Wave Equation

A wave equation suitable for modeling piano strings is given by [80,46,322,521]

$$f(t,x) = \epsilon{\ddot y}- K y''+ EIy''''+ R_0{\dot y}+ R_2 {\ddot y'} \protect$$ (6.1)

where the partial derivative notation $y'$ and ${\dot y}$ are defined on page , and

$$\begin{eqnarray*} f(t,x) &=& \mbox{driving force density (N/m) at position $x$\ ... ... I &=& \mbox{radius of gyration of the string cross-section (m)} \end{eqnarray*}$$

Young's modulus and the radius of gyration are defined in Appendix E.

The first two terms on the right-hand side come from the ideal string wave equation (see Eq.$\,$(G.1)), and they model transverse acceleration and transverse restoring force due to tension, respectively. The term $EIy''''$ 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. The final two terms provide damping. The damping associated with $R_0$ is frequency-independent, while the damping due to the $R_2$ term increases with frequency.

In [47], the damping in real piano strings was modeled using a length 17 FIR filter for the lowest strings, and a length 9 FIR filter for the remaining strings.