REFLECTION AND REFRACTION. 



159 



Fie:. 



Foi- tliis purpose let us assume a 

 plane surface, as MN, bounding a trans- 

 parent medium more dense than air. 

 Let AA' be the point of a plane wave 

 advancing in the direction ]^A, P'A', 

 jjand passing from the air into the medi- 

 um between the points A and 15. AVe 

 have seen that this Avave Avill be par- 

 tially reflected from AB. In order to 

 determine the front of the reflected 

 wave let us suppose, according to the 

 principle of Huyghens, that every point 

 of the surface of AB is the origin of a new spherical wave propagated in every 

 direction from that point. It will be sufficient for the illustration to attend to 

 a few only of these points, as, for instance. A, C, and B: — C being taken half 

 way between the other two. By hypothesis AA' is perpendicular to PA/ 

 Draw CA" parallel to PA. The velocity of the reflected wave being equal to 

 that of the incident, when the wave front AA' reaches the position Gb, the ele- 

 mentary reflected wave from A will have travelled to a distance equal to CA." 

 When' A' reaches B the reflected wave from A will have reached a distance 

 AB' equal to AB. Also, at the same instant, the elementary wave from will 

 have travelled a distance CB" equal to ^Bz=A'i. With A and C as centres de- 

 scribe the small circular arcs shown at B' and B'', and from B draw BB' tangent 

 to the first of these arcs. It will also be tangent to the second. For if AB'Q' 

 be, drawn from A through the point of contact B', and CB"Q" parallel to 

 it, then, because AB is bisected in 0, CB" is half of A B' ; and it is also 

 perpendicular to B'B, consequently B" is a point in the spherical wave front 

 which originates from C. And as similar reasoning may be applied to all the 

 elementary waves reflected from the points between A and B, it follows that 

 the tangent BB' is the resultant reflected wave front moving in the direction 

 AQ' or BQ. Also, from the similarity of the triangles AA'B and AB'B, it 

 is evident that the inclination of the wave front AA', or of the ray PA to the 

 reflecting surface MN, is equal to that of the wave front BB', or of the ray 

 BQ to the same surface. The incident and reflected rays are therefore equally 

 inclined also to the normal to the surface: or, in other words, the angle of re- 

 flection is equal to tbe angle of incidence. 



If we turn our attention to that portion of the incident wave which is propa- 

 gated into the medium beyond the surface, the construction which determines 

 the refracted wave front is analogous to the foregoing. Only, in the second 

 medium, as it is the denser of the two, the velocity of 'jiropagation w^l be 

 diminished. Suppose it to be so in the ratio of n to 1, the circles described 

 from A and C must have their radii, AB'" and Cc reduced below A'B, and 

 5B in the same proportion.* The wave front of the refracted wave Avill then 

 be the common tangent BB'" of these circles. Also, if we observe that A'B ^= 

 AB sin. AAB, and AB"' = AB sin. B" BA, we shall obtain the proportion, 



sin A'AB : sin B"'BA : : A'B : AB'" '.-.%'. 1. 



But tbcse angles, being the inclinations of the wave fronts to the refracting 

 surface, are also the inclinations of the rays themselves to the normal to the 



'' A diminished velocity might be a consequence of a change in the timt of a vibration 

 without a corresponding change in the undulation length; or it might be caused by a change 

 (rf both. But, in the case in hand, wo uic not at liberty to assume any change in the time 

 of the vibration, because the impulses which produce the molecular movements in the second 

 medium are the vibrations themselves of the first. The undulations in the two media are, 

 therefore, isochronous ; and it is only possible to explain the dii'ierenco of velocity of undala- 

 tion progress by siipposing a change of undulation length. 



