Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/269

 has been shown by Lord Rayleigh and others that the velocity ($$U$$) with which a group of waves is propagated in any mediam may be calculated by the formula where $$V$$ is the wave-velocity, and $$\lambda$$ the wave-length. It has also been observed by Lord Rayleigh that the fronts of the waves reflected by the revolving mirror in Foucault's experiment are inclined one to another, and in consequence must rotate with an angular velocity where $$\alpha$$ is the angle between two successive wave-planes of similar phase. When $$dV / d\lambda$$ is positive (the usual case), the direction of rotation is such that the following wave-plane rotates towards the position of the preceding (see Nature, vol. . p. 52).

But I am not aware that attention has been called to the important fact, that while the individual wave rotates the wave-normal of the group remains unchanged, or, in other words, that if we fix our attention on a point moving with the group, therefore with the velocity $$U$$, the successive wave-planes, as they pass through that point, have all the same orientation. This follows immediately from the two formulæ quoted above. For the interval of time between the arrival of two successive wave-planes of similar phase at the moving point is evidently $$\lambda / (V - U)$$, which reduces by the first formula to $$d\lambda / dV$$. In this time the second of the wave-planes, having the angular velocity $$\alpha dV / d\lambda$$, will rotate through an angle a towards the position of the first wave-plane. But $$\alpha$$ is the angle between the two planes. The second plane, therefore, in passing the moving point, will have exactly the same orientation which the first had. To get a picture of the phenomenon, we may imagine that we are able to see a few inches of the top of a moving carriage-wheel. The individual spokes rotate, while the group maintains a vertical direction.