Relaxation Time in Damped Harmonic Oscillators

  • #1
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?
 

Answers and Replies

  • #2
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58
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.
 

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