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 [286], Newtonian mechanics [181], 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 a particular finite-difference recursion. (This only happens for the lossless 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, and various special topics are introduced.

- The Ideal Vibrating String
- The Finite Difference Approximation

- 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

- A 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

- The Dispersive 1D Wave Equation

- Alternative Wave Variables
- Spatial Derivatives
- Force Waves
- Wave Impedance
- State Conversions
- Power Waves
- Energy Density Waves
- Root-Power Waves
- Total Energy in a Rigidly Terminated String

- Scattering at Impedance Changes
- Plane-Wave Scattering
- Plane-Wave Scattering at an Angle
- Longitudinal Waves in Rods
- Kelly-Lochbaum Scattering Junctions
- One-Multiply Scattering Junctions
- Normalized Scattering Junctions
- Junction Passivity

- Digital Waveguide Filters
- Ladder Waveguide Filters
- Reflectively Terminated Waveguide Filters
- Half-Rate Ladder Waveguide Filters
- Conventional Ladder Filters
- Power-Normalized Waveguide Filters

- ``Traveling
Waves'' in Lumped Systems

- Properties of Passive Impedances

- Loaded Waveguide
Junctions

- Two Coupled Strings

- Digital Waveguide Mesh
- The Rectilinear 2D Mesh
- Dispersion
- Further Developments
- 2D Mesh and the Wave Equation
- The Lossy 2D Mesh
- Diffuse Reflections in the Waveguide Mesh

- FDNs as Digital Waveguide Networks

- Waveguide Transformers and Gyrators

- The Digital Waveguide Oscillator
- Additive Synthesis
- Digital Sinusoid Generators
- The Second-Order Waveguide Filter
- Application to FM Synthesis
- Digital Waveguide Resonator
- State-Space Analysis of Waveguide Oscillator
- Eigenstructure
- Summary
- Matlab Sinusoidal Oscillator Implementations
- Faust Implementation

- Non-Cylindrical Acoustic Tubes

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University