I can't seem to figure out how to derive this relation, so a first step or any suggestions would be greatly appreciated. Thank you in advance. 1. The problem statement, all variables and given/known data After four cycles the amplitude of a damped harmonic oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of this oscillator to that of its natural frequency (undamped value). 2. Relevant equations I started off with two equations of motion: 1. x=C*cos(ωt) for simple harmonic oscillator undamped 2. x=C*e^(-Kt)*cos(ωdt) for under-damped since this seems to be the only case where four cycles would occur, although it is not specified. Also we have the equations for frequency for both cases: 1. ω=2∏/T for simple harmonic oscillator 2. ωd = (ω2-K2)^(1/2) for under-damped K=β/2m where β=damping coefficient 3. The attempt at a solution I'm having trouble isolating the frequency from both equations, but I'm not sure there is any need to use the equations of motion for a simple ratio of the frequencies. I know that four cycles indicates 4T = 4*(2∏/ωd) for the under-damped.. but not sure where to go from there.