Constructive and destructive interference

Two waves with the same frequency, but coming from different sources, overlap at a point and the result depends on their phase difference. If the crests of one coincide with the crests of the other, the displacements add up and the amplitude doubles: this is called constructive interference. If instead a crest meets a trough, the two displacements cancel and the amplitude vanishes: destructive interference. Between these two extremes lies a whole range of intermediate situations.

Interference: if the waves are in phase, the resultant has amplitude $2A$ (constructive); if they are in antiphase, they cancel out (destructive).

What decides between constructive and destructive is the path difference $\Delta x$: if the two sources are in phase, what matters is how much extra distance one of the two waves must travel to reach the observation point. An extra path equal to a whole number of wavelengths brings the wave back exactly in phase; an extra path equal to half a wavelength (plus a whole number) brings it into antiphase.

Key formula — Interference conditions

Let $\Delta x$ be the path difference between the two sources:

• Constructive: $\Delta x = n\lambda \quad (n = 0, 1, 2, \ldots)$
• Destructive: $\Delta x = \left(n + \tfrac{1}{2}\right)\lambda$

The link between path difference and phase shift is direct: $\Delta\varphi = \dfrac{2\pi}{\lambda}\,\Delta x$. In terms of phase, the constructive condition is $\Delta\varphi = 2n\pi$ (waves in phase) and the destructive one is $\Delta\varphi = \pi + 2n\pi$ (waves in antiphase).

Summary

In phase ($\Delta\varphi = 2n\pi$): constructive interference. In antiphase ($\Delta\varphi = \pi + 2n\pi$): destructive interference. The intensity of the resultant wave varies as $\cos^2(\Delta\varphi/2)$, passing continuously from the maximum (constructive) to zero (destructive).

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Topics: Onde Concepts: Interferenza Skills: Principio di sovrapposizione

Related exercises: Interferenza di due onde identiche · Problema — Frange di Young · Sfida doppia fenditura ottica