Why Do Two Waves Sometimes Cancel? Adding Peaks and Troughs
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When two waves meet, the result at each point is the sum of both, which means they can reinforce or cancel entirely. That single rule explains noise cancelling headphones, the colours in soap bubbles and how we know light is a wave.
The rule that governs it
Where two waves overlap, the displacement at each point at each moment is the sum of what each wave would produce alone, which is called superposition and holds for sound, light, water waves and most others under ordinary conditions. If two waves arrive with their peaks aligned they add to a larger wave, and if a peak of one aligns with a trough of the other they subtract, cancelling completely when the two are equal in size. What decides which happens at a given point is the difference in the distance each wave travelled to get there, since travelling half a wavelength further puts a wave exactly out of step. The waves pass through each other unchanged and resume their separate courses afterwards, which is worth stating because the cancellation is local rather than destructive in any lasting sense.
Where it is observable
The effect shows up in a great many ordinary situations:
- •Colours in soap films and oil on water, from light reflecting off the front and back surfaces
- •The coating on camera lenses, engineered so reflections cancel and more light passes through
- •Dead spots in a room where two loudspeakers cancel at particular frequencies
- •Noise cancelling headphones, which generate a wave inverted relative to the incoming noise
- •Beats, the slow pulsing heard when two notes are very slightly different in pitch
- •The patterns produced when two sets of ripples cross on water
The experiment that settled an argument
Whether light was a wave or a stream of particles was disputed for over a century, and interference provided the decisive evidence. Thomas Young, around 1801, passed light through two narrow closely spaced slits and observed a pattern of bright and dark bands on a screen beyond, which a particle account cannot produce, since two sources of particles should give more light everywhere rather than dark regions where light plus light gives darkness. The bands appear exactly where the path difference from the two slits is a whole number or a half number of wavelengths, which also allowed the wavelength of light to be measured for the first time. The experiment was resisted initially and became the foundation of the wave theory, and it acquired a second life in the twentieth century when it was found that the pattern still appears when particles are sent through one at a time.
The condition that makes it visible
A stable pattern requires the two waves to hold a constant phase relationship, which is called coherence and which explains why interference is not observed everywhere despite waves overlapping constantly. Light from an ordinary lamp is emitted by enormous numbers of atoms independently, each for a very short interval, so the phase relationship between any two parts of the beam changes far faster than any detector or eye can follow, and the pattern averages out to uniform illumination. The classic two-slit arrangement solves this by deriving both beams from the same source, so whatever the phase does it does identically in both. Lasers produce coherent light directly, which is why interference effects are easy to demonstrate with one and why laser light shows speckle, an interference pattern formed by scattering from a rough surface.
Putting it to work
Interference is used as a measuring tool of extraordinary precision, because the pattern shifts by a visible amount when a path length changes by a fraction of a wavelength. Instruments splitting a beam, sending the halves along different paths and recombining them can detect changes in distance far smaller than any ruler could, and the technique is used to test optical surfaces, to measure refractive index and to calibrate length standards. The same principle detected gravitational waves in 2015, using arms kilometres long and measuring changes smaller than the width of an atomic nucleus. Radio astronomy combines signals from widely separated dishes so that the array behaves like a single instrument the size of the separation, which is how the sharpest astronomical images are made.
The takeaway
Overlapping waves add point by point, so aligned peaks reinforce and a peak meeting a trough cancels, with the path difference deciding which happens where. Soap film colours, lens coatings and noise cancelling all follow from that. Bright and dark bands from two slits cannot be produced by particles, which is how light was established as a wave.