= Solution
If $\rho_1\ll\rho_2$, then $k_1\ll k_2$ and the wave meets a much larger string impedance. The <displacement> and <power> coefficients approach
$$
\boxed{r\to-1,\qquad t\to0^+,\qquad\mathcal R\to1,\qquad\mathcal T\to0}.
$$
The junction behaves approximately as a fixed end. The reflected <displacement> reverses sign, corresponding to a <phase> shift of $\pi$; the small transmitted <displacement> is not inverted. For a large but finite density ratio, $t\simeq2k_1/k_2$ and the transmitted <power> fraction is about $4k_1/k_2$, tending to zero rather than vanishing identically at every finite ratio.
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