172-174] REFLECTION. 281 



where the function F represents the original wave, and /, $ the reflected and 

 transmitted portions respectively. The constancy of mass requires that at 

 the point # = we should have b^u^b^u^ where b lt b. 2 are the breadths at 

 the surface, and k lt h 2 are the mean depths. We must also have at the same 

 point 7 1 =/2) on account of the continuity of pressure*. These conditions give 



*1 



F(t)+f(t): 



We thence find that the ratio of the elevations in corresponding parts of the 

 reflected and incident waves is 



F & 



The similar ratio for the transmitted wave is 



The reader may easily verify that the energy contained in the reflected and 

 transmitted waves is equal to that of the original incident wave. 



174. Our investigations, so far, relate to cases of free waves. 

 When, in addition to gravity, small disturbing forces X, Y act on 

 the fluid, the equation of motion is obtained as follows. 



We assume that within distances comparable with the depth 

 h these forces vary only by a small fraction of their total value. 

 On this understanding we have, in place of Art. 166 (1), 



Pj= f = (9- Y)(y. + v-y) ................ (1), 



and therefore 



1 d di 



The last term may be neglected for the reason just stated, and if 



* It will be understood that the problem admits only of an approximate treat 

 ment, on account of the non-uniform character of the motion in the immediate 

 neighbourhood of the point of discontinuity. The degree of approximation implied 

 in the above assumptions will become more evident if we suppose the suffixes to 

 refer to two sections /Sj and S 2 , one on each side of the origin 0, at distances from 

 which, though very small compared with the wave-length, are yet moderate 

 multiples of the transverse dimensions of the canal. The motion of the fluid will 

 be sensibly uniform over each of these sections, and parallel to the length. The 

 conditions in the text then express that there is no sensible change of level between 

 S 1 and S 2 . 



