(Z_1 - Z_2)/(Z_1 + Z_2) = [sqrt(rho_1) - sqrt(rho_2)]/[sqrt(rho_1) + sqrt(rho_2)],
and transmitted wave in strong 2 is
2Z_1/(Z_1 + Z_2) = 2sqrt(rho_1)/[sqrt(rho_1) + sqrt(rho_2)].
The first animation shows reflection and transmission of a pulse wave of unit amplitude when rho_1/rho_2 = 1/4 (Z_1/Z_2 = 1/2). The reflected wave is negative and its peak is -1/3. The transmitted wave, which propagates slower, has an amplitude of 1 - 1/3 = +2/3, as expected from the equations.
If the string 2 is much heavier than string 1 (as at a fixed end), reflection becomes complete.
The third animation shows the case Z_1/Z_2 = 2, i.e., the incident wave
is in a heavier string. There is no sign reversal in the reflected wave
in this case. The transmitted wave has an amplitude larger than the incident
wave. This does not mean amplification in wave energy. (Why not?)
Finally, reflection at a free end is shown. In this case too, reflection
is complete but without change in the polarity.
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