I am trying to solve for unknowns $R,T$ when I have the following wave ansatz:
$u(x,t)=\Re((e^{iwx}+Re^{-iwx})e^{-iwt}$, for $x<0$, and $u(x,t)=\Re(Te^{iwx}e^{-iwt})$, for $x>0$.
If I impose continuity and energy conservation at $x=0$ I end up with the $2$ conditions $T=1+R$ and $T^2+R^2=1$. These are solved by both $(T,R)=(1,0)$ and $(T,R)=(0,-1)$. I have done similar problems like this but only ending up with one solution for $R$ and $T$ so here is where I am a little confused.
Now I am also told that $u(x,t)$ describes small oscillations of an infinite one-dimensional string, with a mass $m$ attached to the string at $x=0$. Does this further restrict what values $R,T$ can take? It feels like I should choose $(T,R)=(1,0)$ since the other solution $(T,R)=(0,-1)$ means I have no oscillations (i.e. $u(x,t)=0$) for $x>0$, but this is just gut feeling and I don't know how I would justify this.