Two wires of different densities but same area of cross section are soldered together at one end and are stretched to a tension T. The velocity of a transverse wave in the first wire is double of that in the second wire. Find the ratio of the density of the first wire to that of second wire.

Sol. Let v1 = velocity in the 1st string ⇒ v1 = √(T /m1) Because m1 = mass per unit length = (1a1l1 / l1) = 1a1 where a1 = Area of cross section ⇒ v1 = √(T / p1a1) …(1) Let v2 = velocity in the second string ⇒ v2 = √(T /m2 ) ⇒ v2 = √(T / p2a2 ) …(2) Given that, v1 = 2v2 ⇒ √(T / p1a1) = 2 √(T / p2a2 ) ⇒ (T/a1p1) = 4(T/a2p2) ⇒ p1/p2 = 1/4 ⇒ p1 : p2 = 1 : 4 (because a1 = a2)