In the simplest cases (monochromatic or homogeneous light) the disturbance is a simple harmonic function of the time (“simple harmonic vibrations”), so that its components can be represented by
Px = a1 cos (nt + ƒ1), Py = a2 cos (nt + ƒ2), Pz = a3 cos (nt + ƒ3).
The “phases” of these vibrations are determined by the angles nt + ƒ1, &c., or by the times t + ƒ1/n, &c. The “frequency” n is constant throughout the system, while the quantities ƒ1, ƒ2, ƒ3, and perhaps the “amplitudes” a1, a2, a3 change from point to point. It may be shown that the end of a straight line representing the vector P, and drawn from the point considered, in general describes a certain ellipse, which becomes a straight line, if ƒ1 = ƒ2 = ƒ3. In this latter case, to which the larger part of this article will be confined, we can write in vector notation
P = A cos (nt + ƒ),
(4)
where A itself is to be regarded as a vector.
We have next to consider the way in which the disturbance changes from point to point. The most important case is that of plane waves with constant amplitude A. Here f is the same at all points of a plane (“wave-front”) of a definite direction, but changes as a linear function as we pass from one such wave-front to the next. The axis of x being drawn at right angles to the wave-fronts, we may write ƒ = ƒ0 − kx, where ƒ0 and k are constants, so that (4) becomes
P = A cos (nt − kx + ƒ0).
(5)
This expression has the period 2π/n with respect to the time and the perion 2π/k with respect to x, so that the “time of vibration” and the “wave-length” are given by T = 2π/n, λ = 2π/k. Further, it is easily seen that the phase belonging to certain values of x and t is equal to that which corresponds to x + Δx and t + Δt provided Δx = (n/k) Δt. Therefore the phase, or the disturbance itself, may be said to be propagated in the direction normal to the wave-fronts with a velocity (velocity of the waves) v = n/k, which is connected with the time of vibration and the wave-length by the relation