Hearing the string, not the shape

Last week I went to a very interesting summer school about algebraic and topological methods in quantum mechanics. It was a very eclectic crowd between mathematicians and physicists and undergrads, grads, postdocs and professors.

There were lectures, talks and something really interesting called short communications. The idea of these was to encourage participants to present some interesting facts that came from the main lectures. In one of these communications, a group presented the famous topic about hearing the shape of a drum, and their presentation made me think about the mathematical formulation of what does it mean to hear a a sound, and moreover why do we hear certain type of sound and not another.

Suppose we are listening to a string sound, like a guitar. The most basic model of this is the wave equation

$\partial_{tt}\psi=\Delta \psi$

subject to the initial conditions $\psi(0,x)=f(x)$ and $\partial_t \psi(0,x)=g(x)$.

Solving this equation models the behavior of the vibrating string. This PDE can be solved using separation of variables and fourier analysis for the initial conditions, and by these means, the associated wave frequency, can be thought as the separation constant, i.e. when supposing a solution of the form $\psi(t,x)=T(t)X(x)$, the above equation takes the form

$\frac{T''(t)}{T(t)}=\frac{X''(x)}{X(x)}$

Since the RHS is a function of $t$ and the LHS is a function of $x$, the only possible case is that they are equal to a constant (the separation constant) $\lambda$. When applying the initial conditions, the equation in $x$ is easier to solve,as it takes the form of an eigenvalue problem

$X''(x)=\lambda X(x)$

with $X(0)=0$. Notice that this equation does not have a time dependence anymore, and the $\lambda$ parameter is what at the end determines the frequency (frequencies) at which the string resonates. Moreover, the initial position $f(x)$ and initial velocity $g(x)$ do not play a role with the solution of this part of the wave equation, and hence, do not affect with the value of the frequencies $\lambda$. This is the main reason why it really doesn't matter how hard or where to pinch a guitar string, it will always sound the same, maybe a little louder or softer, but the same type of sound. An E string will always sound E, no matter where or how you pinch it.

This means that the sound is an intrinsic characteristic of a material, is not really dependent on the force applied to it but to its shape and physical characteristics.