Find the ratio of the frequency of the damped oscillator

[kreil]

Homework Statement

Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.

Homework Equations

T=\frac{ 2\pi}{w_1}

w_1^2=w_0^2- {\Beta}^2

The Attempt at a Solution

I know the answer is \frac{w_1}{w_0}=\frac{8 {\pi}}{\sqrt{64 {\pi}^2+1}}

However, I think I am missing an equation here because using only the two above i cannot obtain this answer. If someone could point me in the right direction I would appreciate it!

Josh

 

[cristo]

Well, firstly you should explain your symbols. I presume you are using w to mean omega; the frequency of the oscillator. What is the definition of the natural frequency?