Measuring the Directivity Patterns of a Loudspeaker with and without a Baffle

I. Introduction

In this lab exercise you will investigate directivity patterns using a simple loudspeaker as an acoustic source. You will measure and plot the directivity patterns for the loudspeaker in a baffle at different frequencies. You will also compare the power levels and directivity patterns for the speaker with and without the baffle to observe the effect of the baffle. This handout discusses the necessary theoretical resources, experimental apparatus, measurement procedure, and analysis to perform the lab.

II. Theory

(4)

A simple acoustic source is a source whose dimensions are much smaller than the wavelength of the sound being radiated. This relationship between wavelength and dimension for a simple source is usually expressed as ka <<1 where k=2 π / λ is the wavenumber, λ is the wavelength, and a is a characteristic dimension of the source.

Most often, in a treatment of the sound radiated by a simple source, one considers the measurement point to be located in the far-field. This means that the distance from the sound source to the observation point is much larger than any dimension of the source. Following the theoretical development in Reference [1], the far-field pressure radiated by a simple source in free space may be written as

(1)

and the pressure amplitude is then

(2)

where Q is the source strength, ρ is the fluid density, c is the speed of sound in the fluid, k is the wavenumber, and r is the distance from source to observation point. A simple source radiates equally well in all 3-dimensional directions (no θ or φ dependence). The pressure falls off as 1 / r with distance from the source.

C. Baffled Simple Source

When a simple source is mounted on or very close to a rigid plane boundary the acoustic waves generated by the source will reflect from this surface, in such a way that the pressure of the reflected waves and that of the radiated waves are virtually in phase everywhere. The pressure field in the half-space, into which the source radiates, will then be twice that generated by the source (with the same source strength) in free space. The pressure field radiated by a baffled simple source is thus

(3)

Another way of thinking of this is that the source is radiating with the same source strength, but into only half the space, so the pressure output is doubled.

D. Acoustic Dipole

Two simple sources of equal source strength, but opposite phase, and separated by a small distance d (such that kd << 1) comprise an acoustic dipole. The pressure field radiated by an acoustic dipole may be written as

(5)

This is a spherically diverging wave with pressure amplitude

(6)

which may be interpreted as the product of the pressure amplitude radiated by a simple source, a term kd which relates the radiated wavelength to the source separation, and a directivity function which depends on the angle θ. A dipole does not radiate equally in all directions. Its directivity pattern has two lobes indicating strong radiation, and two directions in which no sound is radiated (it looks like a figure-8).

The power radiated by an acoustic dipole can be shown to vary with frequency as ω⁴, whereas the power radiated by a monopole (simple source) varies with frequency as ω². It stands to reason then, that at low frequencies a dipole is much less efficient at radiating sound than is a monopole.

E. Circular Piston in a Baffle

Many realistic acoustic sources may not be treated as simple sources because the relationship between dimension and radiated sound wavelength does not satisfy ka<<1. In such a case, the radiated pressure field may be built up by breaking up the complex source into many small elements, each acting as a simple source (or a baffled simple source), and then adding up (integrating over a surface) all of these simple sources. A classic example of a complex acoustic source, and one which may be derived relatively easily, is a circular piston mounted in an infinite rigid baffle. This source is also a good first approximation to a loudspeaker mounted in a finite baffle.

The complete derivation for the pressure field radiated by a circular piston mounted in an infinite rigid baffle may be found in the references listed at the end of this lab write-up. After all the math, the far-field radiated pressure may be written as

(7)

which may be interpreted as the product of the pressure radiated by a baffled simple source from Eq.(3) and a directivity function specific to the geometry of the circular piston. The function J_1(x) is a first order Bessel function. Figure 1 shows the behavior of the function in brackets in Eq.(7).

Figure 1: Behavior of the directivity function for a baffled piston.

At low frequencies, such that ka<<1, the Bessel function asymptotes to so that the pressure radiated by the baffled circular piston becomes

(8)

which is just the pressure radiated by a baffled simple source, and the radiated sound pattern is omnidirectional (the same in all directions). At higher frequencies, however, one must actually calculate the directivity function for the specific value of ka and also for the specific angle θ. Figure 2 shows calculated directivity patterns for four values of the parameter ka for a baffled circular piston.

Figure2: Radiation directivity patterns for a circular piston in an infinite plane rigid baffle

	ka = 1
	ka = 4
	ka = 7
	ka = 10

E. Directivity Index

The directivity factor D may be defined as the ratio of the intensity of a source in some specified direction (usually along the acoustic axis of the source) to the intensity at the same point in space due to an omnidirectional point source with the same acoustic power. Mathematically, the directivity factor may be written as

(9)

Pressure are rms values. The directivity factor indicates how much more effectively a directional source concentrates its available acoustic power into a preferred direction.

The directivity index DI is simply 10 times the log of the directivity factor.

(10)

III. Equipment

• 3-inch loudspeaker (Radio Shack \#40-243)
• baffle (cardboard or wood, approximately 1 meter on a side, with 3-inch hole cut in middle)
• Function/Waveform Generator (Hewlet Packard 33120A)
• Audio Amplifier
• Sound Level Meter (Radio Shack \#33-2050)
• a large, nonreveberant room, (an anechoic chamber is preferred)

1. Measure the approximate or "effective" radius a of the loudspeaker cone. This radius should include about half of the "surround" (the suspension at the outer edge of the speaker connecting it to the speaker frame).
2. Make sure the speaker is mounted on its stand, about 1m above the floor.
3. Connect the cables between the signal generator, amplifier, and the loudspeaker.
4. Check the battery of the Radio Shack Sound Level Meter. If the measurements are being made in an ordinary room with low frequency background noise, use A weighting and slow response. If measurements are being made in an anechoic chamber use C weighting.

