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Answer :
If the mass per unit length of a violin string is quadrupled, the speed of the wave on the string will become one half of its former value, based on the inverse square root relationship in the wave speed formula.
Hence, the correct answer to the student's question is (d) become one-half of its former value.
If the mass per unit length μ of a violin string is quadrupled, the speed v with which the wave travels on the string will neither become four times as large, nor twice as large, nor will it remain unchanged.
Instead, according to the formula for wave speed on a string, v = √(T/μ), where T is the tension in the string and μ is the mass per unit length, if the mass per unit length is quadrupled, the speed of the wave will become one half of its former value because the square root of four (which is the factor by which the mass has increased) is two, and therefore the wave speed is inversely proportional to the square root of the mass per unit length.
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