The physical modeling of strings and acoustic tubes using digital waveguides has been well studied elsewhere. The wave equation of a vibrating string is reviewed and spectral analysis methods are introduced to show equivalence between the 1-D digital waveguide and the standard difference approximation to the wave equation. Spectral methods are further used to analyze stiffness and loss terms. A method for modeling and calibrating stiffness is presented. A method for modeling and calibrating coupled strings, such as in the piano, is presented. A novel method for modeling membranes, plates, and acoustical space efficiently using a mesh of digital waveguides (first described in 1993) is derived and analyzed through spectral means. Various topologies for the digital waveguide mesh are analyzed. Methods for introducing nonlinear coupling of the modes, as in gongs, and nonlinear excitations, as in the piano hammer or felt mallet, are also derived using traveling-wave decomposition of lumped impedances. A simplified piano model including soundboard, nonlinear hammer, and coupled stiff strings is presented.
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