The 1D Wave Equation

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]

- Definition
- Non-dimensionalized Form
- Initial Conditions
- Dispersion Relation
- Phase and Group Velocity
- Energy Analysis
- Boundary Conditions
- Bounds on Solution Size
- Modes
- Characteristics and Travelling Wave Solutions

- A Simple Finite Difference Scheme
- Initialization
- von Neumann Analysis
- Numerical Dispersion
- Energy Analysis
- Numerical Boundary Conditions
- Bounds on Solution Size and Numerical Stability
- Matrix Form
- Output and Interpolation
- Digital Waveguide Interpretation

- Other Schemes

- A Comparative Study: Physical Modeling Sound Synthesis Methods

- Problems
- Programming Exercises