[ 106 ] 



VII. The Progressive Long Waves of Solitary and Periodic 

 Types in Shallow Water. By R. F. GrWYTHER, M. A* 



IN a paper on " The Classes of Progressive Long Waves " tr 

 I obtained an equation [namely, 



( c 2 _ gll jfy h*f n =—2cf!* + 2 (c*-gh)f» + constant, . (8) 



where /stands for /(a-)], which is the first approximation to 

 the condition that 



may give the velocity-potential and the stream-function of a 

 steady long- wave motion in a fluid of which the equation of 

 the free surface is yfr= —ch. In an Appendix % to that paper 

 I also showed that the method is one of considerable accuracy; 

 and I now attempt to find the whole range of periodic long 

 waves in water of uniform depth, limited by the condition 

 that the uniform velocity c in the direction of the axis of or 

 applied throughout the fluid may make the motion steady. 



The velocity of propagation of the wave-figure will only 

 be precisely given by c, when there is no proper horizontal 

 motion (or drift) essential to the especial wave-system. 

 It will appear that this is not generally the case with the 

 class of waves with which this paper deals. We have also 

 not defined h, since by taking -\|r = — ch as the equation of the 

 free surface, we leave the linear constant h to be defined, in 

 each case, by the form of the equation of the wave- surface 

 arising from the solution, the depth of the undisturbed fluid 

 in periodic motion being the mean depth over a wave-length. 



The result of the investigation gives the velocity-potential 

 of waves of the " New Type of Long Stationary Waves," the 

 existence of which was discovered and to which the name of 

 Cnoidal waves was given by Drs. Korteweg and de Vries §. 

 There is, however, also matter not included in their paper 

 from which this differs both in scope and in the manner of 

 treatment. 



For convenience I number the equations continuously 

 with those of the papers referred to. 



Proceeding with the equation (8) ; if the parameters c and 

 h are connected by the equation c' 2 = gh, we should obtain 



(c"-<//,/3)/r/" 2 =-2c(^W), 



* Communicated bv the Author. 

 t Phil. Mag. Aug. "J900, p. 213. 

 | Phil. Mao-. Sept, 1900, p. 308. 

 § Phil. Mag. May 1895, p. 422. 



