550 VISCOSITY. [CHAP, xi 



Let us first examine the effect of a purely tangential force. 

 Assuming p' yy = 0, we find 



p^y _m (a + 2z/fc 2 ) 2 + <r 2 - 4i/ 2 A?m 

 gpr)~~gk' ~ a + 2vk* - 2vkm 



For a given wave-length, the elevation will be greatest when 

 a= icr, nearly. To find the force necessary to maintain a train 

 of waves of given amplitude, travelling in the direction of x- 

 positive, we put a = icr. Assuming, for a reason already indi- 

 cated, that vk 2 /cr and vkm/cr are small, we find 



P'xylgpv = *vk<r/ff, or P'*y = 4M<"/ M ....... W- 



Hence the force acts forwards on the crests of the waves, and 

 backwards at the troughs, changing sign at the nodes. A force 

 having the same distribution, but less intensity in proportion to 

 the height of the waves than that given by (4), would only retard, 

 without preventing, the decay of the waves by viscosity. A force 

 having the opposite sign would accelerate this decay. 



The case, of purely normal force can be investigated in a 

 similar manner. If p' xy = 0, we have 



(a + 2z/P) 2 4- o- 2 - 



. 



............... ( O ). 



gm g 



The reader may easily satisfy himself that when uhere is no 

 viscosity this coincides with the result of Art. 226. If we put 

 a = icr, we obtain, with the same approximations as before, 



p'yy = 4fifjikcrrj ........................ (6). 



Hence the wave-system 



T] = a sin (lex a~t) ..................... . . .(7) 



will be maintained without increase or decrease by the pressure- 

 distribution 



p' const. + 4>jj,kao- cos (kx cri) ............... (8), 



applied to the surface. It appears that the pressure is greatest on 

 the rear and least on the front slopes of the waves*. 



If we call to mind the phases of the particles, revolving in their 

 circular orbits, at different parts of a wave-profile, it is evident 



* This agrees with the result given at the end of Art. 226, where, however, the 

 dissipative forces were of a different kind. 



