The Wavefront Setup
A wavefront is a surface of constant phase: the locus of points that oscillate together. Energy travels perpendicular to the wavefront, and the direction of that perpendicular is what we call a ray. At a large distance from a source, a small patch of a spherical wavefront is effectively flat, and we treat it as a plane wave. This is the object Huygens' construction works on at an interface.
Huygens' principle states that every point on a wavefront is the source of secondary spherical wavelets that spread with the wave speed of the medium; the new wavefront at a later time is the forward common tangent (envelope) to these wavelets. When the wave crosses from medium 1 into medium 2, the only thing that changes is the radius the wavelets reach in a fixed time, because the speed differs. Holding the time the same across both media while the speeds differ is the entire engine behind the laws of refraction and reflection.
| Symbol | Meaning | Defining relation |
|---|---|---|
v1, v2 | Wave speed in medium 1 and 2 | Set by the medium |
i, r | Angle of incidence, angle of refraction | Measured from the normal |
n1, n2 | Refractive indices | $n = c/v$ |
BC | Distance foot of wavefront travels in medium 1 in time $t$ | $BC = v_1 t$ |
AE | Radius of secondary wavelet in medium 2 in time $t$ | $AE = v_2 t$ |
Refraction of a Plane Wave
Let the surface $PP'$ separate medium 1 (speed $v_1$) from medium 2 (speed $v_2$). A plane wavefront $AB$ arrives at the interface at angle of incidence $i$. The end $A$ has just reached the interface while the end $B$ still has the distance $BC$ to cover in medium 1. In a time $t$, the foot travels $BC = v_1 t$ to land at $C$. During that same interval, the secondary wavelet launched from $A$ has already entered medium 2 and spread out a radius $AE = v_2 t$.
To find the refracted wavefront we draw a sphere of radius $v_2 t$ centred at $A$ in the second medium and draw the tangent plane from $C$ to that sphere. That tangent, $CE$, is the new refracted wavefront. The refracted ray is the normal to $CE$, making angle $r$ with the interface normal.
A plane wave $AB$ incident at angle $i$ refracts at $PP'$. With $v_2 < v_1$, the wavelet radius $AE = v_2t$ is shorter than $BC = v_1t$, so the refracted wavefront $CE$ tilts and the ray bends toward the normal ($r < i$). Adapted from NCERT Fig. 10.4.
Snell's Law from Geometry
Both triangles $ABC$ and $AEC$ share the hypotenuse $AC$ lying along the interface. In triangle $ABC$ the side opposite angle $i$ is $BC$; in triangle $AEC$ the side opposite angle $r$ is $AE$. Reading the sines directly off the geometry:
$$\sin i = \frac{BC}{AC} = \frac{v_1 t}{AC}, \qquad \sin r = \frac{AE}{AC} = \frac{v_2 t}{AC}$$
Dividing the two and cancelling $t$ and $AC$ gives the central result of the construction:
$$\frac{\sin i}{\sin r} = \frac{v_1}{v_2}$$
If $r < i$ (the ray bends toward the normal), then $v_2 < v_1$: the wave is slower in the second medium. This wave-theory prediction is the exact opposite of the corpuscular model, and it was confirmed experimentally by Foucault in 1850 when light was measured to be slower in water than in air. Introducing the refractive index $n = c/v$, with $n_1 = c/v_1$ and $n_2 = c/v_2$, the relation becomes:
$$n_1 \sin i = n_2 \sin r$$
This is Snell's law, now derived from wavefronts rather than postulated.
The secondary-wavelet idea used here is set up in full in Huygens' Principle.
Frequency, Speed and Wavelength
Track a single crest. If the crest from $B$ reaches $C$ in time $t$, the crest from $A$ must reach $E$ in the same time $t$. So if $BC$ equals one wavelength $\lambda_1$ in medium 1, then $AE$ equals one wavelength $\lambda_2$ in medium 2. Therefore:
$$\frac{\lambda_1}{\lambda_2} = \frac{BC}{AE} = \frac{v_1}{v_2}$$
The number of wavefronts crossing the interface per second cannot change, so the frequency stays the same. With $v = \nu \lambda$ and $\nu$ fixed, the wavelength scales exactly as the speed. Entering a denser medium ($v_1 > v_2$): speed decreases, wavelength decreases, frequency unchanged.
Frequency vs. speed vs. wavelength on refraction
Three quantities, three different behaviours, and exam stems mix them deliberately. The colour of light is set by frequency, which is fixed by the source; refraction never changes it. Speed and wavelength both change, and they change in the same ratio $v_1/v_2$.
