Reflection, Transmission and Impedance

The Physical Picture

Imagine a heavy rope tied to a light rope. If you send a pulse down the heavy rope, what happens when it reaches the junction? The heavy rope has a high impedance because it is difficult to move (high mass per unit length). The light rope has low impedance. When the pulse hits the junction, the heavy rope "pushes" back against the light rope more strongly than the light rope can push back. This imbalance causes part of the pulse to bounce back (reflection) and part to continue forward (transmission). If the two ropes had identical impedance, the pulse would pass through without any reflection, as if the junction did not exist. The key insight is that the boundary conditions—continuity of displacement and force—dictate how the wave splits.

Impedance and Boundary Conditions

The wave impedance \(Z\) of a medium is defined as the ratio of the transverse force to the transverse velocity of the wave. For a string under tension \(T\) with linear mass density \(\mu\), the impedance is given by \(Z = \sqrt{T\mu}\). At the junction between two media with impedances \(Z_1\) and \(Z_2\), two physical conditions must be met: the transverse displacement must be continuous (the string cannot break), and the transverse force must be continuous (Newton's third law). These conditions lead to the reflection and transmission coefficients for amplitude.

Here, \(r\) is the amplitude reflection coefficient and \(\tau\) is the amplitude transmission coefficient. Note that \(\tau = 1 + r\): this is simply the continuity of displacement at the junction. If \(Z_1 = Z_2\), then \(r=0\) and \(\tau=1\), meaning perfect transmission. If \(Z_2 \gg Z_1\) (close to a fixed end), then \(r \approx -1\) and \(\tau \approx 0\): almost total reflection with a phase inversion. If \(Z_2 \ll Z_1\) (close to a free end), then \(r \approx +1\) and \(\tau \approx 2\): the transmitted amplitude exceeds the incident amplitude, yet the transmitted power is small because it is carried by a low-impedance medium.

Power Coefficients and Energy Conservation

While amplitude coefficients describe the height of the wave, power coefficients describe the energy flow. The average power carried by a wave is proportional to the square of the amplitude and the impedance of the medium. Therefore, the power reflection coefficient \(R\) and power transmission coefficient \(\mathcal{T}\) are given by:

A crucial check is that energy is conserved. The sum of the reflected and transmitted power must equal the incident power. Indeed, adding the two expressions yields:

This result confirms that no energy is lost at the boundary; it is simply partitioned between the reflected and transmitted waves.

Impedance Matching

In many applications, we want to maximize the transmission of energy from one medium to another. This is achieved by impedance matching, where we insert an intermediate medium with an impedance \(Z_m\) a quarter of a wavelength thick, whose impedance is the geometric mean of the two original impedances: \(Z_m = \sqrt{Z_1 Z_2}\). This removes the reflection completely at that one frequency. By using multiple layers with gradually changing impedance, we can create an "impedance transformer" that minimizes reflection. This principle is widely used in optics (anti-reflection coatings), acoustics (microphone diaphragms), and electronics (transmission lines).