10.2. HUYGENS PRINCIPLE

We would first define a wavefront: when we drop a small stone on a calm pool of water, waves spread out from the point of impact. Every point on the surface starts oscillating with time. At any instant, a photograph of the surface would show circular rings on which the disturbance is maximum. Clearly, all points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points, which oscillate in phase is called a wavefront; thus a wavefront is defined as a surface of constant phase.

A wavefront is a surface over which an oscillating quantity, such as light or sound, has a constant phase. When a small object, like a stone, is thrown into a pond, it creates waves that move outward in circles. These circles represent wavefronts, with each point on the wavefront oscillating together at the same time, ensuring they are 'in phase'. Imagine it like a synchronized swimming team where all members move together, forming a perfect pattern.

Think of wavefronts like the ripples that spread out when you throw a pebble into a calm pond. The circles of the ripples represent wavefronts, and at every point on a particular ripple, the water level is at the same height, all rising and falling together like a synchronized dance.

If we have a point source emitting waves uniformly in all directions, then the locus of points which have the same amplitude and vibrate in the same phase are spheres and we have what is known as a spherical wave. At a large distance from the source, a small portion of the sphere can be considered as a plane and we have what is known as a plane wave.

When a sound or light source emits waves uniformly in all directions, these waves form spherical wavefronts, akin to balloons expanding outward from a central point. In far-away observations, the curvature of these spheres becomes negligible, and they can be approximated as 'plane waves', which are easier to analyze because they simplify the complex behavior of waves into a flat, two-dimensional representation.

Consider how sound travels from a loudspeaker. Close to the speaker, sound waves are spherical as they spread out. However, far away, they can be seen as flat waves, similar to how the Earth's surface appears flat due to its vast size compared to our view.

Now, if we know the shape of the wavefront at t = 0, then Huygens principle allows us to determine the shape of the wavefront at a later time t. Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wavefront at any time allows us to determine the shape of the wavefront at a later time.

Huygens' Principle can be understood as a method for predicting the future position and shape of wavefronts. It suggests that every point on an existing wavefront serves as a source of new waves, known as secondary wavelets. These secondary waves spread out in all directions. By tracing the envelope of these secondary waves, you can determine how the overall wavefront will evolve over time.

Imagine you are at the edge of a large pond and you see the ripples moving outward. If you drop more stones one after another, each ripple can be seen as a new point generating its own set of ripples. By observing how the ripples spread, you can effectively predict the overall shape of the water surface at any future time.

Let us consider a diverging wave and let F F represent a portion of the spherical wavefront at t = 0. According to Huygens principle, each point of the wavefront is the source of a secondary disturbance, and the wavelets emanating from these points spread out in all directions with the speed of the wave.

In this illustration of Huygens' Principle, we start with a known wavefront, termed FF, that is at a specific time. Each point on this wavefront emits new wavelets that expand outward. By considering the radii of these emitted wavelets, we can construct new wavefronts at future times based on their positions after they spread out for a certain distance.

Imagine a group of people standing in a circle, each person starts tossing a ball simultaneously. The balls represent secondary waves. As each ball travels outward, the farthest points reached by the balls can help define a new boundary to show where the next circle of people will form as they get ready to throw.

Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this ad hoc assumption, Huygens could explain the absence of the backwave. However, this ad hoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory.

Huygens introduced an assumption that only waves moving forward carry energy and intensity, hence there are no waves moving backwards. This simplification was useful for his initial theories but lacks complete accuracy for more complex wave behaviors. Rigorously, it has since been concluded through further studies in optical phenomena that while waves typically propagate outward, potential backward waves exist but are not observed under normal conditions.

Think of a train on a track. When the train moves forward, it generates a sound that travels outward, but if the train were to reverse, you wouldn't hear that sound traveling back toward it in the same way. Huygens' initial approach is similar: while he focused on the waves moving away, the full understanding of sound waves considers that sounds can resonate backward too, depending on the environment.

In a similar manner, we can use Huygens principle to determine the shape of the wavefront for a plane wave propagating through a medium.

Applying Huygens' Principle to plane waves allows us to analyze how these waves propagate in the same way as spherical waves. So when a plane wave travels through a medium, we can treat each point on its wavefront as a new emitter of wavelets. The process generates new wavefronts at future times that maintain the behavior and characteristics of the original wave.

Think of a vast field of flowers swaying in the wind. Each bloom represents a point of the wavefront; as the wind moves, every flower sways in a rhythmic motion, simultaneously producing a wave-like effect across the field, allowing us to see the plane wave propagating through the sea of flowers.

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