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In this chapter, we summarize the basic principles of *digital
waveguide models*. Such models are used for efficient synthesis of
string and wind musical instruments (and tonal percussion, etc.), as
well as for artificial reverberation. They can be further used in
modal synthesis by efficiently implementing a quasi harmonic series of
modes in a single ``filtered delay loop''.

We begin with the simplest case of the infinitely long ideal vibrating
string, and the model is unified with that of acoustic tubes. The
resulting computational model turns out to be a simple
*bidirectional delay line*. Next we consider what happens when a
finite length of ideal string (or acoustic tube) is rigidly terminated
on both ends, obtaining a *delay-line loop*. The delay-line loop
provides a basic digital-waveguide synthesis model for (highly
idealized) stringed and wind musical instruments. Next we study the
simplest possible excitation for a digital waveguide string model,
which is to move one of its (otherwise rigid) terminations.
Excitation from ``initial conditions'' is then discussed, including
the ideal plucked and struck string. Next we introduce *damping*
into the digital waveguide, which is necessary to model realistic
losses during vibration. This much modeling yields musically useful
results. Another linear phenomenon we need to model, especially for
piano strings, is *dispersion*, so that is taken up next.
Following that, we consider general excitation of a string or tube
model at any point along its length. Methods for calibrating models
from recorded data are outlined, followed by modeling of coupled
waveguides, and simple memoryless nonlinearities are introduced and
analyzed.

- Ideal Vibrating String
- Wave Equation
- Wave Equation Applications
- Traveling-Wave Solution
- Sampled Traveling-Wave Solution
- Wave Impedance

- Ideal Acoustic Tube
- Rigid Terminations

- Moving Rigid Termination
- Digital Waveguide Equivalent Circuits
- Animation of Moving String Termination
- Terminated String Impedance

- The Ideal Plucked String
- The Ideal Struck String
- The Damped Plucked String

- Frequency-Dependent Damping
- The Stiff String

- The Externally Excited String

- Loop Filter Identification
- General Loop-Filter Design
- Damping Filter Design
- Dispersion Filter Design
- Fundamental Frequency Estimation
- EDR-Based Loop-Filter Design

- String Coupling Effects
- Horizontal and Vertical Transverse Waves
- Coupled Horizontal and Vertical Waves
- Asymmetry of Horizontal/Vertical Terminations
- Coupled Strings
- Longitudinal Waves

- Nonlinear Elements
- Memoryless Nonlinearities
- Clipping Nonlinearity
- Arctangent Nonlinearity
- Cubic Soft Clipper
- Series Expansions
- Arctangent Series Expansion
- Spectrum of a Memoryless Nonlinearities
- Square Law Series Expansion
- Power Law Spectrum
- Arctangent Spectrum
- Cubic Soft-Clipper Spectrum
- Stability of Nonlinear Feedback Loops
- Practical Advice

- Memoryless Nonlinearities

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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