## 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.

**Next Section:**

Mass-Spring Resonators

**Previous Section:**

Early Musical Acoustics