Lock it in: air → glass means $v\downarrow$, $\lambda\downarrow$, $\nu$ unchanged. Wavefronts bend toward the normal because the wave slows in the denser medium.
| Quantity | Air → denser medium | Reason |
|---|---|---|
| Frequency $\nu$ | Unchanged | Forced oscillation of atoms takes up source frequency |
| Speed $v$ | Decreases | $v = c/n$ and $n$ is larger |
| Wavelength $\lambda$ | Decreases | $\lambda = v/\nu$, $v\downarrow$ with $\nu$ fixed |
| Bending | Toward normal | $r < i$ since $v_2 < v_1$ |
Refraction at a Rarer Medium
Reverse the situation: light passes into a rarer medium, so $v_2 > v_1$. The identical construction now produces a wavelet radius $AE = v_2 t$ that is larger than $BC$, so the refracted wavefront tilts the other way and the ray bends away from the normal, giving $r > i$. Snell's law $n_1 \sin i = n_2 \sin r$ still holds.
As $i$ grows, $r$ grows faster and reaches $90^\circ$ at a special angle of incidence called the critical angle $i_c$, defined by:
$$\sin i_c = \frac{n_2}{n_1}$$
When $i = i_c$, $\sin r = 1$ so $r = 90^\circ$ and the refracted wavefront grazes along the interface. For $i > i_c$ no refracted wavefront can be constructed at all, and the wave undergoes total internal reflection.
Into a rarer medium the ray bends away from the normal ($r > i$). At $i = i_c$ the refracted ray grazes the surface; beyond it, total internal reflection. Based on NCERT Fig. 10.5 and §10.3.2.
Reflection of a Plane Wave
For reflection both wavefronts stay in the same medium, so there is a single speed $v$. A plane wave $AB$ strikes a reflecting surface $MN$ at angle $i$. The foot $B$ advances to $C$ in time $t$, covering $BC = vt$. At the same time the wavelet from $A$ spreads a radius $vt$ back into the medium. Drawing the tangent plane $CE$ from $C$ to that sphere gives the reflected wavefront, with $AE = BC = vt$.
Compare triangles $EAC$ and $BAC$: each has a right angle, they share the side $AC$, and $AE = BC$. They are therefore congruent, which forces the angle of incidence $i$ to equal the angle of reflection $r$. That is the law of reflection, obtained without any new assumption beyond the wavelet construction.
Reflection of plane wave $AB$ at surface $MN$. Since $AE = BC = vt$ and the triangles $EAC$ and $BAC$ are congruent, $i = r$. Adapted from NCERT Fig. 10.6.
One consequence ties the chapter together: along any ray, the total time from a point on the object to the corresponding point on the image is the same. When a convex lens focuses light, the central ray travels a shorter geometric path but moves slower through the thicker glass, so it arrives at the same instant as a ray near the rim. The wavefront picture makes this equal-time principle automatic.
Worked Numbers
Monochromatic light of wavelength 600 nm travels from air ($n_1 \approx 1$) into glass of refractive index $n_2 = 1.5$. Find its speed, wavelength and frequency in glass. Take $c = 3.0 \times 10^8\ \text{m s}^{-1}$.
Speed: $v_2 = c/n_2 = (3.0\times10^8)/1.5 = 2.0\times10^8\ \text{m s}^{-1}$.
Wavelength: $\lambda_2 = \lambda_1 (v_2/v_1) = \lambda_1/n_2 = 600/1.5 = 400\ \text{nm}$.
Frequency: unchanged. $\nu = c/\lambda_1 = (3.0\times10^8)/(600\times10^{-9}) = 5.0\times10^{14}\ \text{Hz}$, the same in glass.
Speed and wavelength fall to two-thirds of their air values; frequency holds steady — exactly the pattern Huygens' construction predicts.
Refraction and reflection from wavefronts
- In time $t$ the wavelet radius is $v_2 t$ in medium 2 while the foot covers $v_1 t$ in medium 1; the common tangent is the refracted wavefront.
- Geometry gives $\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2}$, hence $n_1 \sin i = n_2 \sin r$ (Snell's law).
- Bending toward the normal ($r < i$) means $v_2 < v_1$ — the wave slows in the denser medium.
- On refraction frequency is unchanged; speed and wavelength change in the same ratio $v_1/v_2 = \lambda_1/\lambda_2$.
- Rarer medium: ray bends away from normal; critical angle $\sin i_c = n_2/n_1$; beyond it, total internal reflection.
- For reflection, $AE = BC = vt$ makes triangles congruent, so $i = r$.