Mr. J. M c Cowan on the Solitary Wave. 49 



the equations (4) and (5) give the current function and 

 velocity potential, and (7) and (8) the velocity components 

 of a steady motion which, to the degree of accuracy indicated 

 by (15), satisfies the condition that its surface, which is given 

 by (11), may be a free surface. 



Thus to this degree of approximation the form of the 

 solitary wave is determined by (11), and we see that, since 

 by (17) ma is essentially positive, the wave consists solely of 

 an elevation, and that there cannot be a wave of depression 

 capable of propagating itself unchanged with constant velocity: 

 a result in accordance with the observations of Scott Russell. 



We proceed to consider in greater detail the character of 

 the approximation we have adopted for the free surface. 



3. The Surface Pressure in the Approximate Theory. 



The pressure at any point of a liquid in steady irrotational 

 motion is given by the equation 



p = constant — ^pq^—gz, . . . . (19) 



where p and p are the pressure and density respectively. 

 Over the free surface p ought to be constant: hence if op 

 denote the excess of pressure at any point of the surface given 

 by (11) over the pressure at the mean level h in the motion 

 just investigated, we have by (15) and (19) 



Bp = —4:m 3 7} 2 (7] —7))p~U 2 cot mh cosec 2 mh, 

 or 



Bp= —4:gpm 2 rj 2 (r] —r)) cosec 2 mh (20) 



Thus there is a defect of pressure everywhere but at the crest 

 and the mean level. Note, however, how this negative 

 pressure is distributed : — At the crest Bp vanishes, and as it 

 contains the factor rjo—r) and the crest is the point of maxi- 

 mum elevation, it remains very .small over a long range on 

 either side of the crest. Again, Bp vanishes at mean level 

 and remains very small over an infinite range. Finally, 8p> 

 is a maximum at the point where 7} = ^rj , having then the 

 value — ^gpm^rjQ 6 cosec 2 mh (only ^V of what the maximum 

 would have been had we taken 3ma = 2 sin 2 mh) : but this is at 

 the point of inflexion (accurately when r) /h is very small) 

 where rj is increasing most rapidly, and therefore this maxi- 

 mum pressure occurs where it can have least range. Thus we 

 see that the pressure-error, small at its greatest, is so dis- 

 tributed as to be least effective. 



4. The Approximations of Boussinesq and Lord Rayleigh. 

 We have found that to a high degree of approximation 

 Phil. Mag. S. 5. Vol. 32. No. 194. July 1891. E 



