Question

What happens to the speed of wave in the stretched string if its radius is doubled without changing the tension?
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Answer 1

Answer:

$the wave in a string is a transverse wave which depends on the tension and the mass per unit length. If you double the tension the mass per unit length goes down by a small amount (the string gets a bit "thinner" because it gets longer) . Both these things increase the velocity of the transverse wave which is given by</p><p></p><p></p><p>v=√Tρ</p><p></p><p></p><p> Where T is the tension and ρ the mass per unit length. Finally, the fundamental frequency is determined as the reciprocal of the round trip time of the wave along the string:</p><p></p><p></p><p>T=2ℓv</p><p></p><p></p><p>so that</p><p></p><p></p><p>f=1T=v2ℓ=√Tρ2ℓ</p><p></p><p></p><p>So to raise the frequency by an octave you need four times the tension (don't try this - you might break the neck) or a string that's half the diameter (one quarter of the area - so one quarter of the mass per unit length). This explains why different strings on the guitar have different gage - you would need too much tension to get the high range from a long thick string$

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