Semi-infinite string with a mass on the free end

1. The problem statement, all variables and given/known data

Hi! This is probably going to be a silly question, but I think I just need someone to point out my obvious mistake so I can go back and solve it properly.

A semi-infinite string of density ρ and tension T has a mass affixed to the free end which is constrained to move transversely. Determine the amplitude and phase of reflected waves when transverse waves of the form A exp [i(wt-kx)] propogate through the string.

2. Relevant equations

Yxx = Ytt/c2

3. The attempt at a solution

Okay, so I attempted to set up some boundary conditions, but I’m really not sure if they’re correct. I said that the waves on the string will obey:

y = A exp [i(wt-kx)] + Ar exp [i(wt+kx)]

i.e. some initial train and then a reflected train.

Then I decided that the boundary conditions could be written as

y(0,t) = 0 (the end at x = 0 is fixed)

yx(∞,t) = (m/T)(ytt)

i.e. applying Newton’s Second Law to the mass that is constrained to oscillate in the transverse direction.

Now I can see that the problem might be easy to solve if the second boundary condition can be applied alone (it gives us a direct relation between A and Ar, which is what we want) but the first boundary condition is problematic. It implies that A + Ar = 0, which gives us that the amplitudes are equal and the waves are in antiphase… but that will contradict what I get from the second boundary condition (and seems wrong because it doesn’t depend on the mass). How do we take into account the fact that one of the ends is fixed? And, if this is not the case, what do they mean by "the free end"? Any help would be greatly appreciated

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