10.2 Huygens Principle

We would first define a wavefront: when we drop a small stone on a calm pool of water, waves spread out from the point of impact. Every point on the surface starts oscillating with time. At any instant, a photograph of the surface would show circular rings on which the disturbance is maximum. Clearly, all points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points, which oscillate in phase is called a wavefront; thus a wavefront is defined as a surface of constant phase. The speed with which the wavefront moves outwards from the source is called the speed of the wave. The energy of the wave travels in a direction perpendicular to the wavefront.

10-1a

A diverging spherical wave emanating from a point source. The wavefronts are spherical

If we have a point source emitting waves uniformly in all directions, then the locus of points which have the same amplitude and vibrate in the same phase are spheres and we have what is known as a spherical wave as shown in Fig. 10.1(a). At a large distance from the source, a small portion of the sphere can be considered as a plane and we have what is known as a plane wave [Fig. 10.1(b)].

10-1b

At a large distance from the source, a small portion of the spherical wave can be approximated by a plane wave.

Now, if we know the shape of the wavefront at t = 0, then Huygens principle allows us to determine the shape of the wavefront at a later time τ . Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wafefront at any time allows us to determine the shape of the wavefront at a later time. Let us consider a diverging wave and let F 1 F 2 represent a portion of the spherical wavefront at t = 0 (Fig. 10.2). Now, according to Huygens principle, each point of the wavefront is the source of a secondary disturbance and the wavelets emanating from these points spread out in all directions with the speed of the wave. These wavelets emanating from the wavefront are usually referred to as secondary wavelets and if we draw a common tangent to all these spheres, we obtain the new position of the wavefront at a later time.

10-2

F_{1}

F_{2}

represents the spherical wavefront (with O as centre) at t = 0. The envelope of the secondary wavelets emanating from

F_{1}

F_{2}

produces the forward moving wavefront

G_{1}

G_{2}

. The backwave

D_{1}

D_{2}

does not exist.

Thus, if we wish to determine the shape of the wavefront at $t = τ$ , we draw spheres of radius $v τ$ from each point on the spherical wavefront where v represents the speed of the waves in the medium. If we now draw a common tangent to all these spheres, we obtain the new position of the wavefront at $t = τ$ . The new wavefront shown as $G_{1}$ $G_{2}$ in Fig. 10.2 is again spherical with point O as the centre.

The above model has one shortcoming: we also have a backwave which is shown as $D_{1}$ $D_{2}$ in Fig. 10.2. Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this adhoc assumption, Huygens could explain the absence of the backwave. However, this adhoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory.

In a similar manner, we can use Huygens principle to determine the shape of the wavefront for a plane wave propagating through a medium (Fig. 10.3).

10-3

Huygens geometrical construction for a plane wave propagating to the right.

F_{1}

F_{2}

is the plane wavefront at t = 0 and

G_{1}

G_{2}

is the wavefront at a later time τ . The lines A

1_{}

A_{2}

B_{1}

B_{2}

… etc., are normal to both

F_{1}

F_{2}

and

G_{1}

G_{2}

and represent rays