404 SURFACE WAVES. [CHAP. IX 



With varying speeds of the ship the stern-waves may tend parti- 

 ally to annul, or to exaggerate, the effect of the bow-waves, and 

 consequently the wave-resistance to the ship as a whole for a 

 given speed may fluctuate up and down as the length of the ship 

 is increased*. 



229. If in the problem of Art. 226 we suppose the depth to be 

 finite and equal to h, there will be, in the absence of dissipation, 

 indeterminateness or not, according as the velocity c of the stream 

 is less or greater than (gh)l, the maximum wave-velocity for the 

 given depth. See Art. 222. The difficulty presented by the former 

 case can be evaded as before by the introduction of small frictional 

 forces ; but it may be anticipated from the investigation of Art. 227 

 that the main effect of these will be to annul the elevation of the 

 surface at a distance on the up-stream side of the region of 

 disturbed pressure t, and if we assume this at the outset we need 

 not complicate our equations by retaining the frictional terms. 



For the case of a simple-harmonic distribution of pressure we 

 assume 



(f)/c = x -f j3 cosh k (y + h) sin lex, \ ,-. 



x/r/c = y + yS sinh k(y + h) cos kx } " 



as in Art. 222 (3). Hence, at the surface 



y = /3 sinh kh cos kx ..................... (2), 



we have 



2 = m/ J (o 2 c 2 ) = /3 (kc~ cosh kh g sinh kh) cos kx ... (3), 

 P 



so that to the imposed pressure 



coskx.. ...... (4), 



P 

 will correspond the surface-form 



sinhM 



7 ,. 



- COS KX ............ (o). 



&c 2 cosh kh g sinh kh 



* See W. Fronde, I.e., and E. E. Froude, " On the Leading Phenomena of the 

 Wave-Making Eesistance of Ships," Trans. Inst. Nav. Arch., t. xxii. (1881), where 

 drawings of actual wave-patterns under varied conditions of speed are given, which 

 are, as to their main features, in striking agreement with the results of the above 

 theory. Some of these drawings are reproduced in Lord Kelvin's paper in the Proc. 

 Inst. Mech. Eng., above cited. 



t There is no difficulty in so modifying the investigation as to take the frictional 

 forces into account, when these are very small, 



