Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/199

Rh by supposing that we have to do with a system of stationary waves. That the relation of the wave-length and the period is the same for stationary as for progressive waves is evident from the consideration that a system of stationary waves may be formed by two systems of progressive waves having opposite directions.

2. Let $$x, y, z$$ be the rectangular coordinates of any point in the medium, which with the system of waves we may regard as indefinitely extended, and let $$\xi + \xi ', \eta + \eta ', \zeta + \zeta '$$ be the components of electrical displacement at that point at the time $$t;$$ $$\xi, \eta, \zeta$$ being the average values of the components of electrical displacement at that time in a wave-plane passing through the point. Then $$\xi, \eta, \zeta, \xi ', \eta ', \zeta ', x, y, z$$ are perfectly defined quantities, of which $$\xi, \eta, \zeta$$ are connected with $$x, y, z,$$ and $$t$$ by the ordinary equations of wave-motion, while each of the quantities $$\xi ', \eta ', \zeta '$$ has always zero for its average value in any wave-plane. We may call $$\xi, \eta, \zeta$$ the components of the regular part of the displacement, and $$\xi ', \eta ', \zeta '$$ the components of the irregular part of the displacement. In like manner, the differential coefficients of these quantities with respect to the time, $$\dot{\xi}, \dot{\eta}, \dot{\zeta}, \dot{\xi '}, \dot{\eta '}, \dot{\zeta '},$$ may be called respectively the components of the regular part of the flux, and the components of the irregular part of the flux.

Let the whole space be divided into elements of volume $$\text{D}v,$$ very small in all dimensions in comparison with a wave-length, but enclosing portions of the medium which may be treated as entirely similar to one another, and therefore not infinitely small. Thus a crystal may be divided into elementary parallelopipeds, all the vertices of which are similarly situated with respect to the internal structure of the crystal. Amorphous solids and liquids may not be capable of division into equally small portions of which physical similarity can be predicated with the same rigor. Yet we may suppose them capable of a division substantially satisfying the requirements.

From these definitions it follows that at any given instant the average value of each of the quantities $$\xi ', \eta ', \zeta '$$ in an element $$\text{D}v$$ is zero. For the average value in one such element must be sensibly the same as in any other situated on the same wave-plane. If this average were not zero, the average for the wave-plane would not be zero. Moreover, at any given instant, the values of $$\xi, \eta, \zeta$$ may be regarded as constant throughout any element $$\text{D}v,$$ and as representing the average values of the components of displacement in that element. The same will be true of the quantities $$\dot{\xi '}, \dot{\eta '}, \dot{\zeta '}$$ and $$\dot{\xi}, \dot{\eta}, \dot{\zeta}.$$

3. Since we have excluded the case of media which have the property of circular polarization, we shall not impair the generality of our results if we suppose that we have to do with linearly polarized light, i.e., that the regular part of the displacement is everywhere parallel to the same fixed line, all cases not already excluded being