Piano String Wave Equation

A wave equation suitable for modeling linearized piano strings is given by [77,45,320,521]

where the partial derivative notation and are defined on page , and

See, *e.g.*,
[145, p. 64] for a detailed derivation.

The first two terms on the right-hand side of Eq.(9.30) come from
the ideal string wave equation (see Eq.(C.1)), and they model
transverse acceleration and transverse restoring force due to tension,
respectively. The term
approximates the transverse
restoring force exerted by a stiff string when it is bent. In an
ideal string with zero diameter, this force is zero; in an *ideal
rod* (or *bar*), this term is dominant [320,263,170].
The final two terms provide *damping*. The damping associated
with
is frequency-independent, while the damping due
increases with frequency.

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