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What would be the equation for the frequency dependent damping of harmonic oscillator? Is there something like:

$$ \ddot{x}+2\delta\dot{x}+\omega_0^2x = \frac{F}{m}f(t) $$

with frequency dependent damping? Of course, it could be done by stating $\delta=\delta(\omega)$, but

  • is there an example from mathematical physics where something like this arises from equation of motion?
  • what is the physics behind?

Note 1: This question is motivated by sound synthesis task, but I am looking for physical backround. Therefore it's not engineering nor computational science.

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Functionally the solution does not change. You can still use phasors to write the equation asenter image description here

goes to

enter image description here

Where X is the fourier image of x and F is the fourier image of the driving force f(t). because enter image description here does not depend on time, its fourier image does not change. The solution is enter image description here

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