Banded Waveguides on Circular Topologies and of Beating
Modes: Tibetan Singing Bowls and Glass Harmonicas
Georg Essl, Perry R. Cook
Computer Science Department, Princeton University
email: {gessl,prc} @cs.princeton.edu
Abstract
Banded waveguides were originally introduced to allow for efficient physical modelling of bowed bar percussion instruments. Recently the method has been
used to model instruments whose topology is more complex. In this paper we discuss the use of the method on
circular topologies of conical, cylindrical and hemispherical shells, specifically the Tibetan singing bowl
and the glass harmonica. The tibetan bowl poses a
special challenge to modeling as its sound production
is very resonant and also some mode-pairs lie close
together to create beating modes. In this paper we discuss how the simulation can be achieved using banded
waveguides.
1 Introduction
Though banded waveguides were initially introduced to model bar percussion instruments, that is instruments that are well-described by one-dimensional partial differential equations (Essl and Cook 1999), the
idea can be extended to objects of higher dimensionality or more complex topology (Essi and Cook 2001;
Essl and Cook 2002). Banded waveguides provide an
efficient alternative to more general finite element methods (O'Brien, Cook, and Essl 2001), but to achieve
proper spatial representation, the relation of the geometry to closed wavetrains has to be studied. In a purely
modal synthesis approach the spatial information has
to be aquired by measurement over the whole object
(van den Doel, Kry, and Pai 2001). Some work in the
direction of banded waveguides on higher-dimensional
topologies has been made for two-dimensional circular
objects like drums (Essl and Cook 2001) and cymbals
(Serafin, Huang, and Smith 2001) and three-dimensional objects like rubbed wine glasses and glass harmonicas (Essl and Cook 2001).
In this paper we look at circularily symmetric threedimensional structures. These are cylindrical shells,
conical shells, hemispherical shells and the like, with
the additional constraint that the dominating modes of
oscillation have circular paths. In part this is an extension of the work initially reported in (Essl and Cook
2001) for wine glasses. We study also the so-called
"Tibetan singing bowl". The tibetan bowl provides additional challenges as it has more perceptually relevant
resonant modes as well as modes which are close in
frequency and yield a perceptual beating pattern. Finally the Tibetan bowl is very highly resonant, that is
has very weak internal damping and hence will ring for
a very long time.
2 Rubbing a Wine Glass
Figure 1: Benjamin Franklin's glass harmonica, which
he called "armonica", as seen in the Franklin Institute
Science Museum in Philadelphia.
Drinking glasses, in particular wine glasses, can be
made to ring in many different ways. They can be excited by impact, by rubbing the top rim with a wet finger, or by radially bowing with a violin bow. While impact can easily be simulated using modal models, rubbing and bowing cannot. Geometrically, a wine glass
is a three-dimensional object and disturbances travel
along the object in all dimensions. The object is symmetrical, however, and the dominant modes are essentially two-dimensional (Rossing 2000). One is left with
bending modes along the cylindrical axis, which can
be excited by rubbing, plucking or bowing, but most of
the energy really goes into flexual modes of the circumfence of the glass. This is a closed path - essentially
a bar being bent into a circular shape, closing onto it
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