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Perhaps a better way to put this question is how is energy concentrated within a resonant system? As others have already stated energy is always conserved if one considers the universe in tallying where energy comes and goes. But if you are considering a system, defined within spatial boundaries, the system can lose or gain energy through its boundaries. In ...

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Just think about how you might push a child's swing. You apply a push once every oscillation of the swing and thus build up the amplitude of the swing. This is a resonance condition whereas if you pushes the swing at a slightly lower frequency you would not be able to increase the amplitude of the swing as much. Once the swing is at a constant amplitude ...

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The wikipedia article specifically mentions water waves, not sound waves. The speed of a wave in water is approximately $v = \sqrt{\frac{g\lambda}{2\pi} \tanh \left(2\pi\frac d \lambda\right)}$, where $g$ is the acceleration due to gravity, $d$ is the depth of the water, and $\lambda$ is the wavelength of the wave. For shallow water ($d \ll \lambda$) this ...

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All materials have a resonant frequency Well, sort of. In general, complicated structures will have many resonant frequencies where the amplitudes of any oscillations will have local maxima. However, one of the jobs of structural engineers, and I would assume this would apply to aeroplanes too, is to find these frequencies and make sure that either (a) ...

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You say: All materials have a resonant frequency but this is at best an oversimplification. Any system has a set of normal modes and if you apply driving force at a frequency that matches a normal mode then you will get a resonance. However for any system significantly more complicated than a tuning fork there are many normal modes and non-linearities ...

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I do not know that it has been done before, but I have no doubt there is a difference. What is not clear is if it would be noticeable by human ear. The difference is explained theoretically by the fact that the string will vibrate different with the supporting body. Only in the hypothetical scenario where the string is held by ideally unmovable holders ...

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The math of the underdamped oscillator $$\tag{1}\ddot{x}+2b\dot{x}+\omega_0^2 x~=~f\qquad\Leftrightarrow\qquad -\underbrace{(\omega^2+2ib\omega -\omega_0^2)}_{P(\omega)=(\omega-\omega_+)(\omega-\omega_-)}\tilde{x}~=~\tilde{f}$$ is easy to work out. The characteristic frequencies \tag{2}\omega_{\pm}~=~\underbrace{-ib}_{\text{exp. ...

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You already have a good mathematical answer, so I will focus on an answer with almost no equations. I take it you understand the basic mathematics of the simple harmonic oscillator. When you add damping, the amount of energy you lose per cycle depends on the velocity: the faster you go, the more energy you lose (at the same amplitude) because the force ...

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Mathematical demonstration It's straightforward to see why this happens if you use a bit of linear response theory. Consider a generic damped harmonic oscillator. There are three forces, the restoring force $F_\text{restoring} = - k x(t)$, the friction force $F_\text{friction} = - \mu \dot{x}(t)$, and the driving force $F_\text{drive}(t)$. Newton's law says ...

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