A metal wire with density ρ and Young's modulus Y is stretched between rigid support At temperature T the speed of transverse wave is found to be `v_1`. When temperature is increased to `T+ΔT` the speed decreases to `v_2 lt v_1`. If the change in cross section area of the wire is ignored then the coefficient of linear expansion of the wire is

Explanation

If temperature decreases,s tension in wire increases and `v` increases
`v_(1) = sqrt((F)/(mu))`
`F = muv_(1^(2))`
`DeltaF = YAalphaDeltaT`
`v_(2) = sqrt((F + DeltaF)/(mu)) = sqrt((F + YAalphaDeltaT)/(mu))`
`v_(2^(2)) = v_(1^(2)) + (YAalphaDeltaT)/((m//l)) = v_(2^(2)) - v_(1^(2))`
`(YalphaDeltaT)/(rho) = v_(2^(2)) - v_(1^(2))`
`alpha = (rho(v_(2^(2)) - v_(1^(2))))/(YDeltaT)`