Translations:Reflection and refraction – book/3/en

In Figure 12, ST is a plane interface (at right angles to the paper) separating two media. Line PA is perpendicular to this plane. Line AD is the trace (in the plane of the paper) of a plane wave at right angles to the paper and incident on the interface ST at the angle ${\theta }_{i}$. Lines EA and FB are two parallel rays of the incident wave. At the instant that the point A on the incident wavefront strikes the interface ST, a wavelet is initiated, in accordance with Huygens’ principle. Similarly, a new wavelet is initiated at each successive point on ST as the incident wavefront AD advances and strikes ST. Finally, when point D strikes the interface ST at B, the wavelet MCN, with center at A, has the radius AC = DB. The radius of the wavelet at B at that particular instant is zero. Because the line BC is tangent to all the wavelets, it constitutes their envelope. Therefore, BC is a wavefront of the reflected wave. Because BC is tangent to the wavelet MCN at C, it follows that BC is perpendicular to the radius AC. Thus, AG is a reflected ray associated with the reflected wavefront BC. In addition, the right triangles ABC and BAD are similar, so that angle ABC equals angle BAD. But angle ABC equals angle ${\theta }_{r}$, and angle BAD equals angle ${\theta }_{i}$. Therefore, angle ${\theta }_{r}$, the angle of reflection, equals angle ${\theta }_{i}$, the angle of incidence.