= Solution
The reflected and transmitted mean-power fractions are
$$
\mathcal R=\left(\frac{Z_1-Z_2}{Z_1+Z_2}\right)^2,\qquad
\mathcal T=\frac{Z_2}{Z_1}\left(\frac{2Z_1}{Z_1+Z_2}\right)^2
=\frac{4Z_1Z_2}{(Z_1+Z_2)^2},\quad \mathcal R+\mathcal T=1.
$$
For strictly positive finite <linear mass densities>, $\mathcal T>0$; a finite density step cannot produce exact total reflection. The ideal limits are
$$
\boxed{\rho_2/\rho_1\to\infty:\ r\to-1,\ \mathcal R\to1},\qquad
\boxed{\rho_2/\rho_1\to0:\ r\to+1,\ \mathcal R\to1}.
$$
The first behaves as a fixed end; the second as a free end. In the free-end limit the transmitted displacement tends to $2A$, but its <mechanical impedance> tends to zero, so it carries vanishing <power>. Thus total reflected \i[<energy>] does not require a vanishing boundary displacement. “Entirely reflected” at a merely large density contrast is an approximation.
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