Question

State Huygens principle. How did Huygens justify the absence of the backwave on a spherical wavefront ?

Hint:

Rigorous electromagnetic wave theory developed later by Kirchhoff proved that Huygens' obliquity factor $\frac{1}{2}(1 + \cos \theta)$ naturally arises from Maxwell's equations.

Answer 1

Concept:
• Huygens' Principle provides a geometrical construction to determine the position and shape of a wave front at a later instant from its known position at an earlier instant.

Step 1: Statement of Huygens' Principle
Huygens' Principle rests on two fundamental postulates:
1. Primary Wavefront as Secondary Sources: Every point on a given primary wavefront acts as a fresh source of secondary disturbance, emitting tiny spherical wavelets called secondary wavelets that spread out in all directions with the speed of light in that medium.
2. New Wavefront Construction: The tangential envelope or forward surface touching all these secondary wavelets in the forward direction at any subsequent time $t$ gives the new position and shape of the wavefront at that instant.

Step 2: Justification for Absence of Backwave
According to Huygens' original model, secondary wavelets radiate in all directions. This mathematically suggests the presence of a backward wavefront (backwave) traveling back towards the source.
To justify why backwaves do not physically exist, Huygens assumed that the amplitude/intensity of secondary wavelets is directional.
The amplitude of a secondary wavelet at an angle $\theta$ with respect to the forward normal of the primary wavefront is proportional to the directivity factor (or obliquity factor):
\[ F(\theta) = \frac{1}{2}(1 + \cos \theta) \]

Step 3: Evaluating directivity factor
- For the forward direction ($\theta = 0^\circ$): $F(0^\circ) = \frac{1}{2}(1 + \cos 0^\circ) = \frac{1}{2}(1 + 1) = 1$ (Maximum amplitude).
- For the backward direction ($\theta = 180^\circ$): $F(180^\circ) = \frac{1}{2}(1 + \cos 180^\circ) = \frac{1}{2}(1 - 1) = 0$ (Zero amplitude).

Step 4: Conclusion
Since the obliquity factor $\frac{1}{2}(1 + \cos \theta)$ vanishes completely at $\theta = 180^\circ$, no secondary wavelets propagate backwards, accounting for the absence of backwave.