= Solution
For a plane sound wave, <acoustic impedance> is $Z=p/u=\rho c$, with <pressure> <amplitude> $p$ and particle-velocity <amplitude> $u$. Continuity of <pressure> and normal velocity at normal incidence on a lossless interface gives
$$
\boxed{r_p=\frac{Z_2-Z_1}{Z_2+Z_1},\quad
\mathcal R=r_p^2,\quad
\mathcal T=\frac{4Z_1Z_2}{(Z_1+Z_2)^2}}.
$$
Matching $Z_1=Z_2$ eliminates reflection. A quarter-wave intermediate layer with $Z_m=\sqrt{Z_1Z_2}$ transforms the load to the source impedance at its design <frequency>; its bandwidth is limited. For strings, the analogous <mechanical impedance> is transverse force divided by velocity, $Z=Tv^{-1}=\sqrt{T\mu}$; continuity of displacement and transverse force gives the corresponding matching condition. These <amplitudes> use different force/velocity conventions from acoustic <pressure>, so reflection signs must be stated consistently.
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