328 
Proceedings of Royal Society of Edinburgh . [sess. 
! 
Some Results in the Mathematical Theory of Seiches. 
By Professor Chrystal. 
(Read July 18. 1904. MS. received July 29, 1904.) 
(Abstract.) 
I propose in this preliminary communication to lay before the 
Society some results of investigations in the theory of Seiches in 
a lake whose line of maximum depth is approximately straight, 
and whose depth, cross section, and surface breadth do not vary 
rapidly from point to point. 
As the seiche disturbance is small compared with the length of 
the lake, I shall make the assumptions usual in the theory of long 
waves: — viz., that the squares of the displacements and of their 
derivatives are negligible. 
The cc-axis, OX, is taken in the undisturbed level of the lake, 
in the average direction of the line of maximum depth ; the 2 -axis, 
0 Z, is taken vertically upwards. The horizontal and vertical dis- 
placements of a water particle originally in the undisturbed surface, 
at a distance x from the origin, are denoted by £ and £. A(x) and 
b(x) are used to denote the area and the surface breadth of the 
cross section at a distance x from 0. 
We suppose that the vertical disturbance at every point in the 
surface line of any cross section of the lake is the same ; in other 
words, we neglect the dynamical effect of any flow perpendicular 
to 0 X due to the gradual increase or diminution of the area of 
the cross section of the lake. As in the theory of long waves, the 
vertical acceleration is also neglected ; and we also neglect the 
(usually small) effect due to the shelving of the shore. 
With these assumptions, the equations which determine £ and £ 
are found to be 
d 2 u 
dt 2 
= g<r(v) 
bv 2 ’ 
(1) 
( 2 ) 
