Undergrad Relationship between standing wave and resonance?

[TT0]

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In the textbook, it said that standing waves only occur at the resonant frequencies of the medium. I don't understand why.

My understanding of standing waves is that it is when the medium length (slinky) is a multiple of 1/2 of the wavelength. This allows the reflected wave to interfere with the original wave creating the standing waves. What does this have to do with resonant frequency?

Thanks

 

[Orodruin]

What does this have to do with resonant frequency?

Everything. The frequencies corresponding to wavelengths that fit an appropriate number of periods into the medium (note that this depends on the boundary conditions) are the resonant frequencies.

 

[CWatters]

+1

Resonance occurs when the input is at a frequency (or wavelength) that is close to a natural frequency (or wavelength) of the system. The natural frequencies (or wavelengths) of something like an air column depend on the length of the column. They correspond to column lengths that are a multiple of half of the wavelength. Under those conditions you get constructive interference between the waves traveling in each direction in the column.