A wave pulse starts propagating in +x-direction along a non-uniform wire of length L witdt mass per unit length given `mu=m_(0)+alpha x` and under tension of TN. Find the time taken by the pulse to travel from the lighter end (x = 0) to the heavier end.

Explanation

Velocity of transverse wave in a string.
`v=(dx)/(dt)=sqrt((T)/(mu))= sqrt((T)/(m_(0)+alphax)) " " :. sqrt(m_(0)+alpha x)dx= alphaTdt`
Integrating within proper limits
`int_(0)^(L) sqrt(m_(0)+alphadx)= sqrt(T) int_(0)^(1)dt rArr [(2(m_(0)+alphax)^(3//2))/(3alpha)]_(0)^(L)= sqrt(T)t`
`:.t=(2)/(3alpha sqrt(T))[(m_(0)+alphaL)^(3//2)-m_(0)^(3//2)]`