If the tension doubles on a string, the wave speed v on that string will increase by what factor (approximately)?

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Multiple Choice

If the tension doubles on a string, the wave speed v on that string will increase by what factor (approximately)?

Explanation:
The main idea is that waves on a string move at a speed that depends on tension and how much mass is in the string per length. The relationship is v = sqrt(T/μ), where μ is the linear mass density. If the tension doubles while μ stays the same, the new speed is v' = sqrt(2T/μ) = sqrt(2) sqrt(T/μ) = sqrt(2) v. So the wave speed increases by a factor of about sqrt(2). This reflects the string getting stiffer, so disturbances propagate faster, but the increase is only with the square root of the tension. (Assumes μ remains unchanged.)

The main idea is that waves on a string move at a speed that depends on tension and how much mass is in the string per length. The relationship is v = sqrt(T/μ), where μ is the linear mass density. If the tension doubles while μ stays the same, the new speed is v' = sqrt(2T/μ) = sqrt(2) sqrt(T/μ) = sqrt(2) v. So the wave speed increases by a factor of about sqrt(2). This reflects the string getting stiffer, so disturbances propagate faster, but the increase is only with the square root of the tension. (Assumes μ remains unchanged.)

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