178 WAVES IN LIQUIDS. [CHAP. VII. 



so that the complete solution of (5) is 



+ cO (8), 



where F, / denote arbitrary functions. 



The interpretation of (8) is simple. Take first the motion re 

 presented by the first term alone. Since F(x ct) is unaltered 

 when t is increased by T, and x by CT, it is plain that the disturb 

 ance of the particle x at the time t has been communicated, at the 

 time t 4- T to the particle x + CT, so that the disturbance advances 

 as a whole with uniform velocity c relative to the fluid. The first 

 term of (8) denotes then a wave travelling in the positive direction, 

 without change of form, with velocity c. The second term denotes 

 in like manner a wave travelling with the same velocity in the 

 direction of x negative. Any motion whatever of the fluid, subject 

 to the restrictions of the preceding article, may be regarded as 

 made up of waves of these two kinds. 



The velocity of propagation c is, by (7), that due to half the 

 depth of the undisturbed fluid. 



149. Let us examine the motion of a surface-particle as a 

 wave passes over it. To fix the ideas we shall suppose the wave 

 to be one of elevation, so that TJ is everywhere positive, and to be 

 travelling in the positive direction, so that 



g=F(x-ct) (9). 



We shall also suppose the length \ of the wave to be finite. By 

 differentiation of (9) we find 



^f _ ^f 

 d t doc 



or, by (4), 



The particle remains at rest until it is reached by the wave ; 

 it then moves forwards with a horizontal velocity proportional to 

 its elevation above the mean level. Also 



so that the total horizontal displacement at any time is equal to 

 the whole volume (per unit breadth) of elevated fluid which has up 



