Wave Behavior

Interference

reinforcing and cancelling

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

waves Constructive 1+1=2 and destructive 1-1=0 fall out honestly from the point-by-point sum.

Example

In phase, two waves reinforce to double the amplitude; a half wavelength apart they cancel to nothing. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

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

Shift by half a wavelength and the sum changes

Keep the two waves equal. A zero shift lines up crests with crests, a half-wavelength shift lines up crests with troughs, and a full-wavelength shift lines the crests up again.

shiftAAtotal0 m1 m2 m2 m1 m0 m4 m1 m2 m\begin{array}{c|c|c}\text{shift} & A & A_{\text{total}} \\ \hline 0\ \text{m} & 1\ \text{m} & 2\ \text{m} \\ 2\ \text{m} & 1\ \text{m} & 0\ \text{m} \\ 4\ \text{m} & 1\ \text{m} & 2\ \text{m}\end{array}

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

Unequal opposite waves only partly cancel

Complete cancellation needs equal amplitudes. If one wave is larger, the opposite wave subtracts from it but leaves the extra height behind.

yaybytotal1 m−1 m0 m2 m−1 m1 m3 m−1 m2 m\begin{array}{c|c|c}y_a & y_b & y_{\text{total}} \\ \hline 1\ \text{m} & -1\ \text{m} & 0\ \text{m} \\ 2\ \text{m} & -1\ \text{m} & 1\ \text{m} \\ 3\ \text{m} & -1\ \text{m} & 2\ \text{m}\end{array}
Opposite waves of unequal size leave a smaller waveA taller upward wave and a smaller opposite wave; the purple sum is not flat because only part of the motion cancels.partial sum