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I noticed this yesterday in the shower and my physics sense tingled. well im here now !

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The difference is really remarkable, but is similar to the difference of intensity in the sound if we touch the floor or the bucket with our finger.

The sudden impact produces a very small downward displacement in the point of contact, that spreads as a mechanical wave. The maths are shown here.

In the bucket, the energy is mainly dissipated in vibrations (waves). In the floor, it is also dissipated by the scattering of the water drops in several droplets, that bounces and wet the region around.

What can also be seen for a coin for example. It bounces more on the floor than on the bucket.

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Short answer: the acoustic impedance match between the bottom of the bucket and the surrounding air is better than the acoustic impedance match between the floor and air.

Sound waves in air are a very non-dense medium moving with large amplitude/distances. Sound waves in the floor are a small amplitude movement of a dense medium. For maximum energy transfer, the movements should be about the same amplitude; there is a mismatch between "floor" and "air", so very little energy is transferred between them.

I like the impedance match answer because it is useful in other realworld cases. For example, why can you throw a baseball so much further than either a shotput or a pingpong ball?

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Bucket acts as some sort of a drum, where the membrane is resonating up and down. But the floor only squeezes up and down. The bucket has more room for resonating and resonates more than the floor.

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