Piano String Wave Equation

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

$$f(t,x) = \epsilon{\ddot y}- K y''+ EIy''''+ R_0{\dot y}+ R_2 {\ddot y'} \protect$$ (10.30)

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

$$\begin{eqnarray*} f(t,x) &=& \mbox{driving force density (N/m) at position $x$\ and time $t$}\\ \epsilon &=& \mbox{mass density (kg/m)}\\ K &=& \mbox{tension force along the string axis (N)}\\ E &=& \mbox{Young's modulus (N/m$\null^2$)}\\ I &=& \mbox{second moment of area (m$\null^4$)} \end{eqnarray*}$$

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 $EIy''''$ 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; in an ideal rod (or bar), this term is dominant [320,263,170]. The final two terms provide damping. The damping associated with $R_0$ is frequency-independent, while the damping due $R_2$ increases with frequency.