47 Reflection and Transmission

We will now consider the full problem of a wave incident on a boundary where there is a sharp transition in the impedance of the medium.  Here, we will determine the relative amplitudes of the reflected and transmitted waves.  The change in direction of the transmitted wave is determined by Snell’s Law.

We will use two key constraints: First, there can be no net force on the boundary (or it would translate in space). Second, energy must be conserved. (The latter isn’t true in the presence of friction, so we are deriving expressions for a frictionless system.)

(1) We write the wave pressure as P=Acos(kx-\omega t+\phi).  No net force on the boundary means that the sum of amplitudes of the pressure waves on either side of the boundary must be the same:

    \[ A_{Incident}+A_{Reflected}=A_{Transmitted} \]

(2) To think about energy conservation, we equate the energy of the system before the wave encounters the boundary (when only the incident wave is present) to the energy afterward (when both reflected and transmitted waves are present). We use the expression for intensity derived in the previous section, and write Z_1 for the impedance of the medium that contains the incident wave and Z_2 for the impedance of the medium that contains the transmitted wave.

    \[ I_{Incident}=I_{Reflected}+I_{Transmitted} \]

    \[ \frac{A_{Incident}^2}{Z_1}=\frac{A_{Reflected}^2}{Z_1}+\frac{A_{Transmitted}^2}{Z_2} \]

Now, with a bit of algebra (optional reading), we can define the amplitudes of the reflected and transmitted waves, as a ratio over the incident wave amplitude.

Key Takeaways

The reflection (R) and transmission (T) coefficients give the amplitude ratios of the reflected and transmitted waves, relative to the incident wave:

    \[ R=\frac{A_{Reflected}}{A_{Incident}}=\frac{Z_2-Z_1}{Z_1+Z_2} \]

    \[ T=\frac{A_{Transmitted}}{A_{Incident}}=\frac{2 Z_2}{Z_1+Z_2} \]

These coefficients are related by

    \[ 1+R=T \]

Dan Russell’s webpage Reflection of Wave Pulses from Boundaries has some nice animations of reflection and transmission (scroll down to the “Reflection from an Impedance Discontinuity” section. Note: the mathematical expressions above his animations are derived for the case of a wave in a string. Do not use those expressions for this class – use the formulas in the “Key Takeaways” box.