Mechanical impedance ($Z=F/v$)

Mechanical impedance is the ratio of a driving force complex amplitude to the resulting velocity complex amplitude at a specified point and in a specified direction. Its units are $\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$. The direction of the force, the time convention and whether one is describing a travelling component or a total field must be stated. A positive real travelling-wave mechanical impedance transfers mean power $⟨P⟩=Z|v{|}^2/2$ when $|v|$ is the peak velocity wave amplitude.

For transverse waves on two strings at common tension, displacement continuity gives $1+r=t$, while transverse-force continuity gives $(1-r)/c_1=t/c_2$. Solving gives the displayed displacement-amplitude coefficients. A denser transmitted string has lower speed and a negative reflected coefficient. The energy fractions are $r^2$ and $(c_1/c_2)t^2$, summing to1. These are displacement coefficients; force or power-normalized wave amplitudes use different conventions.

At an ideal nondispersive string junction, every temporal feature is scattered with the same constant coefficients. The reflected spatial width equals the incident width, while the transmitted width changes in proportion to wave speed. A reflected asymmetric spatial profile is reversed left-to-right as well as receiving the reflection sign. For a denser second string the transmitted amplitude is smaller but its spatial slopes can be steeper, because its width is compressed.