Delayed models for simplified musical instruments

Most musical instruments contain, at their very basis, a continuous vibrating element (string or air column) which can be treated as a one-dimensional system. Its oscillation is obtained either through an initial condition or by means of a continuous energy input through a nonlinear device. In both cases and as a first approach, the excitation can be localized at one single point, and the continuous system can be considered as a linear one. The coupling between these two elements is often represented through a convolution integral. This convolution will be rewritten here in a way that different phenomena taking place in the continuous element (internal losses, radiation at the ends...) are separated. Different choices in the formulation of these processes and some mathematical manipulation will lead to either algebraic iterative or delayed differential equations. These equations are valid for any form of energy input. Once this energy input is defined, they can be used to simulate the behavior of different instruments in a more efficient way than that of traditional convolution. Moreover, these equations allow an analytical analysis of possible regimes using the tools of nonlinear dynamical systems (NLDS). The case of woodwinds will be emphasized throughout the paper, while that of strings will be presented briefly for the sake of completeness.