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LONDON, EDINBURGH, ann DUBLIN ©”) 
PHILOSOPHICAL MAGAZINE: 
AND 
JOURNAL OF SCIENCE. 

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4 
[SIXTH SERIES.] 
JUNE 1904. SENG Wb ey ae 
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LXVIIIL. On Deep-water Two-dimensional Waves produced by , 
any given Initiating Disturbance. By Lord KeEnyty *. 
et. ONSIDER frictionless water in a straight canal, 
infinitely long and infinitely deep, with vertical sides. 
Let it be disturbed from rest by any change of pressure on the 
surface, uniform in every line perpendicular to the plain sides, 
and left to itself under constant air pressure. It is required 
to find the displacement and velocity of every particle of the 
water at any future time. Our initial condition will be fully 
specified by a given normal component velocity, and a normal 
component displacement, at every part of the surface. 
§ 2. Taking O, any point at a distance h above the undis- 
turbed water level, draw OX parallel to the length of the 
canal, and OZ vertically downwards. Let £, ¢ be the 
displacement-components of any particle of the water whose 
undisturbed position is (z, z). We suppose the disturbance 
infinitesimal ; by which we mean that the change of distance 
between any two particles of water is infinitely small in 
comparison with their undisturbed distance; and the line 
joining them experiences changes of direction which are 
infinitely small in comparison with the radian. Water being 
assumed frictionless, its motion, started primarily from rest 
by pressure applied to the free surface, is essentially irro- 
tational. Hence we have 
‘Ml sak: DR ie GSI tie 
f= dn? —e t) ; c= [Per t); E= de®: c= re, (aa 
where $(z, z, t), or @, as we may write it for brevity when 
* From the Proceedings of the Royal Society of Edinburgh for 
Feb. 1, 1904. Communicated by the Author. 
Phil. Mag. 8. 6. Vol. 7. No. 42. June 1904. 2T 
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