Physical Models

We now turn to the main subject of this book, *physical models
of musical instruments and audio effects*. In contrast to the
non-physical signal models mentioned above, we will consider a signal
model to be a *physical signal model* when there is an explicit
representation of the relevant physical *state* of the sound
source. For example, a string physical model must offer the
possibility of exciting the string at any point along its length.

We begin with a review of physical models in general, followed by an overview of computational subtypes, with some indication of their relative merits, and what is and is not addressed in this book.

- All We Need is Newton
- Formulations
- ODEs
- PDEs
- Difference Equations (Finite Difference Schemes)
- State Space Models
- Forming Outputs
- State-Space Model of a Force-Driven Mass
- Numerical Integration of General State-Space Models
- State Definition

- Linear State Space Models

- Transfer Functions
- Modal Representation
- State Space to Modal Synthesis
- Force-Driven-Mass Diagonalization Example
- Typical State-Space Diagonalization Procedure
- Efficiency of Diagonalized State-Space Models

- Equivalent Circuits
- Impedance Networks
- Wave Digital Filters
- Digital Waveguide Modeling Elements
- General Modeling Procedure

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