Piano String Modeling

Physical picture:

Chaigne and Askenfelt (JASA, Feb. 1995) suggested the following partial differential equation (PDE) to model the piano string:

$$\begin{eqnarray*} \frac{\partial^{2}y}{\partial t^{2}} = c^{2}\frac{\partial^{2}y}{\partial x^{2}}-\epsilon c^{2}L^{2}\frac{\partial^{4}y}{\partial x^{4}}-2b_{1}\frac{\partial y}{\partial t}+2b_{3}\frac{\partial^{3}y}{\partial t^{3}}+f(x,y,t) \end{eqnarray*}$$

In addition, we must specify

• boundary conditions
• initial conditions (usually zero position and velocity).