5. Set the signal generator to generate a 1000 Hz (1 kHz) sine wave.
6. Hold the Sound Level Meter on the same level as the speaker, and at a distance of 1 m. Start with the microphone pointing towards the front of the speaker (on-axis), and call this 0°. Adjust the level of the amplifier until the meter shows a about 70-80 dB. Record the sound pressure level.
7. Move the Sound Level Meter in a complete 360° circular path around the loudspeaker, taking measurements of the sound pressure level at every 10 degrees. Be sure that the Sound Level Meter is always pointing towards the loudspeaker, 1 m above the floor and 1 m from the loudspeaker.
8. Place the baffle so that the speaker just sits in the hole. Measure the sound pressure level 1 m in front of the baffled speaker, and ±20° on either side.

9. Make a table of the frequency f, wavelength \lambda (use c=344 m/s), wavenumber k, and dimensionless frequency ka for the loudspeaker at frequencies of 1.5, 7, 12, and 17 kHz.
10. Change the frequency of the sinewave to 1500 Hz (1.5 kHz). Measure the sound pressure level at a distance of 1 m from the speaker for angles ranging from -90° to +90°, every 10 degrees, with 0° being directly in front of the speaker.
11. Repeat step 10 for frequencies of 7, 12, and 17 kHz. You may need to take measurements every 5 degrees for 17 kHz in order to get accurate enough data.

1. Normalize your data to the on-axis measurement (corresponding to 0°) by taking the negative of the difference of each measurement and the on-axis value (including the on-axis value itself). For example, on-axis measurement of 80 dB becomes 0 dB, and 10° measurement of 78 dB becomes -2 dB.
2. Plot the normalized values on a polar plot with the maximum, on-axis value being 0 dB at 0°. In other words, make a plot like those in Fig. 2 (or Fig. 4-10 of [Beranek, 1986].)
3. Does this directivity plot look like what you would expect for an omnidirectional simple source, or a dipole (figure-8)?
4. Compare the on-axis value for the speaker in free space with the on-axis value for the speaker. Did you hear any difference in the sound level when the baffle was added? Does the pressure double (SPL increase by 3 dB) because of the baffle? If not, try to explain why it doesn't.
5. Comment on how might the monopole vs dipole behavior and the unbaffled vs baffled behavior could explain why loudspeakers are placed in a box in order to improve their sound quality?

6. Normalize all your data to the on-axis measurements as in step 1.
7. Plot the normalized values on a polar plot. Make a separate plot for each frequency.
8. How do your plots compare to those shown in Fig. 2 (or to Fig. 4-10 of [Beranek, 1986])? How well do the calculated directivity plots for a piston in a baffle predict your measured directivity patterns for a real loudspeaker.

9. Determine the on-axis rms pressure P_ax =P_e radiated by the speaker in the baffle at 1000 Hz from Eq.(4). Divide this pressure by 2. This result will serve as a fabricated omnidirectional simple source with rms pressure, P_s.
10. For each frequency (1.5, 7, 12, and 17 kHz), calculate the on-axis rms pressure pressure, P_ax, from the on-axis sound pressure levels, using Eq.(4) for the baffled speaker. Calculate the directivity factor, D = P_ax² / P_s².
11. Calculate the directivity index DI = 10 log D for each frequency. How do your values compare to those shown in the attached figures? What do these values indicate about the directionality of the speaker as a function of frequency?

1. L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics, Third Edition, Section 8.8, pp 176-182, (John Wiley & Sons, 1982).
   
2. D. Reynolds, Engineering Principles of Acoustics: Noise and Vibration Control, (Allyn & Bacon, 1981).
   
3. L. Beranek, Acoustics, (reprinted by The Acoustical Society of America, 1986).

This laboratory exercise has been used very successfully in a rather popular senior level course, "PHYS-580 / ME-530, Acoustics, Noise, and Vibration," which serves as an elective for Mechanical Engineering, Electrical Engineering, and Applied Physics majors at Kettering University. The directivity plots shown below were obtained by senior level students who performed this experiment in a large, uncarpeted, open room, with several large cabinets and pieces of equipment asymmetrically scattered around the the room. The speaker used was a Radio Shack 3 inch cone speaker (a = 3.7 cm). The plots show reasonably good agreement with theoretical predictions for a circular piston in a baffle in Figure 2 of this writeup as well as with Figure 4.10 of [Beranek, 1986]. Performing the measurements in an anechoic chamber would no doubt improve the accuracy of the plots.

NOTE: Even though there was a significant amount of reflection from the floor, cabinets and large pieces of machinery and equipment in the room in which these measurements were made, the overall shapes (and number of side lobes) of the directivity patterns are quite good and match expected patterns within acceptable limits.

Figure2: Student measurements of radiation directivity patterns for a 3-inch loudspeaker with and without a baffle

Unbaffled speaker has a dipole-like directivity pattern
Baffled loudspeaker at 1.5 kHz (ka=0.8)	Baffled loudspeaker at 7 kHz (ka=4)
Baffled loudspeaker at 12 kHz (ka=7)	Baffled loudspeaker at 17 kHz (ka=10)