Two coherent waves, coming from sources at different...

Two coherent waves, coming from sources at different locations, move along the $x$ axis. Their wave functions are $$ E_{1}=(860 \mathrm{V} / \mathrm{m}) \sin \left[\frac{2 \pi x_{1}}{650 \mathrm{nm}}-2 \pi(462 \mathrm{THz}) t+\frac{\pi}{6}\right] $$ and $$ E_{2}=(860 \mathrm{V} / \mathrm{m}) \sin \left[\frac{2 \pi x_{2}}{650 \mathrm{nm}}-2 \pi(462 \mathrm{THz}) t+\frac{\pi}{8}\right] $$ Determine the relationship between $x_{1}$ and $x_{2}$ that produces constructive interference when the two waves are superposed.

Two coherent waves, coming from sources at different locations, move along the $x$ axis. Their wave functions are
$$
E_{1}=(860 \mathrm{V} / \mathrm{m}) \sin \left[\frac{2 \pi x_{1}}{650 \mathrm{nm}}-2 \pi(462 \mathrm{THz}) t+\frac{\pi}{6}\right]
$$
and
$$
E_{2}=(860 \mathrm{V} / \mathrm{m}) \sin \left[\frac{2 \pi x_{2}}{650 \mathrm{nm}}-2 \pi(462 \mathrm{THz}) t+\frac{\pi}{8}\right]
$$
Determine the relationship between $x_{1}$ and $x_{2}$ that produces constructive interference when the two waves are superposed.

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Key Concepts

Constructive Interference

Constructive interference occurs when the phase difference between overlapping waves leads to their amplitudes adding together, resulting in a wave with a larger amplitude. This happens when the phase difference is an integer multiple of 2?, meaning the waves are in phase and reinforce each other.

Phase Difference

The phase difference between waves is the angular difference between their phases at a given point and time. It is crucial for determining the type and degree of interference; when the phase difference is such that the waves’ peaks align, constructive interference occurs, while misaligned peaks lead to destructive interference.

Wave Superposition Principle

This principle states that when two or more waves overlap at a point, the resultant displacement is the sum of the displacements from each individual wave. It is fundamental to understanding interference, as it explains how different wave contributions add together to produce a combined wave pattern.

Coherent Waves

Coherent waves have a constant phase difference and the same frequency, which is essential for producing stable and predictable interference patterns. Because their phase relationship does not vary randomly, coherent waves can constructively or destructively interfere in a consistent manner.

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