Bounds for the rth 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 rth 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 rth 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 rth eigenvalue of a Hermitian matrix.