﻿Problem of the Weir. ll?j 



using. the same restrictions, discussed the case in which the 

 change of level was abrupt. 



"When the fall of the stream-bed is not small compared 

 with the general depth of the stream the problem becomes 

 of some practical importance, and a solution of it might 

 throw light on the theory of the construction of weirs and 

 aqueducts. Although the problem is dealt with in various 

 text-books on Hydraulics*, the treatment is usually somewhat 

 empirical. It seems, therefore, worth while to inquire what 

 information on the subject can be gained from the ordinary 

 hydrodynamical equations when viscosity is neglected. It 

 is true that in the absence of viscosity the problem is inde- 

 terminate, but an attempt may be made to represent approxi- 

 mately the physical conditions by assuming a laminar motion 

 of the fluid. 



In the present note the motion is supposed to be two- 

 dimensional and steady. The upper and lower stream-beds 

 are taken horizontal, but the intervening connexion may be 

 of any form consistent with continuous flow without impacts. 

 Along the upper and lower reaches the flow is assumed to 

 be ultimately horizontal t, and it is at these places that the 

 motion is considered. In practice these places will be more 

 or less close to the weir, or sloping part of the bed, according 

 to its form. 



Given the height of the weir or difference of level of the 

 upper and lower stream-beds, the depth of the upper stream 

 and the distribution of velocity across a vertical section of it 

 where the flow is horizontal, it is proposed to determine . 

 (1) the depth of the lower stream; (2) the distribution of 

 velocity across a vertical section of it where the flow has 

 again become horizontal; and (3) the total horizontal thrust 

 on the weir face or sloping stream -bed. 



It will be seen that the motion is not restricted to be 

 irrotational ; on the contrary, the fluid may be given any 

 desired degree of spin. Thus, while the solution in any 

 particular case is only partial, the problem considered is more 

 general than the irrotational case usually dealt with. 



In an Appendix some notes have been added bearing on 

 the work in the paper. As they deal only with the case 



* E. g., Bovey, Hydraulics, ]904; Merrimai), Hydraulics, 1904. 



t The hypothesis of the ultimate horizontality of the lower stream- 

 surface involves, when viscosity is neglected, a distinct assumption. Sonic 

 remarks in justification are given in §§ 5 and 14. I am indebted to Prof. 

 Lamb for some helpful criticism of this and other points in the paper. 

 Apart from this physical assumption, the results which follow are 

 analytically exact. 



