λ = vT.
(6)
In isotropic bodies the propagation can go on in all directions with the same velocity. In anisotropic bodies (crystals), with which the theory of light is largely concerned, the problem is more complicated. As a general rule we can say that, for a given direction of the wave-fronts, the vibrations must have a determinate direction, if the propagation is to take place according to the simple formula given above. It is to be understood that for a given direction of the waves there may be two or even more directions of vibration of the kind, and that in such a case there are as many different velocities, each belonging to one particular direction of vibration.
7. Wave-surface.—After having found the values of v for a particular frequency and different directions of the wave-normal, a very instructive graphical representation can be employed.
Let ON be a line in any direction, drawn from a fixed point O, OA a length along this line equal to the velocity v of waves having ON for their normal, or, more generally, OA, OA′, &c., lengths equal to the velocities v, v′, &c., which such waves have according to their direction of vibration, Q, Q′, &c., planes perpendicular to ON through A, A1, &c. Let this construction be repeated for all directions of ON, and let W be the surface that is touched by all the planes Q, Q′, &c. It is clear that if this surface, which is called the “wave-surface,” is known, the velocity of propagation of plane waves of any chosen direction is given by the length of the perpendicular from the centre O on a tangent plane in the given direction. It must be kept in mind that, in general, each tangent plane corresponds to one definite direction of vibration. If this direction is assigned in each point of the wave-surface, the diagram contains all the information which we can desire concerning the propagation of plane waves of the frequency that has been chosen.
The plane Q employed in the above construction is the position after unit of time of a wave-front perpendicular to ON and originally passing through the point O. The surface W itself is often considered as the locus of all points that are reached in unit of time by a disturbance starting from O and spreading towards all sides. Admitting the validity of this view, we can determine in a similar way the locus of the points reached in some infinitely short time dt, the wave-surface, as we may say, or the “elementary wave,” corresponding to this time. It is similar to W, all dimensions of the latter surface being multiplied by dt. It may be noticed that in a heterogeneous medium a wave of this kind has the same form as if the properties of matter existing at its centre extended over a finite space.
8. Theory of Huygens.—Huygens was the first to show that the explanation of optical phenomena may be made to depend on the wave-surface, not only in isotropic bodies, in which it has a spherical form, but also in crystals, for one of which (Iceland spar) he deduced the form of the surface from the observed double refraction. In his argument Huygens availed himself of the following principle that is justly named after him: Any point that is reached by a wave of light becomes a new centre of radiation from which the disturbance is propagated towards all sides. On this basis he determined the progress of light-waves by a construction which, under a restriction to be mentioned in § 13, applied to waves of any form and to all kinds of transparent media. Let σ be the surface (wave-front) to which a definite phase of vibration has advanced at a certain time t, dt an infinitely small increment of time, and let an elementary wave corresponding to this interval be described around each point P of σ. Then the envelope σ′ of all these elementary waves is the surface reached by the phase in question at the time t + dt, and by repeating the construction all successive positions of the wave-front can be found.
Huygens also considered the propagation of waves that are laterally limited, by having passed, for example, through an opening in an opaque screen. If, in the first wave-front σ, the disturbance exists only in a certain part bounded by the contour s, we can confine ourselves to the elementary waves around the points of that part, and to a portion of the new wave-front σ′ whose boundary passes through the points where σ′ touches the elementary waves having their centres on s. Taking for granted Huygens’s assumption that a sensible disturbance is only found in those places where the elementary waves are touched by the new wave-front, it may be inferred that the lateral limits of the beam of light are determined by lines, each element of which joins the centre P of an elementary wave with its point of contact P′ with the next wave-front. To lines of this kind, whose course can be made visible by using narrow pencils of light, the name of “rays” is to be given in the wave-theory. The disturbance may be conceived to travel along them with a velocity u = PP′/dt, which is therefore called the “ray-velocity.”
The construction shows that, corresponding to each direction of the wave-front (with a determinate direction of vibration), there is a definite direction and a definite velocity of the ray. Both are given by a line drawn from the centre of the wave-surface to its point of contact with a tangent plane of the given direction. It will be convenient to say that this line and the plane are conjugate with each other. The rays of light, curved in non-homogeneous bodies, are always straight lines in homogeneous substances. In an isotropic medium, whether homogeneous or otherwise, they are normal to the wave-fronts, and their velocity is equal to that of the waves.