Physics: mechanical wave

Let m be the mass of the spring.

By Hooke's law,

F = kx, where x is the extension in the spring.

When the length is 1.5L, x = 1.5L - L = 0.5L

Tension, T = 0.5kL

Mass per unit length, u = m/1.5L

So, v = sqrt(T/u)

v = sqrt(0.75kL2/m) ... (1)

When the spring is stretched to 2L,

Tension, T' = k(2L - L) = kL

mass per unit length, u' = m/2L

Wave speed, v' = sqrt(T'/u')

v' = sqrt(2kL2/m) = v sqrt(8/3)