Relaxation Time in Damped Harmonic Oscillators

[blackwater]

Relaxation time is defined as the time taken for mechanical energy to decay to 1/e of its original value.

Why do we take a specific ratio of 1/e? What is its significance?

 

[Einj]

Because it is the natural time scale of the problem. When you have a damped harmonic oscillator then its amplitude decreases with time like $e^{-t/\tau}$. In this case $\tau$ is the time scale that determines the behavior of the system: if $t\ll\tau$ the system is still oscillating as an ordinary harmonic oscillator, while for $t\gg\tau$ it is already completely damped. This is why $t=\tau$ (the relaxation time) determines some sort of "benchmark" in the state of the system.