148 150.] WAVES IN CANALS. 179 



to that time passed over the particle, divided by the depth of the 

 fluid. The horizontal motion is therefore at first infinitely small 

 compared with the vertical motion, so that the particle moves at 

 first upwards; its horizontal velocity gradually increases, and at 

 tains a maximum when the highest part of the wave is passing it, 

 the vertical motion being then zero. The particle then begins to 

 fall, its horizontal motion at the same time slackening, until it 

 finally comes to rest at its original level, but in advance of its 

 former position by a distance 



_ total volume of elevation, per unit breadth 

 depth of fluid. 



If the wave be one of depression, the motion is simply reversed in 

 every respect. 



If we have a series of waves of alternate elevation and depres 

 sion, the elevations and depressions being equal in every respect, 

 the motion of a surface-particle takes place in a closed curve, a 

 complete revolution being performed during the transit of an ele 

 vation and the following depression. Compare Art. 157. 



The horizontal motion is, in all cases, the same for particles 

 in the same vertical ; the x vertical motion is everywhere small, and 

 may (since it is zero at the bottom), be taken as proportional to 

 the undisturbed height of the particle above the bed. The motion 

 of a particle at a height y from the bottom may therefore be ob 

 tained from that of a surface-particle, by merely reducing the 

 vertical component in the ratio y : h. 



150. We can now examine under what circumstances the 

 assumptions of Art. 147 are justified. The time which a wave (of 



elevation, say), of length X takes to pass any particle is -, so that 



c 



if the slope of the wave-profile be gradual, the vertical velocity of 

 a surface-particle is of the order k+ -, where k is the maximum 

 elevation above the undisturbed level. The horizontal velocity is, 

 by (10), of the order j- ; the ratio of the vertical to the hori 

 zontal velocity is therefore of the order - . The assumption made 



X 



122 



