Huygens Principle — Wavefront Propagation, Reflection, Refraction Derivation

Question

How does Huygens’ principle explain wavefront propagation, and how can we derive the laws of reflection and refraction from it?

Solution — Step by Step

Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at a later time is the forward envelope (common tangent) of all these secondary wavelets.

Two key points:

Only the forward envelope is considered (the backward wavefront is ignored — Huygens didn’t fully explain why, but it works)
The speed of the secondary wavelets equals the wave speed in that medium

Consider a plane wavefront hitting a reflecting surface at angle of incidence $i$ . While point A of the wavefront has already reached the surface, point B is still travelling.

Time for B to reach the surface = time for the reflected wavelet from A to spread:

BC = v \cdot t \quad \text{and} \quad AD = v \cdot t

So $BC = AD$ . In triangles $ABC$ and $ADC$ :

Both are right triangles
$AC$ is common
$BC = AD$

Therefore $\angle i = \angle r$ — the law of reflection.

Now the wavefront enters a denser medium (speed changes from $v_1$ to $v_2$ ). In time $t$ :

BC = v_1 t \quad \text{(in medium 1)}

AD = v_2 t \quad \text{(in medium 2)}

From the geometry:

\sin i = \frac{BC}{AC} = \frac{v_1 t}{AC}

\sin r = \frac{AD}{AC} = \frac{v_2 t}{AC}

Dividing:

\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1}

This is Snell’s law: $n_1 \sin i = n_2 \sin r$ .

graph TD
    A[Original Wavefront] --> B[Each point becomes a secondary source]
    B --> C[Secondary wavelets expand as spheres]
    C --> D[Draw forward common tangent]
    D --> E[New Wavefront]

    F[Application] --> G[Reflection: same medium speed, angle i = angle r]
    F --> H[Refraction: different speeds, Snells law emerges]
    F --> I[Diffraction: wavelets spread into shadow region]

Why This Works

Huygens’ construction is powerful because it is purely geometric — no equations of motion needed. By treating every wavefront point as a new source, we capture wave behaviour (including diffraction) that ray optics misses entirely.

The fact that Snell’s law emerges naturally from this construction — with $n = c/v$ — was historically significant. It showed that light bends toward the normal when entering a denser medium because it slows down, not speeds up (as Newton’s corpuscular theory predicted).

For CBSE boards, the Huygens’ derivation of Snell’s law is a guaranteed 5-mark question. Draw the diagram clearly with the wavefront, the interface, and the two triangles sharing the common hypotenuse $AC$ . Label all angles and distances.

Alternative Method

We can also derive Snell’s law using Fermat’s principle of least time — light takes the path that minimises travel time between two points. Minimising $t = \frac{\sqrt{a^2 + x^2}}{v_1} + \frac{\sqrt{b^2 + (d-x)^2}}{v_2}$ with respect to $x$ yields $\frac{\sin i}{v_1} = \frac{\sin r}{v_2}$ , the same result.

Common Mistake

Students often write $n_1/n_2 = v_1/v_2$ — this is wrong. The correct relation is $n_1/n_2 = v_2/v_1$ (refractive index is INVERSELY proportional to speed). A denser medium has a higher $n$ and a LOWER speed. Getting this ratio inverted flips the bending direction and gives a completely wrong answer.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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