Interference and diffraction

Noise-cancelling headphones make a second sound that is the exact opposite of the noise, and the two add up to near silence.

Waves add up

Two balls thrown at each other bounce off. Two waves that meet do something different: they pass straight through each other, and while they overlap the disturbances simply add. OpenStax calls this superposition. If two identical waves arrive in phase, crest on crest and trough on trough, the result has twice the amplitude. This is pure constructive interference. If they arrive exactly out of phase, crest on trough, they cancel and the amplitude is zero. This is pure destructive interference. Most of the time the result is something in between, and it changes from place to place.

You meet interference often without noticing it. A stereo can sound loud in one spot of a room and dull in another, because sound from two speakers and from the walls adds partly constructively in some places and partly destructively in others. Noise-cancelling headphones listen to the incoming noise and play a wave with its highs and lows reversed. OpenStax notes that complete cancellation only works under the simplest conditions, but noise can be cut by 30 dB or more this way.

Two sources and path difference

Put two sources side by side, vibrating in step with the same wavelength, like two fingers tapping a pond together. At any point, the waves from the two sources have travelled different distances. What matters is the path difference Δr. If Δr is a whole number of wavelengths (0, λ, 2λ, …), the waves arrive in step and add. If Δr is a half-number of wavelengths (½λ, 1½λ, …), crest meets trough and they cancel. The quiet points line up in curved lines that fan out from between the sources, with lively strips in between.

Young and the wave nature of light

In OpenStax’s words, interference is the hallmark of waves: one way to prove something is a wave is to see it interfere. For light this was settled in 1801 by Thomas Young, who passed light through two narrow, closely spaced slits. Each slit acted as a source, and instead of two bright lines the screen showed a row of bright and dark bands. Bright bands appear in directions where d sin θ = mλ, with d the slit spacing and m a whole number. In OpenStax’s example, slits 0.0100 mm apart and a third bright band at 10.95° give λ = 633 nm, the red light of a helium-neon laser.

Diffraction: waves spreading through gaps

When a wave passes through an opening or around an edge, it spreads into the region behind, an effect called diffraction. The amount of spreading depends on the size of the opening compared with the wavelength. A gap much wider than the wavelength lets the wave through as a beam with a fairly sharp shadow on either side. A gap about one wavelength wide spreads the wave out into a wide fan. This is why you hear sound around a doorway (wavelengths around a metre) but cannot see around it (light wavelengths are less than a millionth of a metre). OpenStax gives a radio example: long AM waves go over and around buildings and hills, while FM and TV reception is best with a clear line of sight.

Beats

Two sounds with slightly different frequencies drift in and out of step. For a moment their crests line up and the sound is loud; a little later crest meets trough and it goes quiet. The result is a single tone at the average frequency that swells and fades, a wobble called beats. OpenStax shows that the beat frequency is the difference of the two: f_B = |f₁ − f₂|. Piano tuners use this. If a 256 Hz tuning fork and a string give 2 beats per second, the string is at 254 Hz or 258 Hz, and the tuner adjusts it until the beats stop.