ON THE HYDRODYNAMICAL THEORY OF SEICHES. 629 
If we put ac =a’! = n’a’a"/gh =z, the equation (41) may be written 
1-74, \(1-3%5) ot (i- al ee ay) 
af 1.2a? 3.402 9.3a'2 : 4.5a/2 
| ef) i Oi my a dethc(4ays 
+a( T.2a” 34a? xa) 4 bat ) 
Then the period of the v-nodal seiche is given by 
T,=22/n,=2raa/ j(zgh); . ; 5 ° ; (43) ; 
where z, is the corresponding root of (42), 
For some purposes it is convenient to put a/a’=p. The equation (41) then becomes 
(pK Gaze Dyn ik(eh I) = Oly Ay 
and we have 
T, = 27a / J (gh) 
= 2ml/(1 + p) Vegi), V ; (45). 
=2nl/( J+ Jer) J(gh). § 
The equations for the seiche displacement in the two portions of the lake may now 
be written 
ogee 1)C(c, w) - C(e, 1)S(c, w) }sin net ; 
(46) ; 
© =- afeni St 1)C'(c, w) — Cle, 1)S'(c, w)}sin nt ; 
EN a we, HIE Cle, w’) + Cle’, 1)S(¢, w’) }sin ne ; 
(47). 
€ =- ier aS(¢,1)* (c', YC(c,, w’) + C(c’, 1)S'(¢, w’)}sin nt. 
It is obvious from (44) that when p is given the value of c’ is determined. Now p 
is the ratio of the distances of the deepest section of the lake from the ends. Hence, if 
this ratio remain unaltered, we see that T, is proportional directly to the length of the 
lake, and inversely to the square root of its maximum depth.* 
In particular, it follows that, if the basins of two lakes be geometrically similar, the 
seiche periods are directly proportional to the square roots of the linear dimensions ; a 
result obvious by Newron’s principle of dynamic similarity. 
A graphic picture of the solution of the equation (44) may be obtained as follows :— 
___ If we trace the curves whose equations are y = K(c’, 1), py= —K(p’c’, 1) , c’ being the 
common abscissa, then the values of c’ corresponding to the intersection of these curves 
are the roots of (44). The latter of these curves is deducible from the former by 
diminishing all the abscissze in the ratio 1: p’ and all the ordinates in the ratio 1: p, 
and then taking the image of this deformed curve in the axis of c’. It is thus easy to 
see by drawing a schematic diagram that the effect of increasing p is to diminish ¢’. 
The period depends on the value of (1+p),/c’ or c+ ,/c’; and the effect of the 
increase of p upon this is not so easy to trace by direct analysis. Since, however, the 
* In the general case A is the maximum value of the product of the area of a cross section by its surface breadth ; 
and a and a’ the areas of the lake surface between the corresponding section and the ends. 
