Sturm-Liouville Eigenfunctions

What you are seeing: a clamped string whose mass density $\rho(x)$ you choose. Its normal modes are the eigenfunctions of the regular Sturm-Liouville problem $-(T y')' = \lambda\,\rho(x)\,y$ on $[0,\pi]$, $y(0)=y(\pi)=0$, solved numerically (finite differences, Jacobi). A uniform string recovers the textbook $\phi_n=\sqrt{2/\pi}\sin n x$, $\lambda_n=n^2$. Load it (heavy centre, a density step, a taper) and the mode shapes bend toward the heavy region and the spectrum shifts off $n^2$, yet the modes stay orthonormal under the weighted inner product $\langle f,g\rangle_\rho=\int\rho f g\,dx$. That invariance is the substance of Sturm-Liouville theory.

Top: the vibrating string, line weight tracking the local density, with the density ribbon drawn beneath. Middle: the eigenvalue ladder $\lambda_k$ against the open-tick $n^2$ reference. Bottom: every one of the $N$ modes you ask for (mode $k$ has exactly $k-1$ interior nodes, the Sturm oscillation theorem). Click the string to re-pluck it with a triangular tent.