Question

deduce the expression for the speed of a compressional wave through an extended solid

Answer 1

pulse (compression) travel in the fluid-filled tube. we observed that, after elementary time (Δt) net force =forward force - backward force

                                 F = pA -(p + Δp)A = -pA---------(1)

 now, we know,

             mass = volume × density

                m = V × ρ

 for elementary mass Δm = ΔV × ρ

                                        = A.Δx × ρ

                                        = Av.Δt × ρ

 here, the air occupies a volume V = AvΔt outside the pulse is compressed by an amount of ΔV = AΔv.Δt as it enters in the pulse.

   so, ΔV/V = Δv/v---------------(2)

 now, F = Δm.a = Av.Δt × ρ .Δv/Δt = Aρv.Δv--------(3)

 from (1) (2) and (3)

    pv² = -Δp/[ΔV/V] = bulk modulus = B

   v= √(B/ρ