Question

How is a wave front defined? Using Huygen’s construction

showing the propagation of a plane wave reflecting at the interface of the two

media. Show that the angle of incidence is equal to angle of reflection.​

Answer 1

Answer:

Wave front: A wave front is a locus of particles of medium all vibrating in the same phase.

Law of Reflection: Let XY be a reflecting surface at which a wave front is being incident obliquely. Let v be the speed of the wave front and at time t = 0, the wave front touches the surface XY at A.

After time t, the point B of wave front reaches the point B' of the surface.

According to Huygen's principle each point of wavefront acts as a source of secondary waves.

When the point A of wavefront strikes the reflecting surface, then due to presence of reflecting surface, it cannot advance further; but the secondary wavelet originating from point A begins to spread in all directions in the first medium with speed v.

As the wavefront AB advances further, its points

... etc. strike the reflecting surface successively and send spherical secondary wavelets in the first medium.

First of all the secondary wavelet starts from point A and traverses distance AA' (= vt) in first medium in time t. In the same time t, the point B of wavefront, after travelling a distance BB', reaches point B' (of the surface), from where the secondary wavelet now starts.

Now taking A as centre we draw a spherical arc of radius AA' (= vt) and draw tangent A'B' on this arc from point B'. As the incident wavefront AB advances, the secondary wavelets starting from points between A and B', one after the other and will touch A'B' simultaneously.

According to Huygen's principle wavefront A'B' represents the new position of AB, i.e., A'B' is the reflected wavefront corresponding to incident wavefront AB.

Now in right-angled triangles ABB' and AA'B'

side BB' = side M' (both are equal to vt)

i.e., incident wavefront AB and reflected wavefront A'B' make equal angles with the reflecting surface XY. As the rays are always normal to the wavefront, therefore the incident and the reflected rays make equal angles with the normal drawn on the surface XY, i.e.,

Angle of incidence i = Angle of reflection r

This is the second law of reflection.

Since AB, A'B' and XY are all in the plane of paper, therefore the perpendiculars dropped on them will also be in the same plane.

Therefore we conclude that the incident ray, reflected ray and the normal at the point of incidence, all lie in the same plane.

This is the first law of reflection.

Thus Huygen's principle explains both the laws of reflection.