Decay of Waves in a Canal. 907 



period as those of the type sin - — ). These two modes have 



the same periods and rates of decay that they would have 

 in an unlimited (laterally) mass of liquid of the same depth. 

 The lateral boundary conditions simply determine whether 

 they shall be present or absent. It will be noticed that 

 equations (1) to (6) on page 488 of my paper represent the 

 conditions at the free surface and bottom only, the remainder 

 are satisfied identically. Now in an unlimited mass of liquid 

 these periods and rates of decay are the same, it is simply a 

 question of choice of origin. 



The combination of my analysis with Dr. Houstoun's expe- 

 rimental results shows that the mode p = l, q = (for notation 

 here see my previous paper) undoubtedly exists and is pre- 

 dominant, since this is the only mode whose periods agree 

 with those observed. My explanation is that this mode 

 represents the motion in the body of the water, but that it 

 does not do so near the sides. It cannot do so at the sides 

 since there is no slipping (if this be the assumption made). 



The experimental fact that the logarithmic decrement is 

 increased by only 10 per cent, when the breadth of the box 

 is decreased by one-half does not render invalid my con- 

 tention that the effect of the sides is confined to their neigh- 

 bourhood, but appears to support it. In any case, whatever 

 explanation be given of the increased rate of decay of the 

 observed over the calculated motion, I still maintain that 

 the method of approximation by which Dr. Houstoun arrived 

 at his equation is wrong. 



He says in his former paper, " Let the motion be in one 

 horizontal direction w *. If v be assumed to be zero, then 

 everything follows as he states, except that there is an un- 

 resisted traction tending to force a velocity v into existence, 



which he has neglected ; this traction depends on ^ -. . 



Wit ^?% 



There is no reason why ^ ^ should be neglected and 



— 2 be retained, as in equation (3), on page 155, unless 



the length of the box be small compared with the breadth. 

 It is impossible for u to have the form given in equation (11), 

 page l-')6, and v to have a zero value. If this is so, then 

 Dr. Houstoun's approximation breaks down, and we have here 

 the reason for the discrepancy between the two results. 

 Clare College, Cambridge. 



* Phil. Ma<v. [6] vol. xvii. p. 154 (1900). 



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