 10^{- 6}}) dB SPL / W / m

$Sensitivity = 20 \log_{10} (\frac{\sqrt{Z_{nom} W_{E}} B l S_{D} ρ_{0}}{2 π r (R_{g} + R_{E}) M_{M S} \times 20 \times 10^{- 6}}) dB SPL / W / m$

(6.33)

where Z _nom is the nominal electrical impedance of the drive unit. Theoretically, it is the average value over the loudspeaker's working frequency range, but in practice it is about 10%–30% greater than R _E so that at some frequencies, especially those below resonance, more than 1 W will be supplied at the nominal voltage. Alternatively, by combining Eqs. (6.8), (6.11), (6.26), and (6.33), we may conveniently express the sensitivity in terms of the Thiele–Small parameters V _AS and Q _ES:

Sensitivity = 20 \log_{10} (\frac{1}{r c \times 20 \times 10^{- 6}} \sqrt{\frac{Z_{nom} W_{E} 2 π f_{S}^{3} V_{A S} ρ_{0}}{(R_{g} + R_{E}) Q_{E S}}}) dB SPL / W / m

$Sensitivity = 20 \log_{10} (\frac{1}{r c \times 20 \times 10^{- 6}} \sqrt{\frac{Z_{nom} W_{E} 2 π f_{S}^{3} V_{A S} ρ_{0}}{(R_{g} + R_{E}) Q_{E S}}}) dB SPL / W / m$

(6.34)

Sometimes, we wish to determine how far a diaphragm must travel to produce a target SPL. From Eqs. (1.18) and (6.31), we obtain

η_{peak} = \frac{\sqrt{2} r \times 10^{(\frac{SPL}{20} - 5)}}{π f^{2} ρ_{0} S_{D}}

$η_{peak} = \frac{\sqrt{2} r \times 10^{(\frac{SPL}{20} - 5)}}{π f^{2} ρ_{0} S_{D}}$

(6.35)

Low frequencies

From Eq. (13.101), we see that the magnitude of the pressure at a point in free space a distance r from either side of the loudspeaker in an infinite baffle is that of a point source multiplied by a directivity function:

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

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

(6.36)

where the directivity function D(θ) is given by

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

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

(6.37)

A piston whose diameter is less than one-third wavelength (ka < 1.0) is essentially nondirectional at low frequencies, that is D(θ) ≈ 1 for any value of θ. Hence, we can approximate it by a hemisphere whose volume velocity is

{\tilde{U}}_{c} = S_{D} {\tilde{u}}_{c}

${\tilde{U}}_{c} = S_{D} {\tilde{u}}_{c}$

. It is assumed in writing this equation that the distance r is great enough so that it is situated in the “far field.” Assuming a loss-free medium, the total radiated power distributed over a spherical surface in the far field is

W = 4 π r^{2} I = \frac{4 π r^{2}}{ρ_{0} c} {| \frac{\tilde{p} (r)}{\sqrt{2}} |}^{2}

$W = 4 π r^{2} I = \frac{4 π r^{2}}{ρ_{0} c} {| \frac{\tilde{p} (r)}{\sqrt{2}} |}^{2}$

(6.38)

where I is the intensity at distance r in W/m². From this, we see for a point source radiating to both sides of an infinite baffle (or free space) that

p_{rms} (r) = \sqrt{\frac{W p_{0} c}{4 π r^{2}}}

$p_{rms} (r) = \sqrt{\frac{W p_{0} c}{4 π r^{2}}}$

(6.39)

It is worth noting that it only takes 1 W of acoustic power to produce 6.7 Pa or 109 dB SPL at 1 m, which is as loud as a pneumatic drill! The fact that it takes much more than 1 W of input power to achieve this with a loudspeaker is due to the low efficiency of most loudspeakers.

Medium frequencies

At medium frequencies, where the radiation from the diaphragm becomes directional but yet where the diaphragm vibrates as one unit, i.e., as a rigid piston, the pressure produced at a distance r depends on the power radiated and the directivity factor Q.

The directivity factor Q was defined in Chapter 4 as the ratio of the intensity on a designated axis of a sound radiator to the intensity that would be produced at the same position by a point source radiating the same acoustic power.

For a directional source in an infinite baffle such as we are considering here,

p_{rms} (r) = \sqrt{\frac{W_{1} Q ρ_{0} c}{4 π r^{2}}}

$p_{rms} (r) = \sqrt{\frac{W_{1} Q ρ_{0} c}{4 π r^{2}}}$

(6.40)

where, W ₁ is acoustic power in W radiated from one side of the loudspeaker. Q is directivity factor for one side of a piston in an infinite plane baffle. Values of Q are found from Fig. 4.30. Note that W ₁ equals W/2 and, at low frequencies where there is no directionality, Q = 2, so that Eq. (6.40) reduces to Eq. (6.39) at low frequencies.

We see from Eq. (6.40) that, as frequency increases, Q increases while W ₁ decreases. In other words, the reduction in radiated power is compensated for by the concentration of the radiated sound pressure over a decreasing beamwidth. The transition is so smooth that the frequency response remains flat. This can also be explained by the fact that the on-axis sound pressure is due to an infinite number of point sources over the surface of the piston, the radiation from which arrives in phase, where the frequency response of each point source is flat.

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6.6. Sound pressure produced at distance r

Low frequencies

Medium frequencies

Table of Contents for
Acoustics