= Solution
If $\rho_1\gg\rho_2$, then $k_1\gg k_2$ and the junction behaves approximately as a free end for the incident string. The limits are
$$
\boxed{r\to+1,\qquad t\to2,\qquad\mathcal R\to1,\qquad\mathcal T\to0}.
$$
The reflected <displacement> is not inverted, and at the junction it adds in <phase> to the incident <displacement>, producing nearly twice its <amplitude>. The transmitted <displacement> is also in <phase>. A transmission <amplitude> near two does not create <power>: the lighter string has much smaller impedance, and $\mathcal T=(k_2/k_1)t^2\simeq4k_2/k_1\to0$. These are limiting statements for positive densities, not an ordinary propagating-wave model for an exactly massless second string.
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