Lab Work: Three-Dimensional Geometry
Using a spherical approach in loudspeaker analysis

By Perrin Meyer

The spherical loudspeaker expressly built for research purposes.

As part of ongoing research on measuring and predicting loudspeaker polar patterns, the Research and Development Group at Meyer Sound has built a loudspeaker with a spherical enclosure. A spherical loudspeaker is useful because a sphere is the simplest three-dimensional geometry. This simplicity is necessary because of the complexity of acoustic radiation and diffraction phenomena. Imagine a single woofer. As it vibrates, acoustic energy is radiated from both the front and back of the woofer.

The simplest loudspeaker design mounts a single woofer in a small enclosure, which “seals” the back wave inside the enclosure, and only the forward sound wave from the woofer is radiated into the environment.

However, since the wavelengths of sound are large compared to the size of a small enclosure, the forward sound wave from the woofer actually diffracts around the enclosure and can be heard (at varying strengths) behind the loudspeaker. This diffraction effect around a loudspeaker cabinet varies with frequency.

VARIOUS FREQUENCIES

At very low frequencies, (20 Hz to 100 Hz), the wavelengths of sound are large (10 feet to 50 feet), and the size of a typical loudspeaker is much smaller than this, so the diffraction is approximately uniform. This is why subwoofers are usually thought to be “omnidirectional.”

Polar patterns can be measured in the anechoic chamber for use with MAPP Online.

As the frequency of sound increases (and the size of the wavelength decreases), the diffraction around the loudspeaker cabinet becomes smaller. At “high” frequencies (approximately 10 kHz and above), the wavelength of the sound is less than one inch, and because most loudspeaker cabinets are much larger than one inch, there is very little diffraction around loudspeaker cabinets at high frequencies. However, in the critical frequency range of 200 Hz to 5 kHz, the wavelength of sound is on the same “order of magnitude” as the size of the loudspeaker cabinets, and this makes both measuring and predicting the spatial diffraction patterns extremely hard. A polar pattern is a single two-dimensional circular measurement of a three-dimensional spatial acoustic diffraction field.

But it turns out that from a mathematical point of view, if a loudspeaker has a spherical enclosure, it is possible to derive a mathematical “analytic series solution” to the diffraction pattern. Physicists studying planetary astrophysics and molecular interactions first used these techniques, but the acoustic diffraction of a small woofer set in a large spherical enclosure has a similar formulation.

Technically, it is possible to derive an analytic series solution of the spherical coordinate formulation of the Helmholtz equation utilizing sums of properly weighted Legendre functions and spherical Hankel functions.

MAKE A PREDICTION

What does all this math lead to? Well, it allows engineers to predict numerically the acoustic diffraction pattern of a small woofer set in a spherical enclosure. In fact, we’ve built such an enclosure using a three-inch woofer set in a ridged 10-inch sphere. Using the mathematical formulation describe above, it is possible to predict the spatial diffraction pattern (and consequently, the polar pattern) of this model loudspeaker.

The polar patterns of this model loudspeaker can then be measured with the polar data acquisition system in an anechoic chamber, which is used to measure loudspeakers for our MAPP Online program. The comparison between the measured and predicted data is used to both verify the accuracy of the data acquisition system and the algorithms used in MAPP Online.

The polar response of the spherical loudspeaker can be viewed in MAPP Online; under the “Configure Loud-speaker” heading choose “SPHERE.”

Editor’s Note: To check it out, go to www.meyersound.com and click on “Get the Latest Release of MAPP Online.”

Perrin Meyer is staff scientist with Meyer Sound and can be reached at pmeyer@meyersound.com

August 2003 Live Sound International