Refraction and Reflection of Plane Waves using Huygens Principle

Concept First

Huygens Principle explains how a wavefront propagates. Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront after a time $t$ is the tangent envelope to all these secondary wavelets. Using this idea, we can derive the laws of reflection and refraction geometrically.

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Refraction of a Plane Wave

Consider a plane wave (wavefront $AB$) incident at an angle on a plane surface separating two media (1 and 2). Let the wave speed in medium 1 be $v_1$ and in medium 2 be $v_2$.

• At time $t = 0$, point $A$ on the incident wavefront touches the boundary.
• Point $B$ reaches the boundary after time $\tau$, where $\tau = \frac{BC}{v_1}$.
• During this time $\tau$, the secondary wavelet from $A$ travels into medium 2 a distance $v_2 \tau$.
• The new wavefront in medium 2 is the tangent from $C$ to the circle of radius $v_2 \tau$ centered at $A$. This tangent touches the circle at point $E$.

From geometry:

• In $\triangle ABC$: $\sin i = \frac{BC}{AC} = \frac{v_1 \tau}{AC}$
• In $\triangle AEC$: $\sin r = \frac{AE}{AC} = \frac{v_2 \tau}{AC}$

Dividing the two equations gives Snell's law:

The refractive index of medium 2 relative to medium 1 is defined as:

Thus, Snell's law becomes:

If $n_1$ and $n_2$ are absolute refractive indices (with respect to vacuum), then $n_1 = \frac{c}{v_1}$ and $n_2 = \frac{c}{v_2}$, so:

Hence:

Key point: The frequency of the wave remains unchanged during refraction. Only speed and wavelength change.

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Reflection of a Plane Wave

Consider a plane wave incident on a reflecting surface. Using the same Huygens construction:

• At $t=0$, point $A$ touches the reflector.
• Point $B$ reaches the reflector after time $\tau = \frac{BC}{v}$.
• During $\tau$, the secondary wavelet from $A$ travels back into the same medium a distance $v\tau$.
• The new reflected wavefront is the tangent from $C$ to the circle of radius $v\tau$ centered at $A$.

From geometry: …