414 



SURFACE WAVES. 



[CHAP, ix 



It is obvious from (1) that the path of any particle (a, b) is a 

 circle of radius k~ l ^ b . 



The figure shews the forms of the lines of equal pressure 

 b = const., for a series of equidistant values of 6*. These curves 

 are trochoids, obtained by rolling circles of radii Ar" 1 on the under 

 sides of the lines y=b + k~\ the distances of the tracing points 

 from the respective centres being k~ l ^ b . Any one of these lines 

 may be taken as representing the free surface, the extreme 

 admissible form being that of the cycloid. The dotted lines 

 represent the successive forms taken by a line of particles which 

 is vertical when it passes through a crest or a trough. 



It has been already stated that the motion of the fluid in these waves is 

 rotational. To prove this, we remark that 



. dx 



. dx 



which is not an exact differential. 



* The diagram is very similar to the one given originally by Gerstner, and copied 

 more or less closely by subsequent writers. 



