 Displacement coefficients at a string junction

For transverse waves on two strings at common tension, displacement continuity gives $1+r=t$, while transverse-force continuity gives $(1-r)/c_1=t/c_2$. Solving gives the displayed displacement-amplitude coefficients. A denser transmitted string has lower speed and a negative reflected coefficient. The energy fractions are $r^2$ and $(c_1/c_2)t^2$, summing to1. These are displacement coefficients; force or power-normalized wave amplitudes use different conventions.