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I'm a high school physics student studying about resonance.

Resonance happens as a result of a forced oscillation. If there is reflection within the system when resonance happens, we end up with a standing wave. If the object forcing another object into oscillation isn't doing so at the forced-upon object's natural frequency, you'll get a weird periodic wave pattern that is overlapping. (I think it's called beats.)

Now, let's do a thought experiment where we force an object to oscillate but not at its natural frequency. At what frequency will the forced upon object oscillate? In this case, I am not concerned about reflection within the system whatsoever. I'm purely concerned about the frequency at which the forced upon object is gonna oscillate. I was always taught that the frequency of oscillation depends upon the source, but by this logic we should just get a smooth clean standing wave and we wouldn't need both the forcing and the forced-upon object to be at the same natural frequency.

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  • $\begingroup$ If you force something to oscillate at a frequency then it oscillates at that frequency. This is true by definition - if it isn't oscillating at that frequency then obviously you aren't forcing it to. $\endgroup$ Commented Aug 30, 2017 at 4:46

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It will be the combination of two oscillations with different phases and amplitudes, one at the natural frequency and one at the forced frequency. The shape depends of the initial conditions http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html#c2. (forget the exponential because in your example there is no damping) It can look very different depending on those conditions. For instance:

https://www.wolframalpha.com/input/?i=plot+sin(.5t%2B1)%2B.9cos(t%2B.5)

Or this

https://www.wolframalpha.com/input/?i=plot+sin(.9t%2B1)%2B.9cos(t)

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  • $\begingroup$ In real life, there is always damping, the response at the resonant frequency dies out, and the system oscillates at the forcing frequency. Google for "forced vibration". $\endgroup$
    – alephzero
    Commented Apr 16, 2017 at 20:35
  • $\begingroup$ @alephzero yes, and in real like there are always gravitational waves, but the solar system will not collapse because of that during the galaxy's lifetime. You can google "idealization" $\endgroup$
    – user126422
    Commented Apr 16, 2017 at 20:44

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