The Secret of the Piano 1

We continue our study of the interaction of string and soundboard through a bridge of a piano with focus on the initialisation where the string is hit by the hammer and through the bridge transfers energy to the soundboard.

We model the string-soundboard-bridge system by the following coupled wave equations: Find functions $u(x,t)$ and $U(x,t)$ representing displacements of string and soundboard from initial flat configuration, such that for $0\lt x\lt 1$ and $t\gt 0$

• $\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0$, 
• $\frac{\partial^2U}{\partial t^2}-\frac{\partial^2U}{\partial x^2}=0$,

combined with the following boundary conditions for $t>0$

• $u(0,t)=U(0,t)=0$,
• $\frac{\partial u}{\partial x}(1,t)=-\frac{\partial U}{\partial x}(1,t)=S(u(1,t)-U(1,t))$,

where $S$ is a spring constant representing a springy connection of string and soundboard through a bridge located at $x=1$.

We initialise by setting $u(x,0)=U(x,0)=0$, $\frac{\partial u}{\partial t}(x,0)=1$ for $0.4\lt x\lt 0.6$ and $\frac{\partial u}{\partial t}(x,0)=0$ else, and $\frac{\partial U}{\partial t}(x,0)=0$ for $0\lt x\lt 1$, corresponding to hitting the string with the hammer, and watch the result here with the string red and soundboard blue.

We compare with initial data for $u$ changed to $u(x,0)=\sin(\pi x)$ and $\frac{\partial u}{\partial t}(x,0)=0$ with somewhat different response here.

We see that the motion settles into periodic modes of string and soundboard with a phase difference of half a period with the bridge basically at rest: when the string deflects upward the soundboard deflects downward and vice versa with zero net force on the bridge. 

We have thus recovered the "breathing" motion with the bridge at rest of the previous post as the basic resonance mechanism of a piano allowing long aftersound with slow transfer of energy from string to soundboard.