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I've been reading many questions about why tuning fork has two prongs. Many of them explain that each prong oscillates with opposite phases. As a consequence, the compression and decompression of air gets stronger and the oscillation on the stem is reduced, which results in a "purer" sound and less damping by the stem (this was what I understood).

However, I couldn't figure out how this opposite phases are created. When I hit one of the prongs, it will start oscillating with a phase. Then this oscillation propagates to the other prong because it's in touch (connected) to the first. But why it will have a different phase? Also, I've been wondering that, if this phase difference reduces the oscillation on the stem, how the resonance is going to happen strongly when I touch the stem in a resonance box?

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The centre of mass of the tuning fork does not move if the two prongs are moving $\pi\$ out of phase. The forks moving in phase would result in the centre of mass of the fork trying to move which would result in the damping of those oscillations.

Those two types of oscillation can be thought of as the normal modes for the tuning fork system but only one of those modes lasts for any period of time.

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