A physical model of a musical instrument, such as a vibrating string or membrane, may be described in terms of two sets of data: 1) the PDE description itself and associated boundary conditions, and 2) excitation information, including initial conditions and/or an excitation function and location, and readout location(s). The basic modal synthesis strategy is as outlined in Figure 1.7. The first set of information is used, in an initial step, to determine modal shapes and frequencies of vibration; this involves, essentially, the solution of an eigenvalue problem, and may be performed in a variety of ways. (In the functional transformation approach, this is referred to as the solution of a Sturm-Liouville problem). Generally, this information must be stored, the modal shapes themselves in a so-called shape-matrix. Then, the second set of information is employed: the initial conditions and/or excitation are expanded onto the set of modal functions (which usually form an orthogonal set) through an inner product, giving a set of weighting coefficients. The weighted combination of modal functions then evolves, each at its own natural frequency. In order to obtain a sound output at a given time, the modal functions are projected (again through inner products) onto an observation state, which, in the simplest case, is of the form of a delta function at a given location on the object.
Though modal synthesis had been called a ``frequency domain" method here, this is not quite a correct description of the workings of a modal synthesis algorithm, and is worth clarifying. In particular, temporal Fourier transforms are not employed, and the output waveform is generated directly in the time domain. Essentially, the behaviour of each mode is described by a scalar second-order ordinary differential equation, and various time-integration techniques (some of which will be described in Chapter 3) may be employed to obtain a numerical solution. In short, it is perhaps better to think of modal synthesis not as a frequency domain method, but rather a numerical method for a linear problem which has been diagonalized (to borrow a term from state space analysis ). As such, in contrast with a direct time domain approach, the state itself is not observable directly, except through reversal of the diagonalization process (i.e., the projection operation mentioned above). This lack of direct observability has a number of implications in terms of multiple channel output, time-variation of excitation and readout locations, and, most importantly, memory usage. Modal synthesis continues to develop, with newer research directions focussing on the problem of interaction with nonlinear point excitation mechanisms.
Modal synthesis techniques will be discussed at various points in this book, in a general way towards the end of this chapter, and in full technical detail in Chapters 6 and 10.