Applying to Reflections on String

Consider a string free at one end ($Z_2=0$).

• Using displacement velocity as the wave variable:
  $$\frac{v_r}{v_i} = \frac{Z_1 - Z_2}{Z_1 + Z_2} = \frac{Z_1}{Z_1} = 1,$$
  

• The total velocity at the junction (boundary) is
  $$v = v_i + v_r = v_i + v_i = 2v_i.$$

If the string is fixed at one end, new impedance approaches infinity ( $Z_2 \gg Z_1$):

$$\frac{v_r}{v_i} = \frac{Z_1 - Z_2}{Z_1 + Z_2} \approx -\frac{Z_2}{Z_2} = -1,$$

where $Z_1$ is considered to be negligible.

The velocity at the boundary is therefore

$$v = v_i + v_r = v_i - v_i = 0.$$