## Conical *vs* Cylindrical Bores

### Why do they sound different?

A soprano saxophone and a clarinet are about the same size and are
both single reed instruments, but they sound completely different.
In addition, the range of a clarinet is rougly one octave lower.
While there are other more subtle differences, the main difference
is that a clarinet has a cylindrical bore and a saxophone has
a conical bore. This difference causes a large difference in
the resonant modes, and hence the overtones which can be excited.
For a cylindrical bore, the amplitude of the pressure variations
for resonant modes are well described by sine waves. At the ends
of an open finite cylinder (neglecting end effects),
the pressure variations
should be zero (i.e. the ends are a pressure node).
At a closed end, the pressure variations should be a maximum
(i.e. an "anti-node").

For a conical bore, the amplitude of the pressure variations
are not simple sine waves, but are described by sin(x)/x, where
x represents a distance (in appropriate units) along the cone,
and x = 0 is the apex. As is the case for the closed cylinder,
a pressure anti-node must be present at the closed end of the
cone, which occurs automatically for the function sin(x)/x,
and a pressure node should be present at the open end.

The figure below illustrates (schematically) the pressure variations
for a cylinder open at both ends, an open cone, and a cylinder
closed at one end. The end conditions for the cylinder closed at
one end can only be met by the odd harmonics, and hence all even
harmonics are missing from the sound. On the other hand, the
cone and the cylinder open at both ends contain both the odd and
even harmonics and when the same length, will have the same
resonant frequencies.

The frequency for the nth harmonic is f_{n} = n*f_{1}.

In addition, note that to obtain the *same* fundamental frequency,
f_{1}, the cylinder closed at one end is 1/2 the length
of the cylinder open at both ends. Hence, if the two were the
same length, the closed cylinder would play an octave lower than
the open cylinder.

To a first approximation,
the clarinet can be considered as having a cylindrical bore
closed at one end (by the
reed/mouthpiece), whereas a saxophone has a (truncated) conical bore
(with the apex near the mouthpiece).

For more detail, including the case of a partial cone with
openings at both ends, see:

"The conical bore in musical acoustics," R. D. Ayers, L. J. Eliason,
and D. Mahgerefteh, American Journal of Physics, Vol 53, No. 6, pgs 528-537,
(1985).

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