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In the last paragraph of this article, the author gives an example of the DFT of a song and his window's resonant frequency. As he points out (and it is a bit intuitive), if he listens to this song, the window is not likely to break, despite the song's spectrum being centered around the natural frequency of the window.

The question is, if we are given a range of resonant frequencies (the "dangerous" range around the resonant frequency), can we decide whether "the window would break", so to speak, based on the DFT alone, without a physics model (and therefore regardless of the problem, i.e. a window, a vibrating wire, etc.)?

Also, if we have two different signals and we compute their DFTs, how does the real-world physical effect of those signals (say, on a window) compare to their difference in amplitude for the resonant frequency? That is, if the amplitudes are equal, are the effects on the window's vibrations also equal? If one amplitude is half the other one, are the vibrations proportional the amplitudes' ratio (1/2)?

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  • $\begingroup$ Whether "the window will break" is entirely up to the geometry of the exciter (the speaker) and the excited structure (the window pane).Whether you are at the resonant frequency or not,modulates the transfer of power from the exciter to the structure. Therefore, if you are not AT the resonant frequency, it would appear that you need more power to break the window. Whether you get close to the window pane's limit load has nothing to do with the DFT. I am afraid that the way the question is phrased is off-topic for this board (maybe physics?) $\endgroup$
    – A_A
    Nov 3 '20 at 12:14
  • $\begingroup$ How can we know "the resonant frequency" without a physics model of the glass or window? For that matter, what is the DFT of? It's unclear whether you're referring to a DFT of the music being played, or some sampled room response, or what? $\endgroup$
    – TimWescott
    Nov 3 '20 at 15:47
  • $\begingroup$ @A_A: I agree that this is more a physics question. For starters, how resonant a bit of glass is matters, and for a window, how the thing is mounted affects resonance. A window that's mounted to have a lot of damping would take weaponized sound levels -- just the air surrounding a large window, or a wooden frame of an old-style multi-pane window, would kill resonance enough such that you'd need more of a shock wave than a pure tone. $\endgroup$
    – TimWescott
    Nov 3 '20 at 15:53
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The question is, if we are given a range of resonant frequencies (the "dangerous" range around the resonant frequency), can we decide whether "the window would break", so to speak, based on the DFT alone, without a physics model (and therefore regardless of the problem, i.e. a window, a vibrating wire, etc.)?

No. There is a whole lot of physics between the spectrum of a sound track and what actual force is excerted on a window. Speaker, amplifier, wave propagation in a room, pressure distribution across the pane (both directioin and magnitude), etc.

The whole example is kind of silly. The energy required to break a window is many orders of magnitude larger than what a residential sound system can produce. Your ear drums will break long before the window shows any type of distress, resonance or not.

That is, if the amplitudes are equal, are the effects on the window's vibrations also equal?

Sometimes yes, sometimes no. In order for this to hold we have to require that

  1. All other things are being equal
  2. The physical systems between signal and effect is reasonably linear.
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