Chapter 8.6, Problem 43P

Consider an equation for damped forced vibrations (mechanical or electrical) in which the right-hand side is a sum of several forces or emfs of different frequencies. For example, in (6.32) let the right-hand side be $F_1{e}^{i{\omega }_1^{\prime }t}+F_2{e}^{i{\omega }_2^{\prime }t}+F_3{e}^{i{\omega }_3^{\prime }t}$,

Write the solution by the principle of superposition. Suppose, for given ${\omega }_1^{\prime },{\omega }_2^{\prime },{\omega }_3^{\prime }$ that we adjust the system so that $\omega ={\omega }_1^{\prime }$; show that the principal term in the solution is then the first one. Thus the system acts as a "filter" to select vibrations of one frequency from a given set (for example, a radio tuned to one station selects principally the vibrations of the frequency of that station).