Consider a uniform string in which small fractional changes occur in both the applied tension FT and the linear mass density μ. Let δF represent a small change in tension and δμ a small change in density, so that the fractional changes are δF/FT and δμ/μ respectively.
(a) Derive, to first order in small quantities, an expression for the fractional change in wave speed δv/v in terms of δF/FT and δμ/μ. Clearly state any approximations you make.
(b) Use your expression to estimate the percentage change in wave speed when the tension is increased by 5% while the linear density is decreased by 2%.
(c) Discuss whether the first-order approximation from part (a) would still be reliable if the tension were increased by 25% and the density decreased by 10%. Support your answer qualitatively.