[UPSC CSE 2006] Waves (Q53, 15M)

In the steady state forced vibration a point particle of mass 'm' moves under the influence of an external force (F \sin pt) \hat{i} in addition to the restoring force - (kx) \hat{i} and damping force - (\beta x) \hat{i} . Show that (i) the amplitude is maximum when p = \sqrt{\omega^2 - 2b^2} , where k/m = \omega^2 and (ii) the value of the maximum amplitude is \frac{f}{2b \sqrt{\omega^2 - 2b^2}} . What do you mean by the sharpness of resonance ?