I have a doubt regarding the angular frequency of a harmonic oscillator when there is damping involved. The frequency of the oscillation changes with time in the case of damping, but I haven't seen mention of this anywhere. I would like to find how the angular frequency depends on time (I'm guessing there must be some function $\omega=\omega(\omega_0,t,\beta)$ or something like that, where $\beta$ refers to the damping coefficient and the $\omega$'s refer to frequencies).
I checked with Landau and Taylor; neither of them, as far as I can see, discuss this phenomenon (although of course they talk about the decrease in amplitude and all that).
I'm pretty sure this phenomenon of frequency decreasing with time does occur (I checked quickly with a mish-mash harmonic oscillator), so why doesn't anyone mention it?
Could someone explain to me the time dependence of frequency when there are damped oscillations? Or maybe point out resources I could check out that do talk about this?
Best Answer
An (undriven) damped harmonic oscillator (satisfying $m\ddot{x} + \gamma \dot{x} + \omega_0^2 x=0$) can be solved by the solution(s) $x_0e^{i \omega t}$. For an underdamped oscillator these solutions represent pure oscillations mixed with exponential decay(/growth). Because both solutions for $\omega$ oscillate with the same period, all combinations of them also oscillate with the same period.
I suppose that your confusion arises from an intuitive idea of why a restoring force leads to periodic motion. In the undamped case the phase space trajectories of the particle are closed (i.e. the particle always returns to the same position(s)). In the damped case the trajectories spiral towards rest. But both of these motions are periodic in the sense that they reach their relative extrema (and zero crossing) after a specific interval of time. How can you see this? Perhaps you can take some inspiration from the undamped case: notice that the period is independent of the size of the orbit. Because the equation is linear any change in amplitude is exactly accounted for by a change in the force. In fact all linear (homogenous) ODE's can be satisfied by the anstaz above (i.e. solutions are periodic, damped, or both).