Mr. J. M c Cowan on the Solitary Wave. 51 



5. The Wave-length, 



The solitary wave cannot directly be regarded as having 

 any finite length as the elevation approaches the mean level 

 asymptotically towards a= +co and -co . Practically, how- 

 ever, Scott Russell found its length to be sufficiently definite 

 to admit of his giving measurements of it. To obtain a 

 measure Rayleigh suggested that the wave might be con- 

 sidered to end where its elevation became some definite and 

 fairly small fraction, say 1/10, of its maximum elevation. 

 Comparing, however, the formula (16) for the velocity with 

 that in the ordinary theory of a train of waves, or the corre- 

 sponding formulas for <£, ^, &c, it is natural to take for the 

 wave-length X = 2irhn. If for an approximation we take the 

 value of m given by (23), this gives 



\=27rV{/ i 2 (A + i|9 7 o)/37;o}; . . . (25) 



or, for waves just on the point of breaking, 



X=2?r/m=27rA, ...... (26) 



for in this case, as we shall see later, mh=l. 



Curiously enough the formula (26) is that taken by Russell 

 to represent approximately his experimental results, and it 

 agrees well with (25), for all fairly high waves, such in fact as 

 would be best suited for measurement. He noticed further 

 that low waves were longer than high ones, which is also in 

 accordance with (25) ; and thus altogether his results may be 

 taken as giving a practical basis for the definition we have 

 chosen, in addition to the theoretical one on which it is 

 founded. 



It should be noted, further, that this definition is practically 

 of the kind suggested by Rayleigh, for, taking for the moment 

 (21) as an approximation to the free surface, we see that 

 taking \ = 27r/m is equivalent to regarding the wave as end- 

 ing where 97/7? =sech 2 7r/2 = 0'16, or where the elevation is a 

 little less than one sixth of the maximum. 



6. The Volume and Displacement oj the Wave, 



The wave surface is given by (11), or, expanding by 

 Lagrange's theorem, by 



_ asinmA a 2 d sin 2 mh „ . 



cos mh + cosh nuc |2 dh (cos mh + cosh mx) 2 ' } * ' 



~ E 2 



