 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 givesMatching $Z_1=Z_2$ eliminates reflection. A quarter-wave intermediate layer with $Z_m=\sqrt{Z_1{Z}_2}$ 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, $Z=T{v}^{-1}=\sqrt{T\mu }$; 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.