Reflection and refraction – book
We now consider phenomena related to the propagation of waves through real media. In particular, we shall study the characteristics of waves as they progress through various materials, crossing interfaces and being reflected and refracted in the process. Many of the basic concepts of wave motion, although they are predicated on the phenomenon’s wave aspects, nevertheless are independent of the exact nature of the wave. As we shall see, the most important such concept is again Huygens’ principle.
When an elastic wave strikes an interface separating two media of different physical characteristics, a reflected wave is created. The reflected wave is present in the same medium as the incident wave, and therefore, they will coexist. The incident wave carries energy toward the interface, whereas the reflected wave carries energy away from it. The energy in the reflected wave necessarily must be derived from the incident wave. This does not imply, however, that all of the incident energy is reflected, because a part of it can be refracted (as a refracted wave) into the second medium. The geometric theory is incapable of determining how the energy in the incident wave is distributed between the reflected ray and the refracted ray, but it can determine a geometric relationship between the angles involved. The angular relationship between the incident ray and the reflected ray now will be derived with an application of Huygens’ principle (Figure 12).
In Figure 12, ST is a plane interface (at right angles to the paper) separating two media. Line PA is perpendicular to this plane. Line AD is the trace (in the plane of the paper) of a plane wave at right angles to the paper and incident on the interface ST at the angle . Lines EA and FB are two parallel rays of the incident wave. At the instant that the point A on the incident wavefront strikes the interface ST, a wavelet is initiated, in accordance with Huygens’ principle. Similarly, a new wavelet is initiated at each successive point on ST as the incident wavefront AD advances and strikes ST. Finally, when point D strikes the interface ST at B, the wavelet MCN, with center at A, has the radius AC = DB. The radius of the wavelet at B at that particular instant is zero. Because the line BC is tangent to all the wavelets, it constitutes their envelope. Therefore, BC is a wavefront of the reflected wave. Because BC is tangent to the wavelet MCN at C, it follows that BC is perpendicular to the radius AC. Thus, AG is a reflected ray associated with the reflected wavefront BC. In addition, the right triangles ABC and BAD are similar, so that angle ABC equals angle BAD. But angle ABC equals angle , and angle BAD equals angle . Therefore, angle , the angle of reflection, equals angle , the angle of incidence.
The complete statement of the law of reflection is now as follows:
1) The reflected ray lies in the plane determined by the incident ray and the normal erected on the reflecting interface at the point of reflection.
2) The angle of reflection is equal to the angle of incidence.
Suppose that a wave impinges on the interface separating two media. As we know, a portion of the incident wave will be diverted back in the form of a reflected wave, whereas the remainder will be transmitted across the boundary as a refracted wave. We now seek to determine the general principles governing or at least describing refraction (Figure 13).
Huygens’ principle also can be used to show how a plane wave is refracted at a plane interface. Let us look at Figure 13, in which ST is the trace of a plane interface (at right angles to the paper) separating two media. The velocity in the upper medium is , and in the lower medium it is . Line AD is the trace of a plane wave at right angles to the paper and incident on the interface ST at the angle . Lines EA and FB are two parallel rays of the incident plane wave. At the instant that point A on the incident wavefront strikes the interface ST, a wavelet is initiated in the lower medium, in accordance with Huygens’ principle. As successive points on AD strike the interface, other wavelets are generated. Finally, when point D on the wavefront AD reaches the interface ST at B, the wavelet MCN, with center at A, has attained the radius AC. Energy takes the same time to go the distance DB as it does to go the distance AC, that is,
Thus, AC is equal to DB times . The wavelet at B has a radius of zero. Because the line BC is tangent to all the wavelets, it constitutes their envelope. Therefore, BC represents a refracted wavefront in the lower medium. Line BC is tangent to the circular wavelet MCN at the point C, and therefore it is perpendicular to the radius AC. It follows that AG is a refracted ray associated with the refracted wavefront BC. Now in triangle BAD, we have
where is the angle of incidence. Also in triangle ABC, we have
where is the angle of refraction (i.e., transmission). It follows that
This is Snell’s law of refraction. The complete statement of the law is as follows:
1) The refracted ray lies in the plane determined by the incident ray and the normal erected on the refracting interface at the point of refraction.
2) The angle of incidence and the angle of refraction are related thusly:
From the construction, we see that the above relationship will hold when the velocities in the two media are interchanged. This is equivalent to saying that the refraction process is reversible in the sense that the same relationship between the angles of incidence and refraction is valid regardless of whether the incident ray is in the upper or lower medium - that is, the incident ray can be directed either upward or downward.
When the lower medium has a lower velocity than the upper medium that contains the incident ray, the ray is bent toward the normal. If the lower medium has a higher velocity, the ray is bent away from the normal. In this latter case, a critical value exists for the angle of incidence for which the angle of refraction is equal to . In this particular case, the refracted ray is parallel to and grazes along the refracting interface. The critical angle is given by the relation , because is equal to unity. If the angle of incidence is greater than the critical angle, no refracted ray exists. This implies that all energy in the incident wave is contained in the reflected wave, and the incident wave is said to be totally reflected.
Let us now summarize. Suppose that we have a plane wave incident on the smooth interface separating two media. We can determine the wave’s behavior using Huygens’ construction. In Figure 14, the angles , , and are the angles of incidence, reflection, and refraction, respectively. As we have seen, the application of Huygens’ principle gives
The first two terms give the law of reflection, . The first and third terms give the law of refraction, . In ancient times, Claudius Ptolemy of Alexandria had found the expression , which is approximately correct for small angles. Kepler very nearly derived the law of refraction, but he was misled by erroneous data. Finally, the correct relationship was derived independently by Snell at Leyden and by the French mathematician/philosopher René Descartes (Robinson and Clark, 1987c). In any case, the law of refraction generally became known as Snell’s law.
In the above derivations, the laws of reflection and refraction have been derived under the assumption that the incident wave is a plane wave and that the interface is a plane surface. However, the incident wave need not be a plane wave. It only needs to be a smooth curved surface - continuous and free of sharp bends or breaks. In such a case, we can concentrate our attention on an infinitesimal area of the wave surface, which can be considered plane. The two incident rays EA and FB then will be very close to each other and essentially will be parallel. Thus, the above derivations for finite wavefronts will hold also for infinitesimal wavefronts. The fact that these laws are stated in terms of geometric relationships between rays rather than between wavefronts removes the necessity for specifying that the wavefronts be either plane or infinitesimal in area. By similar reasoning, it can be argued that the reflecting or refracting surface need not necessarily be a plane surface but can be a smooth curved surface. A normal can be erected at any point on a smooth surface, so the various angles can be defined uniquely.
- Robinson, E. A., and R. D. Clark, 1987c, A wave at a boundary: Reflection, transmission/refraction and diffraction: The Leading Edge, 6, no. 9, 38–42.
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Also in this chapter
- Wavefronts and raypaths
- d’Alembert’s solution
- One-dimensional waves
- Sinusoidal waves
- Phase velocity
- Wave pulses
- Geometric seismology
- The speed of light
- Huygens’ principle
- Ray theory
- Fermat’s principle
- Fermat’s principle and reflection and refraction
- Appendix A: Exercise