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www.chrysalis-foundation.org
Acoustic phenomena are not always predictable.
However, once understood, it is possible to develop methods and techniques
that produce desired results. Below, I
describe two memorable experiences, which began in moments of complete surprise and
ended in contemplating unexpected possibilities.
Frequencies and
Amplitudes of Cavity Resonators
Cavity resonators are often called Helmholtz resonators, in honor of Hermann
Helmholtz (1821–1894) who first used them to hear the faint harmonics of strings
and organ reeds. Unfortunately,
Helmholtz’s original frequency equation, and many other related equations give
reasonably accurate results only if the
walls of the resonator are absolutely rigid.
Many cavity resonators fulfill this requirement. When we blow across the opening of a glass
bottle to produce a musical sound, the walls of the bottle do not vibrate in
response to changes in air pressure inside the cavity. Tests show there is fair agreement between
the theoretical frequency and the actual frequency of the bottle. However, when the walls of the cavity are
not rigid — as in the hollow bodies of violins, guitars, and bass marimba
resonators — the actual resonant frequencies are significantly
lower than the calculated frequencies.
When I first began to design and build the five cavity resonators for the Bass
Marimba, they all sounded much lower than predicted by theory. One day, while I was striking a resonator
with a mallet to hear the resonant frequency, my knee accidentally pushed against
one of the large sides of the resonator.
While my knee contacted the side, the resonant frequency increased by more
than a semitone. I quickly began
installing 1-inch diameter dowels between the two large sides to inhibit their
motion. To my delight, six dowels
increased the resonant frequency by more than a minor third.
I also discovered that increases in the stiffness of the sides dramatically
increased the amplitude of the resonator.
Since less wave energy is spent in vibrating the sides, the amplitude of a
tuned resonator with dowels is much greater than an untuned resonator without
dowels. To understand this process,
imagine riding a bicycle with flexible springs for pedals. Most of the energy supplied by your legs
would be lost in compressing and expanding the springs, and very little energy
would actually go into turning the front sprocket and driving the chain. Similarly, when a pressure wave encounters a moveable surface, a great deal of
the wave energy is lost in bending the surface, instead of compressing and
rarefying the air.
Decreasing and
Increasing the Frequencies of Snifter Glasses
With the exception of the first glass in the lower left hand corner of the
Glassdance, all others are brandy glasses called snifters. The remarkable acoustic properties of these
crystal glasses enabled me to tune two octaves from a single kind of glass. Even though the short glasses in the upper
rows look completely different from the long glasses in the lower rows, they are
all identical with respect to manufacturer and
model number.
An unanticipated moment occurred when I attempted to tune a snifter. Equipped with a precision diamond blade
band saw, I sliced a ¾-inch
high ring from the rim of a glass.
While absorbed in this delicate operation, my intention was to increase the
fundamental frequency of the glass. To
my amazement, the fundamental decreased
by about a minor third. After a
further removal of a 1-inch wide ring, the fundamental, as expected,
increased by about a fifth. For a second
glass, I sliced a ¼-inch high ring from the rim and the frequency
decreased by about a major second; and after a further removal of a ¾-inch
high ring, the fundamental decreased by
only about a semitone. Subsequent
removals all increased the fundamental.
To understand why the frequency of the fundamental decreased after reducing the mass of the glass, note that stiffness acts as the only restoring force which returns a vibrating glass to its equilibrium position. Because the walls of a snifter constrict at the opening, the restoring force due to stiffness has an especially high value at the rim of the glass. In the upper portion of the glass, a removal of circular sections has a greater effect on the restoring force than on the mass. Removing rings of glass in this area causes the walls to become less stiff, or more flexible. Consequently, the walls vibrate less rapidly, which, in turn, decreases the fundamental frequency of the glass. However, after slicing three or four narrow rings from the top, the removal of material has a greater effect on the mass than on the restoring force. This causes the walls in the lower portion to vibrate more rapidly, which, in turn, increases the fundamental frequency of the glass. |