Piano String Model, Continued

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

• $y(t,x)$ represents transverse displacement along the string at time $t$ seconds and position $m$ meters
• $L$ is the string length in meters
• The PDE takes into account
  • dispersion (fourth space derivative)
  • frequency-dependent loss from 3rd time derivative
  • hammer excitation denoted by $f(x,y,t)$
• A derivation of the wave equation for stiff strings is given in Morse 1948 and in many acoustics texts. (Cremer 1984 is especially good.)
  (See the Course Bibliography for full citations.)