The 1D Wave Equation

In this chapter, the one-dimensional wave equation is introduced; it is, arguably, the single most important partial differential equation in musical acoustics, if not in physics as a whole. Though, strictly speaking, it is useful only as a test problem, variants of it serve to describe the behaviour of strings, both linear and nonlinear, as well as the motion of air in an enclosed acoustic tube. It is well worth spending a good deal of time examining this equation as well as various means of arriving at a numerical solution, because the distinctions between well-known synthesis techniques such as modal synthesis, digital waveguides and time-domain methods such as finite differences appear in sharp contrast.

In §6.1, the one-dimensional wave equation is defined, in the first instance, for simplicity, over the entire real line. Frequency domain analysis is introduced, followed by a discussion of phase and group velocity. The Hamiltonian formulation for the wave equation appears next, followed, finally, by a brief look at traveling wave solutions, which form the basis of digital waveguide synthesis. A simple finite difference scheme is presented in §6.2, followed by frequency domain (or von Neumann) analysis, yielding a simple (Courant-Friedrichs-Lewy) stability condition [61], and information regarding numerical dispersion, as well as its perceptual significance in sound synthesis. The section is concluded by a brief look at the matrix form of the finite difference scheme, and, finally the very important special case of the digital waveguide. Other varieties of finite difference schemes appear in §6.3. Finally, in §6.4, various synthesis methods, specifically modal synthesis, lumped networks, and digital waveguides are compared in the special case of the wave equation.

References for this chapter include: [208,22,210,51,153,154,29,228,209,223,103,130,157,61,134]