If you discretize the string into a bunch of tiny masses, linked together by a bunch of tiny springs, you can build a mass Matrix (diagonal) M and a Stiffness m

If you discretize the string into a bunch of tiny masses, linked together by a bunch of tiny springs, you can build a mass Matrix (diagonal) M and a Stiffness matrix K (element ij = stiffness of spring that links mass I and mass j). I can't remember the next part exactly, you can look it up in a textbook, but you multiply the matrices KMK, or similar, and the eigenvalues of this are the natural frequencies of the string. The eigenvectors represent the mode shapes, ie the displacement of each mass element. The same technique is used in Finite Element Analysis to find the modes and modeshapes of complex structures (a car frame, a bridge, etc)