Huygens Principle

The Wavefront: A Surface of Constant Phase

Drop a stone in a calm pool: circular ripples spread out. At any instant, all points on one ripple-circle oscillate in phase (they are equidistant from the source). Such a locus is a wavefront:

A wavefront is a surface of constant phase. The wave's speed is the speed at which the wavefront advances; energy travels perpendicular to the wavefront.

Shapes worth knowing on sight:

• Point source → spherical wavefronts (diverging).
• Far from any source (e.g. starlight) → a small portion of the giant sphere is effectively a plane wavefront.
• Line/slit source → cylindrical wavefronts.

Huygens' Principle: The Wavefront-Forecasting Machine

Given the wavefront now, where is it a moment later? Huygens' two-part recipe:

1. Each point of a wavefront acts as a fresh source of secondary wavelets, spreading out in all directions with the speed of the wave.
2. The new wavefront is the forward envelope (common tangent surface) of all these secondary wavelets at the later time.

For a plane wave: wavelets of radius $v\tau$ drawn from every point of the wavefront have a plane forward envelope — the wavefront advances by $v\tau$, staying plane. A sphere stays spherical. Simple, geometric, powerful.

The backwave problem: the same construction also suggests a wavefront travelling backwards — which is never observed. Huygens had to assume the back-directed amplitude is zero (in the rigorous wave theory the obliquity factor makes it vanish). NCERT flags this honestly — and so do examiners.

[NEET Important] One-liners that recur: wavefront ⟂ ray; spherical / plane / cylindrical shapes matched to source type; 'envelope of secondary wavelets' as the exact phrase; the absent backwave.

Solved Examples

Example 1: Shape spotting [NEET pattern]

State the wavefront shape for: (a) a point source, (b) light from a distant star, (c) a long narrow slit, (d) a parallel beam.

Solution:

1. (a) Spherical (diverging).
2. (b) Plane — a tiny patch of an enormous sphere.
3. (c) Cylindrical.
4. (d) Plane (that's what 'parallel beam' means in wave language).

Example 2: Wavefront vs ray

How are rays related to wavefronts?

Solution:

1. Rays are perpendicular to wavefronts, pointing along the energy flow.
2. Spherical wavefronts ↔ radial rays; plane wavefronts ↔ parallel rays.
3. Bend the wavefront (lens, prism) and the rays re-aim accordingly — two pictures, one physics.

Example 3: Advancing a plane wave [JEE Numerical]

A plane wavefront in air advances for 2 ns. Using Huygens' construction, how far does the new wavefront lie from the old, and what is its shape?

Solution:

1. Each point emits a wavelet of radius $v\tau = 3\times10^8 \times 2\times10^{-9} = 0.6$ m.
2. The forward envelope of equal-radius spheres centred on a plane is a parallel plane 0.6 m ahead.
3. Plane stays plane; the construction reproduces straight-line propagation.

Example 4: The sphere stays a sphere

Show via Huygens that a spherical wavefront from a point source remains spherical.

Solution:

1. Every point of the sphere (radius R) emits a wavelet of radius $v\tau$.
2. The forward envelope is the concentric sphere of radius $R + v\tau$.
3. Symmetry does all the work — the construction preserves the shape dictated by the source.

Example 5: Phase on a wavefront

Two points lie on the same wavefront. What is the phase difference between their oscillations, and why?

Solution:

1. Zero — a wavefront is defined as a surface of constant phase.
2. For a point source this is automatic: equal distances from the source mean equal travel times.
3. This zero-phase property is what makes wavefront points legitimate 'in-phase secondary sources' in every later derivation.

Example 6: The missing backwave

Why doesn't Huygens' construction produce a wave travelling backwards?

Solution:

1. The raw construction does suggest a backward envelope.
2. Huygens assumed zero amplitude backwards (no backwave is ever observed).
3. Rigorous wave theory later justified it (the amplitude of secondary wavelets vanishes in the backward direction). Quote the assumption honestly — it earns the mark.

Example 7: Starlight's flatness [JEE Numerical]

A star is $10^{17}$ m away. Over a 1 m telescope aperture, estimate how much its spherical wavefront deviates from a perfect plane.

Solution:

1. Sagitta of an arc: $s \approx \frac{a^2}{2R} = \frac{1}{2\times10^{17}}$.
2. $s = 5\times10^{-18}$ m — about a hundredth of a proton's radius!
3. 'Plane wavefront' is not an approximation to apologise for; it is fabulously exact.

Example 8: Energy direction

In which direction does a wave's energy travel relative to its wavefront, and what does this mean behind a lens?

Solution:

1. Perpendicular to the wavefront, always.
2. A convex lens turns an incoming plane wavefront into a converging spherical one; perpendiculars to those spheres all aim at the focus.
3. Energy follows the perpendiculars — which is exactly why the focus is bright (Section 3 draws these pictures).

Example 9: Wavelets at an aperture (preview) [JEE pattern]

A plane wave hits an opaque screen with a small hole. Using Huygens, what does the emerging wavefront look like?

Solution:

1. Only the wavefront points within the hole act as secondary sources.
2. With few wavelets, the envelope is no longer plane — it bulges and spreads beyond the geometric beam.
3. That spreading IS diffraction (Section 6); Huygens' principle contains it from the start.

Example 10: Why the construction is 'forward-looking'

Summarise Huygens' principle in two sentences suitable for a Board answer.

Solution:

1. Every point on a given wavefront is a source of secondary wavelets, which spread in all directions with the speed of the wave.
2. The new wavefront at a later instant is the forward envelope (common tangent) of all these secondary wavelets; the backward wave is absent (assumed zero amplitude).
3. Two sentences, full marks — and the engine of the next section's derivations.