Bounds for the r th characteristic frequency of a beaded string or of an electrical filter

The fundamental mode of vibration of a beaded string has a shape without change of sign. The r th higher normal mode of vibration has r changes of sign. Given any virtual shape of the string with r changes of sign, an algorithm is found that gives upper and lower bounds for the r th characteristic frequency as a function of the virtual shape. By making a certain transformation it is found that this algorithm holds for the characteristic frequencies of an inductor-capacitor network. Other transformations show that it applies to the r th eigenvalue of a Hermitian matrix.