Waves Question! My brain hurts, need help!?

Question

Three pieces of string, each of length L, are joined together end-to-end, to make a combined string of length 3L. The first piece of string has mass per unit length u1(mu subscript 1), the second piece has mass per unit length u2 = nu1, and the third piece has mass per unit length u3 = u1/n.

If the combined string is under tension F, how much time does it take a transverse wave to travel the entire length 3L? Give your answer in terms of L, F, and u1.

Conceptual questions are my weakness! Can someone help me out with this one?

Answer 1

First, you have to know the equation for the speed of the transverse wave on a string:
c = sqrt(F/mu).
Since every piece of string is under the same tension, we write three speeds:
c1 = sqrt(F/u1), c2 = sqrt(F/nu1) and c3 = sqrt(F/ (u1/n)). We use math to write c2 and c3 in terms of c1:
c2 = sqrt(1/n) x c1 and c3 = sqrt(n) x c1.
Then we have time = distance / speed for every piece of the string, so:
t1 = L / c1 = L / sqrt(F/u1)
t2 = L / c2 = L / ( sqrt(1/n) x c1) = L x sqrt(n) / c1
t3 = L / c3 = L / ( sqrt(n) x c1)
The total time equals t = t1 + t2 + t3. A least common denominator for all three fractions is sqrt(n) x c1 and we have:
t = (L x sqrt(n) + L x sqrt(n) x sqrt(n) + L) / sqrt(n) x c1 =
= (L x sqrt(n) + n x L + L) / sqrt(n) x c1 =
= L x (sqrt(n) + n + 1) / c1 = L x (sqrt(n) + n) / sqrt(F/u1)