382 Frederick Guthrie on Stationary Liquid Waves, 



in air will oscillate for hours, and remembering that rate of de- 

 cadence is nearly proportional to individual delay, I do not think 

 that the above numbers are at all in disaccord with the asser- 

 tion that the fundamental or binodal circular undulations in an 

 infinitely deep circular vessel are isochronous with those of a pen* 

 dulum whose length is equal to the radius of the vessel. 



In hemispherical and cycloidal vessels I anticipate that a 

 still nearer identity would be obtained. The exact form of the 

 cup of revolution which would give the' nearest approach to the 

 pendulum-rate depends of course upon the shape of the wave ; 

 and here we are again referred to the complex influence of vis- 

 cosity and cohesion. 



§ 25. Comparison of rectangular troughs with pendulum. — 

 Making a similar comparison to the above with rectangular 

 troughs having binodal undulations, let us find what are the 

 lengths of pendula isochronous with the water-waves in the 

 troughs Z, Y, X, W respectively. The number of liquid undu- 

 lations being n } the formula 



-© 



994 



gives us the required lengths of the pendula in millims. 





Trough- or 

 wave-length. 



Number of 

 liquid un- 

 dulations. 



Length of 

 isochronous 

 pendulum. 



2 



— X trough- 



7T 



length. 



z 



Y 



767 

 619 

 463 

 308 



845 



94 5 



110-4 



135-8 



501194 

 400-482 

 293-627 

 194028 



488-4 

 394-1 

 294-8 

 196-1 



X 



w 





It appears, therefore, that the binodal undulations in a rectan- 

 gular trough are isochronous with the oscillations of a pendulum 



2 



whose length is— of the trough or wave-length. 



§ 26. Comparison between circular and rectangular troughs in 

 binodal vibration. — Briefly, it has appeared that the differences 

 between the wave-systems in circular and rectangular troughs, 

 both being binodal, are as follows : — In rectangular troughs the 

 nodes are one fourth of the trough -length from the ends. In cir- 

 cular troughs they are one sixth of the diameter. In rectangular 

 troughs the amplitude is a very little more at the centre than at 

 the edges. In circular troughs it is very little more than double 

 that at the edges. The rate of progression in both varies as the 

 square root of the wave-length ; and the rate of sequence, or n s 

 varies inversely as the square root of the wave-length. The 

 waves of the circular system are isochronous with a pendulum 



