### 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
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 . 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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A Software Delay Line

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General Modeling Procedure