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Huygens’ Principle
The Dutch scientist Christiaan Huygens suggested a graphical method of predicting the future position of a wave-front, knowing its current position.
His principle is stated as follows.
 Each point on the existing wave-front can be considered to act as a source of waves (sometimes referred to as secondary wavelets).

For physical evidence to support this suggestion, just consider the phenomenon of diffraction of waves by a small aperture.
When a small part of a plane wave-front is isolated, it does behave like a point source... I rest my case (well, Huygens' case)...

In the following examples it is perhaps easiest to imagine waves on the surface of water (as observed in a ripple tank), but the results can be applied to any two (or three) dimensional waves.

First, a rather trivial example.
Consider the set of plane (straight) waves shown below.
The velocity of propagation of the waves is v.
To apply the principle, we must pretend that we cannot guess where the wave-front will be, say, t seconds later!

First chose a point (any point), A, on the wave-front and draw an arc of radius vt.
This is the distance the secondary wavelets will have moved in t seconds.

Now chose another point, B, at random, on the wave-front and repeat the process.

The new wave-front is the tangent to the two curves…well what a surprise, it's just where we expected!
However, we will now consider two situations where the principle can help make useful predictions.

Reflection of Waves Using Huygens’ Principle
Consider a set of plane waves moving towards a reflecting surface, indicated by the line x-x’.
At time t = 0, point A on the wave-front reaches the reflecting surface.
The red arrow is a "ray" showing the direction of motion of the waves. A ray is always at 90° to the wave-front.

We will try to find the position of the wave-front at time t, the instant when point B reaches the reflecting surface.

First, draw an arc of radius equal to the distance B C (see next diagram).
The secondary wavelets from point A will have travelled this far by the time the waves at point B reach point C.
The new wave-front is the tangent to this arc which passes through point C.

Now, using the observed fact that the direction of propagation of a wave is always at 90° to the wave front, we can predict the direction of motion of the waves after reflection.

The angle of incidence is the angle between the direction of propagation of the waves and a normal to the reflecting surface before reflection.

The angle of reflection is the angle between the direction of propagation of the waves and a normal to the reflecting surface after reflection.

We therefore wee that Huygens' method predicts that waves obey the familiar law of reflection, easily observed using a light beam and a mirror.

Refraction of Waves Using Huygens’ Principle
When waves travel across a boundary between two different media, the speed of propagation changes.
For example, the speed of light in a vacuum is 108ms-1, whereas in glass its speed is about 108ms-1.
The change in speed can result in a change in direction of propagation of the waves.
This change in direction is called refraction.

The diagrams below show how to use Huygens’ principle to predict the position of the wave-front when waves move from a medium in which they have speed v1 to a medium in which they have speed v2.
In this case, v2 < v1
In these diagrams the line x x’ represents the boundary between the two media.

At time t = 0, point A on the wave-front reaches the boundary.
Consider secondary wavelets emitted from A at time t = 0.
At time t seconds later, point B reaches the boundary.
At time t, the secondary wavelets emitted from A have moved a distance v2t.

The position of the new wave-front is shown by line C D.
The situation at a later time is shown in the next diagram.

Notice that the change in speed of the waves inevitably produces a change in the wavelength, as explained here.
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