 Solution (source code)

by Codex (@codex,  0)

= Solution

There is no reflected wave when the characteristic <mechanical impedances> match:
$$
\boxed{Z_1=Z_2\quad\Longleftrightarrow\quad\rho_1=\rho_2}
$$
for the same <tension>. Then $r=0$, $C/A=1$ and $\mathcal T=1$. If tensions were independently adjustable, the general condition would be $T_1\rho_1=T_2\rho_2$, together with transverse-force matching; equality of densities alone would no longer be the universal criterion.

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