Interference of two identical waves

Problem

Two identical waves meet at a point. Under what conditions do they produce maximum constructive interference? And total destructive interference? Where does the energy go in the second case?

Solution

Superposition principle. At the meeting point the resultant displacement is the instant-by-instant sum of the two displacements.

Maximum constructive interference. The two waves must have the same frequency and arrive in phase (zero phase difference, $\Delta\varphi = 0$, i.e. paths differing by an integer number of wavelengths). The amplitudes add: if they are equal ($A$), the resultant amplitude is $2A$ and the intensity is $\propto (2A)^2 = 4$ times that of a single wave.

Total destructive interference. Requires the same frequency, same amplitude and a relative phase $\Delta\varphi = \pi$ (paths differing by half a wavelength, or odd multiples thereof). The waves cancel: the resultant amplitude is zero.

Where the energy goes. Energy does not disappear: destructive interference at one point is always accompanied by constructive interference at other points. Averaged over the whole region, the total energy is conserved, simply redistributed spatially (fringe maxima and minima).

$\ev{\text{Max constructive: same } f,\ \Delta\varphi=0.\quad \text{Total destructive: same } f \text{ and } A,\ \Delta\varphi=\pi.\quad \text{Energy is redistributed.}}$

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Topics: Onde Concepts: Interferenza Skills: Principio di sovrapposizione