## History of Modal Expansion

While the traveling-wave solution for the ideal vibrating string is
fully general, and remains the basis for the most efficient known
string synthesis algorithms, an important alternate formulation is in
terms of *modal expansions*, that is, a superposition sum of
orthogonal basis functions which solve the differential equation and
obey the boundary conditions. Daniel Bernoulli (1700-1782)
developed the notion that string vibrations
can be expressed as the superposition of an infinite number of
harmonic vibrations [103].^{A.5}This approach ultimately evolved to Hilbert spaces of orthogonal basis
functions that are solutions of Hermitian linear operators--a
formulation at the heart of quantum mechanics describing what can be
observed in nature [112,539]. In computational acoustic
modeling, Sturm-Liouville theory has been used to give improved models
of nonuniform acoustic tubes such as horns [50], and to
provide spatial transforms analogous to the Laplace transform
[360].

In the field of computer music, the introduction of *modal
synthesis* is credited to Adrien [5]. More generally, a
modal synthesis model is perhaps best formulated via a diagonalized
state-space formulation
[558,220,107].^{A.6} A more recent
technique, which has also been used to derive modal synthesis models,
is the so-called *functional transformation method*
[360,499,498].
In this approach, physically meaningful transfer functions are mapped
from continuous to discrete time in a way which preserves desired
physical parametric controls.

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Mass-Spring Resonators

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Early Musical Acoustics