Digital Waveguide Theory

In this appendix, the basic principles of digital waveguide acoustic modeling are derived from a mathematical point of view. For this, the reader is expected to have some background in linear systems and elementary physics. In particular, facility with Laplace transforms [290], Newtonian mechanics [183], and basic differential equations is assumed.

We begin with the partial differential equation (PDE) describing the ideal vibrating string, which we first digitize by converting partial derivatives to finite differences. This yields a discrete-time recursion which approximately simulates the ideal string. Next, we go back and solve the original PDE, obtaining continuous traveling waves as the solution. These traveling waves are then digitized by ordinary sampling, resulting in the digital waveguide model for the ideal string. The digital waveguide simulation is then shown to be equivalent to the finite-difference recursion obtained earlier. (This only happens for the ideal vibrating string with a particular choice of sampling intervals, so it is an interesting case.) Next digital waveguides simulating lossy and dispersive vibrating strings are derived, and alternative choices of wave variables (displacement, velocity, slope, force, power, etc.) are derived. Finally, an introduction to scattering theory for digital waveguides is presented.

• The Ideal Vibrating String
• The Finite Difference Approximation
  • FDA of the Ideal String
  
  
• Traveling-Wave Solution
  • Traveling-Wave Partial Derivatives
  • Use of the Chain Rule
  • String Slope from Velocity Waves
  • Wave Velocity
  • D'Alembert Derived
  • Converting String-State to Traveling-Waves
  
  
• Sampled Traveling Waves
  • Digital Waveguide Model
  • Digital Waveguide Interpolation
  • Relation to the Finite Difference Recursion
  
  
• The Lossy 1D Wave Equation
  • Loss Consolidation
  • Frequency-Dependent Losses
  • Well Posed PDEs for Modeling Damped Strings
  • Digital Filter Models of Damped Strings
  • Lossy Finite Difference Recursion
    • Frequency-Dependent Losses
  
  
• The Dispersive 1D Wave Equation