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Your question has been answered by following standard mathematical procedures. Nonetheless, your concern about intuitive sense is quite fair. On one hand, the idea that energy transfer should be maximum at a frequency below the natural frequency $\omega_0$ is not wrong at all. The work done by the external force over one period of the stationary oscillation ...


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One way to explain it is to look at the string as a waveguide. You can send wave packets over the string, and they're reflected at the ends: Now, you can send in wave packets repeatedly rather than just once: This is essentially exciting the string with a (non-sinuoidal) periodic signal of pulses. But it is not resonance: the new pulses don't amplify the ...


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In addition to mike stone's answer. A damped driven oscillator can oscillate with any frequency. However the oscillation amplitude will be the largest only if the driving force frequency is close to the resonance frequency. The less damping, the more peaked the dependence is. Search information about driven oscillator, e.g. https://en.wikipedia.org/wiki/...


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Are you familiar with the steady-state solutions to the equation $$ \frac{d^2 x}{dt^2}+\Omega_0^2 x= F \sin (\Omega t)? $$ (If not, look for a solution $x(t) \propto \sin (\Omega t)$ and find the constant of proportionality) When you solve this you can immediately see that you can excite an oscillator with frequency $\Omega$ even when $\Omega$ is not ...


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The fact that sound does not propagate in vacuum was tested for the first time in the 17th century, when vacuum pumps became available. Athanius Kircher and Robert Boyle are some of the first experimenters. At that time it was an imprtant experiment, to decide if the sound need a medium or (according to alternative theory) is due to some special particles ...


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Your observations are mostly correct. "a defect can be introduced to create a defect/resonance mode and enable transmission" - that is, when the photonic crystal is of finite thickness. In an infinite PhC, the defect only introduces localised states centered at that defect. "if it is a half-wavelength layer it will enable a resonance ...


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You are mixing up two different frequencies. The frequency of the string you are referring to is the fundamental frequency of the string. The frequency of the source, which vibrates the string, is independent of the string itself. The way these two frequencies are associated with each other is in the amplitude of the strings vibration. If the source ...


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