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Our Plan

A logical organization for this book would be along the lines above, starting with Newton's laws, differential equations, digitization methods, and so on through the various methods. However, we proceed instead as follows:
  1. Delay-line and digital-waveguide acoustic modeling
    1. Time-invariant case (acoustic multipath, artificial reverberation)
    2. Time-varying case (phasing, flanging, chorus, Leslie, vibrating strings and air columns)
  2. Impedance modeling and equivalent circuits
  3. Finite differences and transfer functions
  4. Application examples
  5. Appendices on C++ software, history, Newtonian physics, digital waveguide theory details, more about finite difference schemes, and wave digital filters
The reason for this ordering is that it starts out really simple, with no calculus required, and plenty of signal processing and practical applications along the way. It is easily possible to fill a first course without reaching impedance models (which require calculus and frequency-domain facility for both continuous- and discrete-time systems). The appendices provide both supporting and relatively advanced material.

A Simplified Starting Theory

For delay-line and simple digital waveguide modeling, we can accept it as an experimental fact that ``traveling waves happen''. That is, traveling waves are simply observed to propagate at speed $ c$ through air and stretched strings, and they are also observed to obey the superposition principle (traveling waves pass through each other as if they were ghosts). Lossy and dispersive propagation can similarly be observed to correspond to a fixed linear filtering per unit length of propagation medium. When the propagation path-length is doubled, the filter transfer function is squared. The simple paradigm of using a ``filtered delay line'' as a computational modeling element for a single ``ray'' of acoustic propagation is a quite general building block that can be pushed far without getting into the full theory. However, to take digital waveguide modeling to the next level, and to include lumped modeling elements such as masses and springs, we ultimately get to impedance concepts and simple differential equations built around Newton's $ f=ma$. This gives us a ``deeper theory'' in which sound speed can be predicted from basic physical properties, as we saw above. We also then know how distributed and lumped elements should interact, such as when a mass (computed as a first-order digital filter) ``strikes'' a vibrating string (computed as a digital waveguide).
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Elementary Physical Modeling Problems
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Physical Models