What is a wavefront?
Light is a wave. At any moment, join all the points where the wave is in the same phase (for example, all the crests). The surface you get is a wavefront. The wavefront moves outward with the speed of the wave.
- Spherical wavefront: from a point source; the wavefronts are spheres centred on the source.
- Cylindrical wavefront: from a line source such as a narrow slit lit from behind.
- Plane wavefront: from a very distant source (like the Sun); a small part of a very large sphere is almost flat.
A ray is the direction the wave travels. It is always at right angles (normal) to the wavefront. Parallel rays = plane wavefront; spreading rays = spherical wavefront.
Huygens' principle
Christiaan Huygens (1678) gave a simple way to find where a wavefront will be after a time t:
- Every point on a wavefront acts as a new source of secondary wavelets. These spread in the forward direction with the speed of the wave.
- After time t, each wavelet has radius v t. The surface that touches all of them in the forward direction (the forward envelope) is the new wavefront.
Why not a backward wave? Huygens simply ignored it; later theory (by Fresnel and Kirchhoff) showed that the backward wave cancels out.
Reflection of a plane wave: proof of i = r
A plane wavefront AB meets a mirror MN at angle i. End A touches the mirror first.
Step 1
B still has to travel distance BC to reach the mirror. It takes time t, so BC = v t.
Step 2
In that same time, the wavelet from A grows to radius AE = v t (in the same medium, same speed).
Step 3
Draw the tangent CE from C to this wavelet. CE is the reflected wavefront.
Step 4
Triangles ABC and CEA are both right-angled (at B and E), share the side AC, and have BC = AE. So they are congruent, which gives angle BAC = angle ECA.
The angle between the incident wavefront and the mirror is i; the angle between the reflected wavefront and the mirror is r. So i = r. Both rays and the normal lie in one plane, which is the second law.
Refraction of a plane wave: proof of Snell's law
A plane wavefront AB in medium 1 (speed v₁) meets the surface into medium 2 (speed v₂) at angle i.
Step 1
End B travels BC = v₁ t in medium 1 to reach the surface.
Step 2
Meanwhile, the wavelet from A travels in medium 2: AE = v₂ t.
Step 3
The tangent CE is the refracted wavefront. It makes angle r with the surface.
Step 4
In triangle ABC: sin i = BC/AC = v₁t/AC. In triangle AEC: sin r = AE/AC = v₂t/AC.
Step 5
sin i / sin r = v₁ / v₂. Since n = c/v, v₁/v₂ = n₂/n₁, so n₁ sin i = n₂ sin r — Snell's law.
If v₂ < v₁ (denser medium), r < i: the ray bends towards the normal. The frequency stays the same in both media (the source decides it), so the wavelength changes: λ₂ = λ₁ v₂/v₁.
Denser to rarer, and what stays the same
If light goes from a slower (denser) medium into a faster one, A's wavelet in medium 2 is bigger than BC, so r > i. When the wavelet from A is so big that the tangent from C cannot be drawn (v₂t > AC), no refracted wavefront forms — this is total internal reflection, and the limit gives sin C = v₁/v₂.
Wavefronts also explain how lenses and mirrors work: a convex lens slows the middle of a plane wavefront more (thicker glass), so the wavefront curves and converges to the focus.
Try it: ripples in a tray
Fill a flat steel plate or tray with a little water. Touch the centre with a fingertip: circles spread — spherical (circular) wavefronts. Now dip a ruler along one edge and push gently: straight ripples — plane wavefronts. Put a thick glass plate in one half to make the water shallow there: the ripples slow down and turn as they cross, just like refraction. In the 3D, move the time slider and watch the wavelets build the new wavefront.
Key formulas and definitions
- Ray ⟂ wavefront
- New wavefront = forward envelope of wavelets of radius v t
- Reflection: i = r
- Refraction: sin i / sin r = v₁/v₂ = n₂/n₁ = λ₁/λ₂
- Frequency same in both media; λ₂ = λ₁ v₂/v₁ = λ₁ n₁/n₂
- n = c / v
Worked examples
1. What shape of wavefront comes from (a) a small bulb, (b) a lit narrow slit, (c) the Sun?
Step 1: a point source gives spherical wavefronts. Step 2: a line source (slit) gives cylindrical wavefronts. Step 3: the Sun is very far, so the part reaching us is a plane wavefront.
2. Light of wavelength 600 nm in air enters water (n = 4/3). Find its speed, wavelength and frequency in water.
Step 1: frequency f = c/λ = 3 × 10⁸ / 600 × 10⁻⁹ = 5 × 10¹⁴ Hz (same in water). Step 2: speed v = c/n = 2.25 × 10⁸ m/s. Step 3: λ = λ₀/n = 600 × 3/4 = 450 nm.
3. A plane wave in air hits glass (n = 1.5) at 60°. Using the wavefront result, find the angle of refraction.
Step 1: sin i / sin r = v₁/v₂ = n₂/n₁ = 1.5. Step 2: sin r = sin 60°/1.5 = 0.866/1.5 = 0.577. Step 3: r ≈ 35.3°.
4. In the Huygens construction for refraction, BC = 6 cm in air. How far does A's wavelet travel in glass (n = 1.5) in the same time?
Step 1: AE/BC = v₂/v₁ = 1/1.5. Step 2: AE = 6/1.5 = 4 cm. The wavelet in glass is smaller, so the wavefront turns.
5. Light goes from glass (v = 2 × 10⁸ m/s) into water (v = 2.25 × 10⁸ m/s). Find the critical angle using wavefronts.
Step 1: sin C = v₁/v₂ (slower to faster). Step 2: = 2/2.25 = 0.889. Step 3: C ≈ 62.7°.
6. A convex lens receives a plane wavefront. Explain with Huygens' idea what shape leaves it.
Step 1: the middle of the wavefront passes through the thickest glass and is slowed the most. Step 2: the edges pass through thin glass and get ahead. Step 3: so the wavefront bends into a spherical shape curving towards a point — it converges to the focus.
Common mistakes
- Thinking the wavelength and frequency both change in a new medium. Frequency stays the same; only speed and wavelength change.
- Drawing rays parallel to the wavefront. Rays are always at right angles to it.
- Writing sin i / sin r = v₂/v₁. It is v₁/v₂ (speed in the first medium on top).
- Including the backward envelope as a new wavefront. Only the forward envelope counts.