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High-end violin bows have been made for centuries out of permanbouc (a now rare and endangered brasilian wood) or equivalent woods and now synthetic material. Permambouc wood is praised for is stiffness and therefore the high speed of sound/vibrations along it.

Why is it so important?

What does it bring to the sound and to the capabilities of control of the bow ?

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When you draw a bow across the strings, you are imparting energy into the combined physical system of the violin and the bow. What you want is for as much of that energy as possible to be transmitted to the strings, which transmit it to the bridge, which transmit it into the violin and then out to the air.

Like anything else, a bow isn't 100% efficient; inevitably, it will absorb some of the energy and reflect some of the energy. The stiffer the bow, the less energy it absorbs and the more it reflects back into the string. So a stiffer bow is preferable because more of your energy goes into producing sound.

This is the same principle as with graphite tennis rackets and baseball bats. The stiffer the material, the more the energy is transmitted where you want it.

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That is true up to a point. But it's not the whole story, because otherwise no one would use a wooden bow but rather one of graphite, or steel, or glass, all of which exist and all of which make a more brilliant, some would say harsher, sound than wood, because they have less hysteresis or internal friction and thus damp the higher frequencies less. So it's a matter of what kind of sound you want too. – Scott Wallace Mar 13 at 20:57

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