165-166] APPROXIMATE TYPES. 271 



Moreover, if the constrained type differ but slightly from a normal type 

 (), <r 2 will differ from c 8 /a 8 by a small quantity of the second order. This gives a 

 valuable method of estimating approximately the frequency in cases where 

 the. normal types cannot be accurately determined*. 



The modifications which are introduced into the theory of 

 small oscillations by the consideration of viscous forces will be 

 noticed in Chapter xi. 



Long Waves in Canals. 



166. Proceeding now to the special problem of this Chapter, 

 let us begin with the case of waves travelling along a straight 

 canal, with horizontal bed, and parallel vertical sides. Let the 

 axis of x be parallel to the length of the canal, that of y 

 vertical and upwards, and let us suppose that the motion takes 

 place in these two dimensions x, y. Let the ordinate of the free 

 surface, corresponding to the abscissa x, at time t, be denoted by 

 V + 2/o > where y Q is the ordinate in the undisturbed state. 



As already indicated, we shall assume in all the investigations 

 of this Chapter that the vertical acceleration of the fluid particles 

 may be neglected, or (more precisely) that the pressure at any 

 point (x, y) is sensibly equal to the statical pressure due to the 

 depth below the free surface, viz. 



p-p^gptyo+y-y} ..................... (i), 



where p is the (uniform) external pressure. 



This is independent of y t so that the horizontal acceleration is the 

 same for all particles in a plane perpendicular to x. It follows 

 that all particles which once lie in such a plane always do so ; in 

 other words, the horizontal velocity it, is a function of x and 

 t only. 



The equation of horizontal motion, viz. 

 du du 1 dp 



_ r I ^ . _ _. _ _ _ J._ 



dt dx p dx ' 

 is further simplified in the ease of infinitely small motions by the 



* These theorems are due to Lord Eayleigh, " Some General Theorems relating 

 to Vibrations," Proc. Lond. Math. Soc., t. iv., p. 357 (1873); Theory of Sound, c. iv. 



