Theory of Long Waves. 253 



initially in such a plane will remain so, having a common 

 horizontal velocity. 



If, finally, we differentiate (4) with respect to a?, we get 



V (i), 



d* <j>(h) d_ $(z) dz m 



dt 2 cj>iz) " 9 dx(j>(h) dx ' 

 or 



I (f§> <">:!= >•->!• • • • < 5 ' 



The particular case in which the cross section of the 

 channel is rectangular is specially important. Here <j>{z) is 

 proportional to z, and therefore the equations (1) to (5) 

 reduce to 



df _ * (!/) 



dx z 



§=« • • • ^ 



dV[___z_dp m 



dt*~ hdx V } 



dt* 9 hdx v ; 



d. A 2 dz _ d dz _,. 



dt z* dt dx"~ dx' 



Equations (1) to (5), with the initial and boundary con- 

 ditions, completely determine the motion. It is, however, 

 very necessary to use great care in interpreting these 

 equations, remembering that they are subject to the limitations 

 assumed in obtaining them. We shall see later that though 

 the initial circumstances of the motion satisfy these conditions, 

 still the motion will gradually alter in character till they are 

 no longer fulfilled, and therefore the further motion will not 

 be given by these equations. It is therefore important to 

 note clearly what are the fundamental conditions to be satisfied 

 in order that any wave-motion may be included in the theory 

 of long waves. 



The essential feature of the theory is the motion in parallel 

 sections ; that is, the motion must be such that all particles 

 initially in a plane, perpendicular to the length of the channel, 

 remain in such a plane. The investigation shows that the 

 motion will have this character provided that initially all 

 particles in such planes receive a common displacement and 



Phil. Mag. S. 5. Vol. 33. No. 202. March 1892. T 



