Huygens Principle

A wavefront is a surface of constant phase.

If a point source emits waves uniformly in all directions, the locus of points which have the same amplitude and vibrate in the same phase is a sphere. This is known as a spherical wave.

At a large distance from the source, a small portion of the spherical wave can be considered as a plane. This is known as a plane wave.

• Christiaan Huygens (1629–1695) was a Dutch physicist, astronomer, mathematician, and the founder of the wave theory of light.
• His book, Treatise on Light, is cited as an important work.
• He explained double refraction shown by calcite, in addition to reflection and refraction.
• He was the first to analyse circular and simple harmonic motion and designed improved clocks and telescopes.
• He discovered the true geometry of Saturn’s rings.

• When a small stone is dropped 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.
• 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.
• 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.

Main idea

• If the shape of the wavefront at t = 0 is known, Huygens' principle allows the shape of the wavefront at a later time τ to be determined.
• Huygens' principle is essentially a geometrical construction.
• Given the shape of the wavefront at any time, it allows the shape of the wavefront at a later time to be determined.

Statement:

• According to Huygens' principle, each point of the wavefront is the source of a secondary disturbance.
• The wavelets emanating from these points spread out in all directions at the speed of the wave.
• These wavelets are referred to as secondary wavelets.
• If a common tangent is drawn to all these spheres, the new position of the wavefront at a later time is obtained.

• Let F₁F_2​ represent a portion of the spherical wavefront at t = 0.
• Each point of this wavefront acts as the source of a secondary disturbance.
• To determine the shape of the wavefront at t = τ, spheres of radius vτ are drawn from each point on the spherical wavefront, where v represents the speed of the wave in the medium.
• A common tangent drawn to all these spheres gives the new position of the wavefront at t = τ.
• The new wavefront, shown as G₁G₂, is again spherical with point O as the centre.

• The model has one shortcoming: it also gives a backwave.
• 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 explained the absence of the backwave.
• However, this adhoc assumption is not satisfactory.
• The absence of the backwave is really justified by more rigorous wave theory.

• In a similar manner, Huygens' principle can be used to determine the shape of the wavefront for a plane wave propagating through a medium.
• Huygens' geometrical construction for a plane wave propagating to the right.
• F₁F₂​ is the plane wavefront at t = 0.
• G₁G₂ is the wavefront at a later time τ.
• The lines A₁A₂, B₁B₂, and others are normal to both F₁F₂​ and G₁G₂ and represent rays.