Use Huygens principle to verify the laws of refraction.

Proof of Snell’s law of Refraction using Huygens wave theory is: 

As seen in the fig. above let XY be a surface separating the two media ‘1’ and ‘2’. Let v₁ and v₂ be the speeds of waves in these media.

A plane wavefront AB in the first medium is incident obliquely on the boundary surface XY and its end A touches the surface at A at time t = 0 while the other end B reaches the surface at point B after time-interval t.

Here, 

As the wavefront AB advances, it strikes the points between A and B¢ of boundary surface.

According to Huygens principle, secondary spherical wavelets emanate from these points, which travel with speed v₁ in the first medium and speed v₂ in the second medium.

Secondary wavelet starting from A, traverses a distance AA’ = v₂t in second medium in time t. In the same time, point of wavefront traverses a distance in first medium and reaches B’, from where the secondary wavelet starts. 

So,   and AA’ = v₂t. 
Assuming A as centre, we draw a spherical arc of radius AA’ (= v₂t) and draw tangent B’A’ on this arc from B’. As the incident wavefront AB advances, the secondary wavelets start from points between A and B’, one after the other and will touch A’B’ simultaneously. 

According to Huygens principle A’B’ is the new position of wavefront AB in the second medium. Hence A’B’ will be the refracted wavefront.

First law: As AB, A’B’ and surface XY are in the plane of paper, therefore the perpendicular drawn on them will be in the same plane. As the lines drawn normal to wavefront denote the rays, therefore we may say that the incident ray, refracted ray and the normal at the point of incidence all lie in the same plane.

This is the first law of refraction.

Second law: Let the angle made by incident wavefront be i and angle made by the refracted wavefront A’B’ be r.

In , 
 
 
                                                     ... (1)

Similarly, in right-angled triangle,  <AA'B,
                                   
Therefore, 
                               
                                    

Now, dividing equation (1) by (2), we have 

The ratio of sine of angle of incidence and the sine of angle of refraction is a constant and is equal to the ratio of velocities of waves in the two media. This is the second law of refraction, and is called the Snell’s law.