Transverse Wave in a String Reflection and Transmission

artongue

A string with a density/length of rho and tension T has an elastic spring with stiffness k at x=0. There is a step pulse (fi = H(t)) incident on this discontinuity. Determine the reflected and transmitted waves.

Im not sure what boundary condition to use at the location of the spring. All I have now is:

BCs: (y(x,t) is deflection, fi(x-ct) is incident wave, gr(x+ct) is reflected, ft(x-ct) is transmitted)

y(0-,t) = y(0+,t) ----> fi(-ct) + gr(ct) = ft(-ct)

Not sure what other BC to use. Maybe Ty' = ky at x = 0? Where do I go from there?

artongue

bump :(