### Damping of waves at the walls of a conical tube

__Cornelis Johannes Nederveen__^{a}, Timo Grothe^{b} and Johannes Baumgart^{c}

^{a}Pijnacker

^{b}Hochschule für Musik Detmold

^{c}Dresden

### Saturday, September 14, 2019 from 14:00 to 14:01

#### in Summer Theater

Abstract :

Resonators of reed wind instruments are tubular ducts with one open
and one closed end. The ratio of pressure
response to flow excitation at the closed end is the input impedance.
Resonance frequencies of the duct are near
to peaks in the impedance spectrum. Damping due to visco-thermal
effects at the walls influences the frequency
and the magnitude of the impedance-spectrum peaks, which
influence
intonation, playing behaviour and timbre.
For cylindrical instruments theory to account for wall losses is
available
and experimentally confirmed. The
wave equation in a conical tube while accounting for dissipative
effects
at the walls appears to be complicated.
Four approximative solutions are compared: (1) Nederveen (1969)
presented an approximate analytical solution
while neglecting some higher order terms. (2) a transmission line
method mimicking the conical pipe as a series
of short conical (or cylindrical) pipes, (3) directly solving the equation
with a Runge-Kutta procedure, (4)
applying a finite difference method. For a “simplified bassoon” (a
perfect cone of 3000 mm length, input
diameter 4.2 mm, output diameter 46.9 mm) the four methods give
different results. Measurements are planned,
but the narrow tube entrance and smoothness requirements make a
high accuracy difficult. Suggestions are
welcome.

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