** Huygens Principle**

• Key Idea: Each point on a wavefront acts as a new source of secondary waves, and these waves interfere to produce the new wavefront.

-Visualization:

Imagine dropping a pebble in a calm pond. The waves that spread out from the point of impact are secondary waves. Each point on the wavefront acts as a new source of secondary waves, which spread out in all directions. The envelope of these secondary waves gives the new wavefront.

      O        O         O
     / \      / \       / \
    /   \    /   \     /   \
   /_____\_/_____\_/_____\_
              O              O

-Mathematical Description:

The Huygens principle can be mathematically described by the Fresnel-Kirchhoff integral, which gives the amplitude (U(P,t)) of a wave at a point (P) in terms of the amplitude (U(Q)) of the wave at all points (Q) on a surface (S): $$U(P,t)=\frac{1}{4\pi}\iint\limits_S \frac{U(\mathbf{Q},t_r)}{r}\cos(\mathbf{n},\hat{\mathbf{r}})d\sigma$$

where

• (d\sigma) is an area element of (S) -(\mathbf{r}= \overrightarrow{QP}) is the vector from (Q) to (P) -(\hat{\mathbf{r}}= \frac{\overrightarrow{QP}}{|\overrightarrow{QP}|}) is the unit vector in the direction of (r) -(t_r= t- \frac{r}{v}) is the retarded time -(v) is the speed of the wave

• (r=|\overrightarrow{QP}|) is the distance from (Q) to (P).

• Applications:

• The Huygens principle can explain the laws of reflection and refraction.

• The Huygens principle can also explain the phenomena of diffraction and interference.