A wavefront is a surface whose points have the same phase. Huygens' principle gives a geometrical way to propagate a wavefront forward in time:

Every point on the current wavefront can be treated as the source of a small secondary wavelet, and the new wavefront is the envelope tangent to those wavelets after a short time.

If the wave speed is $v$, then after a time $\Delta t$ each secondary wavelet has radius

$$v\Delta t.$$

Plane-wave propagation

For a plane wave in a uniform medium, begin with a straight wavefront. Construct equal-radius wavelets centered at many points along it. Their forward envelope is another parallel wavefront a distance

$$v\Delta t$$

away.

The normal to successive wavefronts defines the ray direction, so this construction reproduces straight-line propagation in a uniform medium.

Spherical-wave propagation

For a point source, the initial wavefront is spherical. Every point advances by the same distance $v\Delta t$, so the next envelope is a larger concentric sphere. This reproduces the outward expansion of a spherical wave.

Why wavefronts bend at an interface

Suppose a plane wave reaches an interface obliquely and enters a medium in which the wave speed is lower. One side of the wavefront enters the slower medium first and begins advancing more slowly while the other side is still moving faster in the first medium.

The new envelope therefore rotates. The ray, being perpendicular to the wavefront, bends toward the normal when the wave slows down and away from the normal when the wave speeds up.

This construction gives a wave interpretation of refraction rather than treating Snell's law as an isolated ray rule.

Geometric relation

During the same time interval $\Delta t$, the wave travels distances

$$v_1\Delta t$$

and

$$v_2\Delta t$$

in the two media. The geometry of the Huygens construction gives

$$\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}.$$

Using the refractive index $n=c/v$,

$$\boxed{n_1\sin\theta_1=n_2\sin\theta_2},$$

which is Snell's law.

Connection with diffraction

At an aperture, different parts of the exposed wavefront contribute secondary waves. Their superposition allows the disturbance to spread into regions that simple straight rays would leave in shadow. Huygens' principle therefore provides useful geometric intuition for diffraction as well as refraction.

The principle is a construction for the evolution of wavefronts. By itself it does not determine the detailed amplitudes of all secondary contributions; more complete wave theories supply that information. Its value is that it connects wave propagation, rays, refraction and diffraction in one reusable geometric picture.