Lab Work: Three-Dimensional Geometry
Using a spherical approach in loudspeaker analysis
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.
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
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.
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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
August 2003 Live Sound International