7.5 Diffraction at an edge and through an aperture

A plane wave meets an obstacle: a wall with an opening, a screen with a sharp edge, a barrier shorter than the wavefront. What happens at the boundary of the geometric shadow?

In geometrical optics, behind a sharp opaque screen there is a perfectly defined shadow — no light gets in. In wave acoustics (and wave optics) there is sound in the shadow, with a pattern of light and dark fringes near the boundary of where the shadow “should” be. This pattern is diffraction, and its scale is set by the wavelength $\lambda$ .

Knife-edge diffraction

A plane wave approaches a half-infinite screen — say, the wall stops at $x = 0$ , with empty space below the wall. In the geometric-shadow region (behind the wall), the field is not zero: secondary wavelets from the edge spread into the shadow. In the illuminated region above, the field is the incident wave plus diffracted contributions from the edge, producing a series of bright and dark fringes parallel to the edge.

The mathematical structure: the field intensity as a function of position $y$ near the shadow boundary follows the Fresnel diffraction integral, with characteristic ripples on the illuminated side and a smooth decay into the shadow. The pattern depends on a single dimensionless parameter — the Fresnel number — which combines the geometry and the wavelength.

Single-slit diffraction

A plane wave hits a screen with a slit of width $d$ . On the far side, the field at angle $\theta$ from the original propagation direction has amplitude (in the far-field, Fraunhofer regime)

p(\theta) \;\propto\; \frac{\sin\!\left(\tfrac{\pi d \sin\theta}{\lambda}\right)}{\tfrac{\pi d \sin\theta}{\lambda}}.

This is the sinc function pattern: a central peak with width $\sim \lambda/d$ , surrounded by side lobes. The first zero of the sinc is at $\sin\theta = \lambda / d$ — for a slit much wider than the wavelength, the diffraction angle is tiny and the wave passes through nearly unaffected. For a slit comparable to the wavelength, the wave spreads dramatically into the geometric shadow region.

The same formula governs the directional pattern of a finite acoustic source emitting a plane wave (the aperture analogy): the radiation pattern of an emitter of width $d$ has a main lobe of width $\lambda/d$ .

Circular-aperture diffraction (Airy pattern)

For a circular aperture of radius $a$ , the diffraction pattern is the Airy pattern:

I(\theta) \;\propto\; \left[\frac{2 J_1(k a \sin\theta)}{k a \sin\theta}\right]^2,

with central peak and concentric rings. The first dark ring is at $\sin\theta = 1.22 \lambda / D$ where $D = 2a$ is the aperture diameter — the Rayleigh resolution criterion.

This is the same pattern as the piston-in-baffle radiation pattern from lesson 6.5. A circular vibrating piston is a coherent emitter from a circular aperture; its emission pattern is its diffraction pattern.

When diffraction matters in audio

Frequency	$\lambda$ in air	Diffraction around a wavelength-scale obstacle…
100 Hz	3.4 m	Bass wraps around furniture, doorways, couches
1 kHz	0.34 m	Voices wrap around heads (just barely); speech audibility around corners
10 kHz	3.4 cm	Sharp shadows behind a hand; localisation cues sharpened
20 kHz	1.7 cm	Strongly directional; small objects cast sharp shadows

This is why bass instruments seem “omnidirectional” indoors (wavelength-scale diffraction around everything) while high-frequency content is “directional” (less diffraction, more like rays).

The same diffraction is the reason a thin curtain or a small barrier doesn’t block low-frequency sound: the wave goes around obstacles smaller than its wavelength.

In rooms

Diffraction is a major contributor to indoor acoustic propagation. Sound waves entering a room don’t just bounce off walls — they wrap around corners, spread out from doorways, and fill spaces that ray-tracing would predict to be in shadow. Architectural acoustics software combines ray-tracing (for the short-wavelength components and direct paths) with diffraction calculations at edges (for the long-wavelength components and shadow zones).

The unifying picture

We have, in five lessons, covered:

Reflection (7.1, 7.2) — boundary condition gives back the angle of incidence.
Refraction (7.2) — Snell’s law from wavefront-matching across an interface.
Transmission through a slab (7.3) — interference of multiple reflections.
Huygens construction (7.4) — every point on a wavefront is a secondary source.
Diffraction (this lesson) — finite apertures or edges interrupt the Huygens superposition.

All five are consequences of the same wave equation, governed by the same boundary conditions. The next four lessons take up the modes — what happens when waves bounce around inside bounded regions and settle into stable standing patterns.