Textbook 'The Physics of Waves': Lifetime of SHM Oscillators

[brettng]

Homework Statement

The lifetime of the state in free oscillation is of order $\frac 1 { \Gamma }$.

Relevant Equations

$$\tau=\frac { \ln(4) } { \Gamma }$$

Reference textbook “The Physics of Waves” in MIT website:
https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/

Chapter 2 - Section 2.3.2 [Page 47] (see attached file)


Question: In the content, it states that the lifetime of the state in free oscillation is of order $\frac 1 { \Gamma }$. To my understanding, it means particularly the half-life of underdamped oscillators with position $x\left( t \right)$:
$$x\left( t \right)=Ae^{ -\frac {\Gamma t} {2} }\cos(\omega t-\theta)$$

However, could I consider the "half-life" of overdamped oscillators and critically damped oscillators?

If so, would it still be of order $\frac 1 { \Gamma }$?

 

[DrClaude]

In the content, it states that the lifetime of the state in free oscillation is of order $\frac 1 { \Gamma }$.

Do you understand what is meant here by "lifetime"? Also, what do the equations for $x(t)$ look like over damped and critically damped oscillators?

 

[brettng]

Do you understand what is meant here by "lifetime"? Also, what do the equations for $x(t)$ look like over damped and critically damped oscillators?

I understand that lifetime means that the oscillation amplitude reduces by a factor of 2.

So, why can’t we still consider the “lifetime” of overdamped and critically damped oscillators by using the same definition?

 

[DaveE]

In practice, people only refer to "oscillations" wrt underdamped systems. Otherwise we tend to use "rise time"; usually defined as 10-90%, t_r = 2.2τ. But it is the equation that holds the real answers, the rest is just jargon.

 

[DrClaude]

I understand that lifetime means that the oscillation amplitude reduces by a factor of 2.

That would be the half-life. Lifetime itself is actually more vague. For instance, in this case one could take $\tau = 1/\Gamma$, which would be the time for the oscillation amplitude to be reduced by a factor $1/\sqrt{e}$, or $\tau = 2/\Gamma$, for a reduction of the amplitude by $1/e$. They are all of order $1/\Gamma$, as the text states, and as @DaveE stated, they all relate to a decrease of the amplitude of oscillations, and different terminology is used for critically and over-damped oscillators, which do not actually oscillate.

 

[brettng]

Thank you so much for your help!!!!!!