168-170] AIRY'S METHOD. 277 



The requisite conditions will of course be satisfied in the 

 general case represented by equation (9), provided they are 

 satisfied for each of the two progressive waves into which the 

 disturbance can be analysed. 



170. There is another, although on the whole a less con- 

 venient, method of investigating the motion of 'long' waves, in 

 which the Lagrangian plan is adopted, of making the coordinates 

 refer to the individual particles of the fluid. For simplicity, we 

 will consider only the case of a canal of rectangular section*. The 

 fundamental assumption that the vertical acceleration may be 

 neglected implies as before that the horizontal motion of all 

 particles in a plane perpendicular to the length of the canal will 

 be the same. We therefore denote by cc + ? the abscissa at time 

 t of the plane of particles whose undisturbed abscissa is x. If rj 

 denote the elevation of the free surface, in this plane, the equation 

 of motion of unit breadth of a stratum whose thickness (in the 

 undisturbed state) is Sac, will be 



where the factor (dpjdx) . Sx represents the pressure-difference for 

 any two opposite particles x and x + 8x on the two faces of the 

 stratum, while the factor h + 77 represents the area of the stratum. 

 Since the pressure about an}^ particle depends only on its depth 

 below the free surface we may write 



dp drj 



>-**&' 



so that our dynamical equation is 



dt* 



The equation of continuity is obtained by equating the volumes 

 of a stratum, consisting of the same particles, in the disturbed and 

 undisturbed conditions respectively, viz. we have 



i+?- i+SE (2). 



* Airy, Encyc. Metrop., " Tides and Waves," Art. 192 (1845) ; see also Stokes, 

 " On Waves," Camb. and Dab. Math. Journ.,t. iv. (1849), Math, and Phys. Papers, 

 t. ii., p. 222. The case of a canal with sloping sides has been treated by McCowan, 

 " On the Theory of Long Waves...," Phil. Mag., March, 1892. 



