Wave Reflection at Impedance Discontinuity

The impedance for mechanical waves is defined as the ratio between the force wave and velocity wave and in the form

Z = sqrt(mass density * elastic modulus).

For transverse waves in a string with linear mass density rho (kg/m) and tension T (N), the impedance is

Z = sqrt(rho * T).

If two strings with mass densities rho_1 and rho_2 are connected with common tension, the impedance discontinuity causes wave reflection. For an incident displacement wave of unit amplitude in string 1, the reflected wave is

(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.

To repeat animation, click "Back" then "Forward" buttons.