54 Sir William Thomson on Stationary 



necessary to allow the motion of the water in every part of 

 the canal to be so nearly two-dimensional, that our formulas 

 for two-dimensional motion in a straight canal shall be prac- 

 tically applicable to the water in the curved canal. 



Now let there be any integral number n of equidistant 

 ridges in the circuit, and let a be the distance from ridge to 

 ridge. Superposition by simple addition of solutions of the 

 formula (2) gives, for the surface effect, 



. =M 4A/a/Icos{(^+^) 



K? — m — • • ■ (4) * 



The consideration of cases of different values of n, even or 

 odd, leads to interesting illustrations both of mathematical 

 principles and of practical results in dynamics ; but for the 

 present I confine myself to the case of n=l, for which (4) 

 becomes identical with (2). 



Eemark, now, that if M(e* — £"*)/( e* +e~ { ) is an integer, 

 the denominator of (2) vanishes for the case of i equal to 

 this integer. This is the case in which the length of the 

 circuit of the canal is an integral number of times the wave- 

 length of free waves in water of depth D. The interpre- 

 tation is obvious, and is interesting both in itself and in 

 its relation to corresponding problems in many branches of 

 physical science. 



Meantime remark only that, when the value of 

 M(e , '—e~ , ')/(6 i +e~ i ) approaches very nearly to any integer j, 

 the chief term of (2) is that for which i=j, and all the other 

 terms are relatively very small. Thus the chief effect is 

 forced stationary waves of wave-length a/j. Thus, if we con- 

 sider different velocities of flow approaching more and more 

 nearly to the velocity which makes M(<?* — e~ l ) / [e { -Y e~ l ) an 

 integer, the magnitude of the forced stationary waves is 

 greater and greater for the same magnitude of ridge, but the 

 motion is still perfectly determinate. Suppose, now, we 

 make the ridge smaller and smaller, so that the wave-height 

 of the stationary wave may have any moderate value ; as the 

 velocity approaches more and more nearly to that which makes 

 M(e* — e~ x )j{e i +e~ l ) an integer, the magnitude of the ridge 

 must be smaller and smaller, and in the limit must be zero. 

 Thus, with no ridge at all, we may have stationary waves 

 of any given moderate value, in the limiting case, — that in 

 which the velocity of the flow equals the velocity of a wave of 

 wave-length a/j. 



