664 



EXPLORATION GEOPHYSICS 



wave ray is propagated in a direction normal to such a boundary, no change 

 in direction occurs. 



The basic law of refraction is usually referred to as Snell's Law, This 

 law was discovered experimentally, but it actually is a consequence of 

 Fermat's principle, which states that a given ray follows a particular path 

 so that it travels from a given point to a second point in a minimum 

 amount of time. If the given point is in a medium characterized by a 

 propagation velocity different from the medium containing the second 

 point, then the ray path between these two points will not be a straight line. 

 As a consequence of this principle we have Snell's law of refraction, which 

 states that refraction or bending of the ray path occurs when traversing 

 a boundary between media of different velocities such that (a) the incident 

 ray and the refracted ray lie in the same plane and (b) the sine of the 

 angle of incidence in the first medium is to the sine of the angle of refrac- 

 tion in the second medium as the velocity in the first medium is to the 

 velocity in the second medium. These relationships have been expressed 

 mathematically below. 



Another useful principle in the study of wave propagation is that of 

 Huygens. It states that in an isotropic medium each point on a wave front 

 may be considered as a source of new waves with spherical wave fronts. 



The ensuing wave front then be- 

 comes the envelope of all wave 

 fronts, which may be considered 

 the surface tangent to the new 

 spherical wave fronts at any given 

 instant of time. 



^i>V 



Fig. 409. — Sketch illustrating the refraction of a 

 plane wave A'B' at a boundary MM'. 



The law of refraction may 

 be proved directly by use of ray 

 paths.f Referring to Figure 409, 

 MM^ is a horizontal boundary 

 separating two homogeneous 

 media in which the elastic wave 

 velocities are Vi and V2, respectively. A^B^ is a trace of a plane wave 

 front which is perpendicular to the plane of the paper, ai is the angle 

 between the ray A A' and the normal NN' to the boundary, and ag is the 

 corresponding angle in the second medium. Let t be the ti me r equired 

 for the wave to travel from 5' to B^\ Then, the distance B'B^' equals 

 tVi. In this same interval of time the wave will travel a distance A'A'^ 

 in the second medium where A'' is the point corresponding to ^' on the 

 wave front at this later time. The distance A' A'' is equal to the product 



t A proof of the laws of reflection and refraction utilizing Huy gen's construction may be 

 found in any standard text on optics. See, for example, R. A. Houston, A Treatise on Light, 5th 

 Edition, pp. 187-128 (Longmans, Green and Co., London) 1928. 



