On the other hand, working with lumped elements in order to describe a distributed system is somewhat cumbersome, especially when an extremely elegant mathematical representation of the distributed system is available (this is often the case in musical acoustics). The distinction between the use of lumped and distributed modeling can be described as physical, in the former case, and mathematical in the latter. The relative advantages of the two approaches was well-expressed many years ago by Ames [2]:
Ames was not speaking about musical systems, but replacing ``heat conducting slab" and ``heat conducting rods" above by ``membrane" and ``masses and springs," one can translate his point to the musical sound synthesis setting. As one might expect, when a lumped network is intended to behave as a distributed object, an equivalent finite difference scheme can be shown to exist--see §6.4.1 and §10.1.3 for two examples. One could proceed farther along these lines, but it quickly becomes difficult to deal with important numerical issues such as accuracy, or global approximations to derivatives, or complex boundary conditions; a fully distributed model is indispensable in such cases. In many ways, however, it is not fair to compare the lumped network formalism with that of direct simulation, because the goals are somewhat different.