2017 Cristiano M.L. Forster
All rights reserved.
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
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
model number and manufacturer.
One of the most surprising moments in building the Glassdance 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, I intended to increase the
fundamental frequency of the glass.
To my complete astonishment, the fundamental
decreased by about a minor
third. After a further removal
of a 1-inch-high 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. All subsequent
increased the fundamental.
To comprehend why the frequency of the fundamental decreased after
reducing the mass of the glass, note that stiffness acts as the
only restoring force
that 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.