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In my book it's written that when the tuning fork A is is struck on a rubber pad then it starts vibrating with natural frequency (though not written but if it's not vibrating with natural frequency then it wouldn't be able to produce resonance to the other tuning fork B). But the vibrations are damped as they are in medium so how is natural frequency obtained through damped vibration because as much I know and read books,the frequency of damped vibration is less than natural frequency.

So my question was that whether the book has written that the tuning fork A has produced natural frequency just for making us understand the concept or whether they really produce natural frequency and if they produce natural frequency then how?

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The tuning fork keeps its frequency, when the amplitude changes. So fork B is in resonance. Maybee you consider only pure sin vibrations as natural frequency? But for example de time distances for going through 0 is the same all the time. So it is not a sin function, but a sin-function with an amplitude A(t) decreasing , while the frequency of the sin is constant.

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A tuning fork works because the arms are stiff and springy. When the arms bend inward a little, the springiness throws them outward. They go out so far that the springiness pulls them back inward. And repeat.

The fork is rigidly attached to the wooden box. The vibrating fork makes the box vibrate. This distorts the box a little, making the volume slightly bigger and smaller. Air is sucked in and pushed out, making the air vibrate. Vibrating air is sound.

This all works because energy is carried away very slowly. There is very little friction. It takes very little energy to puff air around. But it does take some, so the vibrations slowly die away.

You can hear these vibrations because your ears are fantastically sensitive. Years ago a very small ant got in my ear and walked across my eardrum. That was the loudest sound I have ever heard. Vibrations from normal sound are far weaker.

If you point the the sound box at a second tuning fork, air rushing in and out expands and contracts its box a little. The box vibrates, and this makes the tuning fork vibrate.

In the case of two identical tuning forks, the incoming air vibrations are at the same frequency as the fork naturally vibrates. That means air rushes in and bends the fork slightly one way. The air rushes out just as the fork bends the other way. Air rushes in just as the fork bends again. So air rushing in is timed just right to push it a little farther. This builds until the second fork is ringing loud enough to hear. This is called resonance.

If the frequencies were different, the vibrations might start to build. But pretty soon the fork would be moving one way when air pushes it the other. The vibrations would die back down.

Likewise, if the second tuning fork was mounted on rubber, the vibrating box would not be as effective at making the fork vibrate. And there would be more friction. If the fork did start vibrating, the vibrations would die away more quickly. So even if the vibrations were at the right frequency, they would die away as quickly as they built up.

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