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 oftransverse wave in a string.
`v = (dx)/(dt) = sqrt(T/m) = sqrt(T/(m_0 + alphax))`
`:. sqrt(m_0 + alphax) dx = sqrtT dt`
Integrating within proper limits,
`int_0^L sqrt(m_0 + alpha x) dx = sqrtT int_0^t dt`
`rArr [(2(m_0 + alphax)^(3//2))/(3alpha)]_0^L = sqrtTt`
`:. t = 2/(3 alpha sqrtT) [(m_0 + alphaL)^(3//2) -m_0^(3//2)]`