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I, on a lark, emailed the German physicist whose web page I mentioned in an
earlier post. He emailed back the next day! Here are his remarks (and his
English is not bad at all):
----- Forwarded by Lindsay Graham/Pavestone on 10/19/00 03:40 PM -----
Ernst
Terhardt To: lindsay@pavestone.com
<terhardt@ei. cc:
tum.de> Subject: Re: Fragen
Sent by:
ter@ei.tum.de
10/19/00
03:58 PM
Dear Mr. Graham.
Essentially I can conceive two quite different types of sound generation
with
a bowl-like
device in combination with striking somewhere near the opening.
The first is mechanical oscillations of the bowl's body including sound
emission from the
walls. This is what ordinarily is called a bell (and bell sound,
respectively). This essentially requires
a kind of bowl whose form, by and large, is bell-like. Regarding the pitch
of
such a sound, it ordinarily
is of the bell-strike-note type (see my web-page), i.e. it is
"psychological".
Even so, that pitch to be
quite low (e.g., in the 100-Hz range or even below) requires the "bowl" to
have considerable dimensions
(more than 1 m) and weight. Smart theorists can compute the oscillations of
such a thing, provided that
all its dimensions and material are known fairly accurately. However the
math
is complicated and can
practically be done only on a computer. I can hardly imagine that the
low-pitched sound you are describing is produced this way.
The second type of sound generation is with a "bowl" that in fact has the
form
of a bottle or vase.
This type of sound generator is known in acoustics as a
"Helmholtz-oscillator". In this device it
essentially is the oscillations of the enclosed air (particularly that
inside
the open neck of the bottle) that form the source of sound. Sound emission
is
from the open neck. When that kind of air-sound oscillations are excited by
striking somewhere near the open neck, the wall's mechanical oscillations
are
not as such the source of sound but merely serve as a transmission line for
the excitation to hit the air inside. The
main and basic oscillation frequency of such a Helmholtz oscillator may
indeed
be quite low. It essentially
depends on the volume of the air within the main body and on the inner
diameter of the neck. When the volume is enlarged, resonance frequency goes
down; when the the neck is made narrower it goes down as well (!).
Computation
is quite simple and easy. The formula can be found in any text book on
acoustics.
Even a bottle of a handy size can be given a form such that the resonance
frequency is quite low. (Just
try with a beer- or wine-bottle.)
So my guess is that it is the second type of mechanism that accounts for
the
particularly low-frequency aspect of the sound of singing bowls. This
aspect
thus probably is physical rather than "psychological".
It appears plausible to assume that the entire sound resulting from
striking
the singing bowl's wall is
"enriched" by the mechanical oscillations (i.e. bell-type oscillations)
which
of course
are inevitably excited as well. However, as mentioned above, it is hard to
see
how the latter type of
oscillation could produce an extremely low pitch, assuming that the singing
bowl is of reasonable
(i.e. portable) size and weight.
Sincerely,
Ernst Terhardt