Wave Behavior

Interference

reinforcing and cancelling

In phase, two waves reinforce to double the amplitude; a half wavelength apart they cancel to nothing.

Example

In phase, two waves reinforce to double the amplitude; a half wavelength apart they cancel to nothing.

highlighted = computed this step

Crest on crest: they reinforce

If the two waves line up crest-on-crest — in phase — every point adds to a bigger displacement. Two equal waves of amplitude 1 give a combined amplitude of 2 metres. This is constructive interference.

Atotal=1+1=2 mA_{\text{total}} = 1 + 1 = \hl{2}\ \text{m}
In phase: amplitude doublesTwo equal in-phase waves and their purple sum at double the amplitude.sum: double

Crest on trough: they cancel

If instead one wave is shifted half a wavelength — crest-on-trough, exactly out of phase — then at every point one wave pushes up by as much as the other pushes down. They cancel completely: the combined displacement is 0 everywhere. This is destructive interference. Complete cancellation like this needs the two waves to have equal amplitude; waves of different sizes only partly cancel.

Atotal=1−1=0 mA_{\text{total}} = 1 - 1 = \hl{0}\ \text{m}
Half a wavelength apart: they cancelTwo equal waves a half wavelength apart; their purple sum is a flat line at zero.sum = 0
waves Constructive 1+1=2 and destructive 1-1=0 fall out honestly from the point-by-point sum.