ON THE HYDRODYNAMICAL THEORY OF SEICHES. 613 
EA ae 
MATHEMATICAL THEORY. 
GENERAL THEORY OF A SMALL LONGITUDINAL SEICHE IN A LAKE 
OF VARYING DEPTH AND CROSS SECTION. 
§ 20. Let O X be a longitudinal axis in the undisturbed surface of the lake. Obser- 
vation seems to show that this axis should be as nearly as possible in the average 
direction of the channel of greatest depth. Take OZ vertical, and O Y horizontal and 
perpendicular to O X. 
Consider any cross section at a distance O P= from the origin. Let the area of 
this section be A(x), and its breadth at the surface-b(x). Take a section parallel to 
A(x) at a distance dx from A(x). The volume of this slice (S) will be, to the first 
order of small quantities, A(x)dz. 
Suppose that, after a time, ¢, the slice, S, has moved into a new position, so that the 
distance of its posterior face from O is now x+& ‘Then the breadth of S in its new 
position will be dx(1+<é/ax); and the part of its volume below the normal level of the 
lake will be A(x + &)dx(1 + 0é/dz). 
If we suppose the rise in level of the slice to be the same throughout, say ¢, which 
involves the assumption that there is no flow parallel to OY, and that all the water 
particles in the same transverse vertical plane have the same velocity parallel to the 
plane ZO X, then we may take the increment of the slice owing to the rise of the water 
above the original level to be o(x)fda(1+<é/éx). In so doing we neglect the effect of the 
shelving of the shore ; so that our calculation would certainly not apply in cases where 
the seiche causes a large horizontal displacement of the high-water mark. 
With these assumptions the equation of continuity is 
A(x)dx = {A(x + €) + 0(x)E}dx(1 + d€/ax) : 
that is, 
Cb(x) = A(a)/(1 + 0&/ax) — A(x + &) : : - : (L) 
Since the amplitude of the seiche is small, we neglect vertical and consider only 
horizontal acceleration. The difference between the pressures on the two sides of the 
slice in its disturbed position will therefore be simply that due to the difference of 
level at its two ends, viz., ged per unit of area, p being the density of the liquid.* 
The equation of motion for S, regarded as a whole, is therefore 
dix(1 +2¢/2z)p- <5 = - gp de 
aes) <<: Bere + 
* In order that these assumptions may be justified, the square of the ratio of the depth to the wave length must 
be negligible at every part of the lake. See Lamb’s Hydrodynamics (1895), § 169. 
that is 
