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 .A.5This 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 [112,541]. In computational acoustic modeling, Sturm-Liouville theory has been used to give improved models of nonuniform acoustic tubes such as horns , and to provide spatial transforms analogous to the Laplace transform .
In the field of computer music, the introduction of modal synthesis is credited to Adrien . More generally, a modal synthesis model is perhaps best formulated via a diagonalized state-space formulation [560,221,107].A.6 A more recent technique, which has also been used to derive modal synthesis models, is the so-called functional transformation method [363,501,500]. In this approach, physically meaningful transfer functions are mapped from continuous to discrete time in a way which preserves desired physical parametric controls.