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 drawing this diagram in an exam, the absolute most critical detail is rigorously ensuring that the line segment $AD$ is drawn completely perpendicular to the line $CD$, forming a clean $90^\circ$ tangent.

Answer 1

Concept:
• Huygens principle boldly states that every single point on an advancing wavefront acts as a fresh source of secondary spherical wavelets.
• The new, updated wavefront at any later time is formed geometrically by taking the forward envelope (common tangent) of all these expanding secondary wavelets.

Step 1: Establish the geometric setup
Imagine a completely flat, plane wavefront named $AB$ propagating steadily through a medium with wave speed $v$.
This wavefront obliquely strikes a perfectly flat reflecting surface $XY$ at an angle of incidence $i$.
Point $A$ of the wavefront firmly touches the reflecting mirror surface first. At this precise instant, point $B$ is still physically distant from the mirror, separated by a distance $BC$.

Step 2: Apply Huygens construction over time $t$
It precisely takes time $t = \frac{BC}{v}$ for the trailing edge of the wavefront at point $B$ to physically travel and strike the mirror surface at point $C$.
During this exact same time interval $t$, the secondary spherical wavelet originating from point $A$ has been expanding rapidly outward into the same upper medium.
The radius of this newly generated expanding spherical wavelet from $A$ is precisely $AD = vt$.
Because both events happen in the exact same medium over the same time, it is an absolute geometric certainty that distance $AD$ perfectly equals distance $BC$ ($AD = BC = vt$).

Step 3: Construct the reflected wavefront
To completely trace the new reflected wavefront, we boldly draw a common tangent line from point $C$ directly to the edge of the spherical wavelet at point $D$.
The resulting straight line segment $CD$ actively represents the new plane reflected wavefront.
The angle between this outgoing reflected wavefront $CD$ and the mirror surface $XY$ geometrically defines the angle of reflection $r$.

Step 4: Prove the law of reflection mathematically
We carefully inspect the two right-angled triangles rigidly formed on the mirror surface: $\Delta ABC$ and $\Delta ADC$.
In these two distinct triangles:
Side $AC$ is completely shared as the common hypotenuse ($AC = AC$).
Side $BC$ perfectly equals side $AD$ ($BC = AD = vt$), as established by uniform wave propagation.
The angle $\angle ABC = 90^\circ$ (since $AB$ is the incident wavefront and ray is normal).
The angle $\angle ADC = 90^\circ$ (since $CD$ is a tangent to the spherical wavelet at $D$).
By the RHS (Right-Angle Hypotenuse Side) congruence criterion, $\Delta ABC \cong \Delta ADC$ perfectly.
Because the triangles are strictly congruent, their corresponding matching angles must be exactly equal.
Therefore, $\angle BAC = \angle DCA$.
By geometric construction, $\angle BAC = i$ and $\angle DCA = r$.
This rigorously proves that the angle of incidence equals the angle of reflection ($i = r$), flawlessly verifying the universal law of reflection.