Vol. 6, 1920 
ENGINEERING: W. F. DURAND 
441 
However, too much faith cannot be placed on intercepts taken so near the origin, 
with small deflections. 
1 Thomson, J. J., London Phil. Mag., Sept., 1915. 
2 Rutherford, E., Proc. Cambridge Phil. Soc, 9, 1898 (401). 
3 Franck, J., Verh. Deuts. Phys. Ges., 12, 1910 (613). 
4 WelHsch, E. M., Phil. Mag., 34, 1917 (199); also New Haven, Amer. J. Sci., July, 
1917. 
5 Haines, W. B., Phil. Mag., Oct., 1915; also July, 1916. 
6 Thomson, J. J., Rays of Positive Electricity, Monographs on Physics, Longmans, 
Green & Co., 1913. 
7 Bohr, N., Meddelanden fran k. Vetenskapsakademiens Nobel Institute, 5, 1919, No. 28. 
SHOCK OR WATER RAM IN PIPE LINES WITH IMPERFECT 
REFLECTION AT THE DISCHARGE END AND IN- 
CLUDING THE EFFECTS OF FRICTION AND 
NON-UNIFORM CHANGE OF VALVE 
OPENING 
By W. F. Durand 
LbivAnd Stanford Jr. University 
Read before the Academy, April 27, 1920 
The classic treatment of the problem of shock in pipe conduits, as de- 
veloped by Joukovsky, Allievi and others, assimilates in effect the condi- 
tion of the water in the pipe line during the manifestation of the phenom- 
ena in question to that of a column of air in a closed organ pipe in longi- 
tudinal vibration, the reservoir end of the line corresponding to the mouth 
end of the pipe and the valve end of the line to the closed end of the pipe. 
On this basis the theory has been developed in some detail, especially by 
Allievi. 
In the treatment thus developed and in subsequent study of the prob- 
lem generally, it has been common to omit the following factors, the ex- 
istence of which must affect the result in actual cases : 
(1) The influence of the velocity head v'^/2g. 
(2) The influence of friction. 
(3) The loss of energy through the discharge valve considered as a 
nozzle. 
(4) The influence due to a time rate of valve area closure irregular, or 
other than uniform. 
(5) The influence due to the fact that the valve end of the line is not 
closed completely so long as the valve is partly open (as in the operation 
of opening or closing). The analogy with the air in a closed organ pipe 
is, therefore, imperfect and in particular the reflection at this end must, 
under these conditions, be incomplete rather than complete, as commonly 
assumed. 
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