7.3 Transmission through a thin layer

A wave in medium 1 encounters a slab of medium 2 of thickness $L$ , beyond which it re-enters medium 1 (or some other medium 3). Reflections happen at both interfaces, and within the slab the waves bounce back and forth. The interference between all of these reflections can give a transmission coefficient that depends sensitively on $L$ , $\omega$ , and the impedance ratios. This is the basis of acoustic impedance matching.

The setup

Three regions:

$x < 0$ : medium 1, impedance $Z_1$ , wavenumber $k_1$ .
$0 \le x \le L$ : layer (medium 2), impedance $Z_2$ , wavenumber $k_2$ .
$x > L$ : medium 3, impedance $Z_3$ , wavenumber $k_3$ . (Often $Z_3 = Z_1$ , the slab is sandwiched between two identical media.)

Incident wave from the left, no incoming wave from the right.

The result

Solving the boundary conditions at $x = 0$ and $x = L$ yields, for the special case $Z_1 = Z_3$ (slab in homogeneous medium),

T_P \;=\; \frac{1}{1 + \tfrac14\!\left(Z_2/Z_1 - Z_1/Z_2\right)^2\, \sin^2(k_2 L)}.

The power transmission coefficient is periodic in $k_2 L$ with period $\pi$ . Two extremes:

$k_2 L = 0, \pi, 2\pi, \ldots$ (slab thickness is a half-wavelength, or integer multiple). $\sin(k_2 L) = 0$ , so $T_P = 1$ — perfect transmission. The slab is invisible.
$k_2 L = \pi/2, 3\pi/2, \ldots$ (slab is a quarter-wavelength, or odd multiple). $\sin(k_2 L) = 1$ . The transmission depends on the impedance ratio.

Quarter-wave matching

The most useful special case: the slab is a quarter-wavelength thick ( $k_2 L = \pi/2$ ) and has impedance $Z_2 = \sqrt{Z_1 Z_3}$ (the geometric mean of the impedances on each side). Then it turns out — straightforward algebra from the general formulas — that

T_P \;=\; 1.

A quarter-wave layer with the right impedance produces perfect transmission between two otherwise mismatched media. The mechanism: the wave reflected from the back interface returns to the front interface a full half-wavelength out of phase with the wave reflected from the front, exactly cancelling. All the energy ends up transmitted.

This is quarter-wave impedance matching. It is used in:

Anti-reflective coatings on optics. A $\lambda/4$ layer of MgF₂ on glass minimises reflection of visible light.
Acoustic transducers. A $\lambda/4$ matching layer between a piezo crystal ( $Z \sim 30 \cdot 10^6$ Pa·s/m) and water ( $Z \sim 1.5 \cdot 10^6$ ) made of a polymer with $Z \sim \sqrt{30 \cdot 1.5} \cdot 10^6 \approx 7 \cdot 10^6$ Pa·s/m gives much better acoustic coupling than direct contact.
The middle ear. The 22:1 ossicular lever in mammals is not literally a quarter-wave matcher (the timescales are wrong), but it serves the same function: convert a high-impedance fluid load to a manageable air-side load. The hearing book covers the details.

Bandwidth

A quarter-wave matcher works only at the design frequency. Move away from it, the slab thickness is no longer $\lambda/4$ , and the cancellation degrades. The bandwidth depends on the impedance ratio: high contrast (like air-to-water) is harder to match over a wide band than low contrast. For broadband matching, you stack multiple quarter-wave layers in sequence, each with a different impedance — a transformer. The same trick is used in microwave engineering and antireflection coatings.

Anti-reflection by absence (not always)

The other extreme of the transmission formula — slab thickness equal to a half-wavelength — also gives perfect transmission, but for a different reason: the reflections from the two faces interfere destructively when the slab is transparent, regardless of $Z_2$ . A half-wave slab of any impedance is invisible at the design frequency. Useful for impedance “spacers” that need to be present but acoustically transparent.

What this gives us

The thin-layer transmission formula is the most general statement of interference-based filtering. By choosing layer thicknesses and impedances we can build acoustic filters that pass certain frequencies and block others — the acoustic analogue of optical interference filters. This is also the framework underlying acoustic metamaterials, where periodic structures of mismatched layers create bandgaps and unusual dispersion.

Next: the Huygens construction unifies reflection, refraction, and diffraction as three faces of one primitive.