Huygens’ principle

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	1
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

A wavefront is a surface over which a wave disturbance has a constant phase. As an illustration, consider a small portion of a spherical wavefront emanating from a monochromatic point source S in a homogeneous medium. Clearly, if the radius of the wavefront at a given time is r, then at some later time t, the radius will simply be $r{\rm {+}}vt$ where v is the phase velocity of the wave. But suppose instead that the wave passes through a nonuniform sheet of material so that the wavefront itself is distorted. How can we determine its new form? Or for that matter, what will the waveform look like at some later time if it is allowed thereafter to continue unobstructed? The major step toward the solution of this problem appeared in print in 1690 in Traité de la Lumière (Huygens, 1690), which had been written 12 years earlier by the Dutch physicist, Christiaan Huygens (Figures 8 and 9). In that work, Huygens enunciated what has become known as Huygens’ principle (Robinson and Clark, 2006b).

At about the same time that Newton was emphasizing the particle theory of light in England, Christiaan Huygens on the continent was greatly extending the wave theory. Unlike Descartes, Hooke, and Newton, Huygens correctly concluded that light effectively slows down on entering denser media. He derived the laws of reflection and refraction and even explained with his wave theory the double-refraction phenomenon observed in a specimen of the mineral calcite (Robinson and Clark, 1986). While Huygens was working with calcite, he discovered the phenomenon called polarization. In his words, “as there are two different refractions, I conceived also that there are two different emanations of the waves of light” (Huygens, 1690, Chapter 5, Section 18).

Figure 8.  Christiaan Huygens, after a painting in the Haags Gemeentemuseum, the Hague.

Figure 9.  Pictures from Traité de la Lumière. From Huygens (1690).

Huygens’ principle embodies a simple but powerful physical concept that serves to explain in a lucid manner how an arbitrary wavefront progresses from one point to another (Robinson and Clark, 2006b). According to this hypothesis, each individual point on an advancing wavefront is disturbed. It therefore acts as a source of a distinct expanding disturbance of its own. Thus, each point of the wavefront at time t excites a secondary spherical wavelet that travels outward in all directions from this point. At a later instant $t{\rm {+}}\Delta t$, the new composite wavefront is the summation of the separate effects of all the wavelets originating in the earlier wavefront at the earlier time t. Graphically, the advanced wavefront is the envelope of all the individual wavelets. Figure 10 shows three examples of such summations.

The generality and flexibility of Huygens’ principle and the fact that it lends itself readily to a geometric construction are its chief merits. By its use, many problems that might otherwise be very difficult to understand can be visualized readily. When irregular discontinuities exist in a nonuniform medium, they give rise to complex reflection, refraction, and diffraction phenomena. Huygens’ principle is particularly useful for treating problems of this type. In the next section, we use it to derive the laws of reflection and refraction at an interface.

In summary, Huygens’ principle states that every point on a primary wavefront serves as the source of spherical secondary wavelets and that the primary wavefront at some later time is the envelope of these spherical wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wavelet at each point in space. If the medium is homogeneous, the wavelets can be constructed with finite radii, whereas if the medium is inhomogeneous, the wavelets must have infinitesimal radii.

Thus far, we merely have stated Huygens’ principle without any justification or proof of its validity. Figure 11 shows a wavefront as well as several spherical secondary wavelets that after a time $\Delta t$ have propagated out to a radius of $v\Delta t$. The envelope of all of these wavelets then is claimed to be the advanced wavefront. This figure represents a graphic application of the essential ideas and as such is known as Huygens’ construction. However, as we see in the figure, there is also an envelope of the secondary wavelets to the rear of the wavefront.

The elementary statement of Huygens’ principle, as given above, is more intuitive than it is rigorous. Because we drew the wavelets as spheres, we might conclude that a backward-traveling waveform also exists, but that is not observed. It is not at all clear why the wavelets do not combine to produce a wavefront that proceeds backward as well as the wavefront that progresses forward. Nor is it obvious, without discussing phase relationships and concepts going beyond the geometric theory, just why the new wavefront coincides only with the geometric envelope of the wavelets. It is evident that we need a slight reformulation of the principle.

Figure 10.  (a) A plane wavefront at a certain instant in time. At a later instant, the advanced wavefront is the envelope of all the small wavelets excited by each point of disturbance on the original wavefront. (b) A similar construction for an advancing spherical wavferont. (c) It is evident that such a construction will be valid for a wavefront of any arbitrary shape, such as the one shown here.

Figure 11.  A wavefront, an advanced wavefront, and a nonexistent backward wavefront.

These difficulties were taken care of theoretically by Fresnel and Kirchhoff, who showed that only the forward-moving wave exists. Fresnel (1818) further analyzed Huygens’ principle, and Kirchhoff (1883) showed that Huygens’ principle results directly from the wave equation, thereby putting it on a firm mathematical basis. We are justified to use merely the forward wavefronts when we apply Huygens’ construction.

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Table of Contents (book)

Also in this chapter

• Introduction
• Wavefronts and raypaths
• d’Alembert’s solution
• One-dimensional waves
• Sinusoidal waves
• Phase velocity
• Wave pulses
• Geometric seismology
• The speed of light
• Reflection and refraction
• Ray theory
• Fermat’s principle
• Fermat’s principle and reflection and refraction
• Diffraction
• Analogy
• Appendix A: Exercise

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