The systems examined so far in this book have been one-dimensional. Certain musical instruments, in particular those of the percussion family, employ vibrating structures which can only be well-described in two dimensions. As one might expect, simulation complexity increases accordingly. At the time of writing, there has been, so far, relatively little work on two-dimensional problems in sound synthesis, partly because, until recently, real time synthesis on small computers was not possible. Another reason has been that percussion instruments have seen much less fundamental investigation from the point of view of musical acoustics. On the other hand, such problems have a long research history in mainstream simulation, and, as a result, there is a wide expanse of literature and results which may be adapted to sound synthesis applications. For the linear problems discussed in this chapter, the systems may be directly generalized from their 1D counterparts, described in Chapter 7, and for this reason, the development here is somewhat abbreviated.
A good starting point is the 2D wave equation, introduced in §10.1, which serves as a useful test problem for the vibration of membranes, and also as another good point of comparison for the various physical modeling synthesis techniques, including finite difference schemes, modal methods, lumped networks, and digital waveguide meshes. The lossless Kirchhoff thin plate model, as well as several variants which serve to describe more realistic plate vibration in a musical context, are discussed in §10.2 and §10.3. Finally, as examples, the audio effect known as plate reverberation is simulated in §10.4, and piano soundboards in §10.5.
References include: [22,29,21,250,224,28,236,192,235,88,89,191,194,96,95,27,97]