582 Dr. J. Larmor on the 



along the face, as we require. If the medium has the 

 dispersive quality fully developed, for all disturbances however 

 sharp, i. e. if the differential equation determining dispersive 

 vibration has no limits to its full application to such dis- 

 turbances, the resolution into trains of waves must be granted 

 as a necessary consequence of the analysis for a steady 

 travelling source. 



This rationale of the dispersive refraction, at a plane 

 surface, of an obliquely incident thin plane pulse, is one of 

 those obvious things that when once grasped form a permanent 

 addition to our stock of physical imagery *. One can picture 

 its application to water-waves. A tract of water may be 

 imagined, of small uniform depth h, in which therefore all 

 waves travel with the same velocity \/gli, separated from a 

 region of deep water, in which the velocity of a train depends 

 on its wave-length, by a straight boundary. A disturbance 

 consisting of a thin plane ridge can advance obliquely towards 

 the deep water without change of form ; the successive parts 

 of the ridge reach the boundary in the manner of a maintained 

 local disturbance running along the boundary with uniform 

 speed, of which a definite fraction is transmitted across into 

 the deep water, the rest being reflected back. The mode of 

 this transmission is, by symmetry, the same as if that part of the 

 disturbance were doubled and the deep water were unlimited 

 on both sides : regular wave-trains are shed off dispersively 

 in the different directions, as in the case of the boat described 

 above, with wave-lengths such that their velocity can just 

 keep up with the travelling source, while the distribution of 

 intensities between the various directions depends on the 

 character of the moving source, i. e. of the incident travelling- 

 ridge of water. The waves must, in fact, form a steady pattern 

 travelling with the source : thus the velocity of free propa- 

 gation of the component train travelling in any direction 

 must be the component in that direction of the velocity of 

 the source. But as the travelling source has finite size, this 

 component train, though nearly homogeneous in wave-length, 

 is not quite so ; being nearly homogeneous, the dispersive 

 quality of the medium will make its waves travel in groups, 

 which progress in the known manner with only half the 

 velocity of their component waves. Thus after the train is 

 well formed, the groups of disturbance will recede from the 



* Lord Itayleigh considers this explanation of the refraction of a pnlse 

 into a dispersive medium to be less simple than his first case of the 

 propagation of a plane pulse in such a medium : but owing to the 

 difficulty described above, regarding the maintenance of such a pulse, I 

 have failed to appreciate the argument in that case. 



