199, 200J CANAL THEORY OF TIDES. 539 



in straight canals. Let the motion be in the plane of XY, as in 

 198, and let Ji, the depth of the canal, be small in comparison 

 with the wave-length. We shall suppose the displacements of all 

 the particles, with their velocities and their space -derivatives, to be 

 small quantities whose squares and products may be neglected. We 

 shall also neglect the vertical acceleration, so that the equation for y 

 is that of hydrostatics, giving the pressure proportional to the distance 

 below the surface. If the ordinate of the free surface is h -\- y, this 

 gives 



173) P 



174) = ' 



ox y * ox 



while the equation of motion, the first of equations 6), is 



175) | x_i|. 



Ot Q OX 



Combining these two equations, we have 

 17\ cu v dr] 



176 ) w =x ~^' 



Q 



and if X is independent of y y since ^ is also, this shows that u 



depends only on x and t, or vertical planes perpendicular to the 

 XY plane remain such during the motion. 

 Integrating the equation of continuity 



du cv A 



Wx + Wy = " (} > 



with respect to y from the bottom to the surface, 



1 nn\ tdu ^ /7 . N du 



1<7) v- -J^dy- -(h + r,)^, 



or approximately, at the surface, 



-, rro\ $n 7 ^ U 



178 ) v = Tt=- ]l TT x 



O 



Now putting u = 3! ? the equation of continuity 178) becomes, 

 ot 



1 7Q^ ^^ 



= ~ 



and on integration with respect to the time, 



180) 1~-*H- 



Substituting in 176) we have for the horizontal displacement 



