A wave equation suitable for modeling linearized piano strings is given by [77,45,320,519]
Young's modulus and the radius of gyration are defined in Appendix B.
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.