Question

A plane wave is incident on a reflecting surface. Using Huygens principle, show how it is reflected from the surface. Hence, verify the law of reflection.

Hint:

When applying Huygens' principle for reflection, remember that because the medium does not change, the speed ($v$) and frequency ($f$) of the wave remain perfectly identical before and after reflection.

Answer 1

Concept: Huygens' Principle states that every point on a primary wavefront acts as a source of secondary spherical wavelets, which spread out in all directions with the speed of the wave in that medium. The new envelope touching these secondary wavelets tangentially at any subsequent instant forms the new secondary wavefront. To prove the law of reflection ($\angle i = \angle r$), we construct a geometric framework using a plane wavefront striking a smooth reflecting barrier.

Step 1: Constructing the geometry of incidence.
Let $MN$ be a smooth plane reflecting surface. Consider a plane wavefront $AB$ incident obliquely on this surface at an angle $i$. The wavefront first touches the reflecting surface at point $A$ at time $t = 0$. The other end of the wavefront, point $B$, is still traveling through space and takes a finite time $t$ to hit the surface at point $C$. If $v$ is the velocity of the wave wavelets in the medium, the distance traveled by the wavefront from $B$ to $C$ in time $t$ is: $$\text{Distance } BC = v \cdot t \quad \cdots (1)$$

Step 2: Activating Huygens' wavelets at the surface.
According to Huygens' principle, as soon as the wavefront hits point $A$, point $A$ starts acting as a source of secondary wavelets. In the same time interval $t$ during which the incident disturbance travels from $B$ to $C$, the secondary spherical wavelet originating from point $A$ expands to a radius equal to: $$\text{Radius } AD = v \cdot t \quad \cdots (2)$$ To find the new reflected wavefront, we draw a sphere of radius $vt$ centered at $A$, and then construct a tangential line or plane $CD$ from point $C$ to this spherical arc. This tangent plane $CD$ represents the newly formed reflected wavefront.

Step 3: Proving congruence of the wave triangles.
Let us examine the two right-angled triangles formed on the base surface $AC$: $\triangle ABC$ and $\triangle ADC$.
• $\angle ABC = \angle ADC = 90^\circ$ (Rays are always perpendicular to their respective wavefront surfaces $AB$ and $CD$).
• $\text{Hypotenuse } AC = AC$ (Common base side shared by both triangles).
• $\text{Side } BC = AD = v \cdot t$ (From equations 1 and 2, distances are equal). By the Right angle-Hypotenuse-Side (RHS) congruence criterion: $$\triangle ABC \cong \triangle ADC$$ Since the triangles are congruent, their corresponding matching angles must be exactly equal by CPCTC (Corresponding Parts of Congruent Triangles are Congruent): $$\angle BAC = \angle DCA \quad \cdots (3)$$ From the definition of angles in the diagram:
• $\angle BAC = i$ (Angle of incidence, which is the angle between the incident wavefront and the reflecting surface).
• $\angle DCA = r$ (Angle of reflection, which is the angle between the reflected wavefront and the reflecting surface). Substituting these definitions into equation (3): $$i = r$$ This proves that the angle of incidence is equal to the angle of reflection, successfully verifying the fundamental law of reflection using wave theory.