**Introduction**

In this lesson, we will address the second approach to the concept of light: waves. We will study the superposition principle, diffraction, Huygens' principle, constructive and destructive interference, diffraction gratings, and polarization.

**Superposition Principle**

We have studied harmonic motion and the time dependence of perturbation. We will now deepen our understanding of waves by studying their interaction. Because each wave is a time and spatial variation of a perturbation and because the appearance of the wave might be different depending on the relative distance between the point of study and the source, we are looking for an expression that connects all these elements together: time, displacement, and propagation. The expression is

*y*(*x*, *t*) = *A* · sin (*k* · *x* – ω · *t*)

where *A* is the amplitude, ω is the angular speed, *t* is the time, and *x* is the distance over which propagation occurred in time *t*, and *k* is called the *wave number* and is defined as

where λ is the wavelength, and α = ω · *t* is called the *phase* and represents an angular displacement. In SI units: *x*, *y*, and *A* are in meters, ω is in radians/seconds, *t* is in seconds, and *k* is in *m*^{–1} .

If a source produces waves at different times, then at the same position in space, there will be differences between the displacements because of the time delay. We call this *initial phase*. An example would be the case of a wave that was produced at time *t* before the time considered to be the origin, which would be *t* at 0 seconds. This means that at time 0 seconds, the wave has already reached some displacement, while the second wave starts from 0 meters displacement (see Figure 20.1).

The *y* (*x*,*t*) is called the *harmonic wave function* and is a characteristic of all periodic waves, giving the displacement and propagation for a certain moment in time (see Figure 20.2).

In the case of a three-dimensional wave, such as the water ripples in an aquarium or a lake or the waves coming from a light source, we can define a concept helpful for future developments: *wavefront*. Throw a stone in the water, and ripples forming from the source of perturbation will soon be visible. You can see concentric circles propagating out in space. A crosssection through water would show that the surface of the water is displaced up and down. At different positions compared to the source of perturbation, the displacement is the same: as for instance, different points are at maximum displacement (amplitude) at the same time. We call these points in phase and the circles that connect these points *in phase* are called *wavefronts*. The radius from the source to the wavefront is called a *ray*.

Figure 20.3 shows the wave propagation in free space, but what will happen if the space contains more than one source or obstacles? Let us answer the first part of the question. The *principle of superposition* says that when two or more waves are present at the same point in space and at the same time, the result of the superposition is the sum of the individual waves.

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