| 
ON THE HYDRODYNAMICAL THEORY OF SEICHES. 60T 
kept pace with their accumulation. In the main, the original theory of Foren has 
been clearly established, viz., that a seiche is a standing oscillation of a lake, usually 
in the direction of its longest dimension, but occasionally transverse. In a motion of 
this kind every particle of the lake oscillates synchronously with every other, the 
periods and phases being the same for all; and the orbits similar (in fact, rectilinear), 
but of different dimensions, and not similarly. situated. Taking, for simplicity, a 
longitudinal seiche in a lake of uniform breadth and rectangular section, but vary- 
ing depth, the horizontal and vertical displacements of any particle on the surface 
originally at a distance x from a fixed point of reference would be given by 
E=$,(x) sin n(t—T), C=x,(x) sin n(t—7); where ¢ is the time measured from any 
fixed epoch, 7 an arbitrary constant determining the phase of the oscillation, and 
T=2r/n is the period of the oscillation. 
For a Jake of given configuration, an infinite number of different values of n (but: 
not any value) are admissible, say n,, n., 23, . . . ; and the functions ¢,(x) and x,,(a) 
are determined when 7 is given. 
For any given value of n, say n,, the function x, (v) vanishes for v different values 
of z. At these points, which are called nodes, the level of the surface is unaltered by 
the seiche. Corresponding to v=1,2, 38, etc., we have uninodal, binodal, trinodal, 
etc., seiches. Any number of these may coexist; and the total seiche displacement is 
obtained by adding these. When only one of these harmonic components is present 
we shall call the seiche pure. 
For a number of values of « , intermediate between the nodal values, $,(x) vanishes, 
and there is no horizontal motion of the surface particles. These points are called ventral 
pots. Four times the distance between a node and the next ventral point is called the 
wave length. Obviously the wave length is not in general the same at all points of the 
lake. When the wave length is large compared with the depth, which is always the 
case in a seiche, the wave is spoken of as a long wave; and the hydrodynamical theory 
in that case admits, as is well known,* of considerable simplification. 
§ 3. When the depth of the lake is constant, the theory of long waves leads to the 
well-known result 
é=A, sin = sin see) (t{-r), 
c= BEA, cos MF sin = VID (tn) ; 
where / is the length of the lake, d its depth, 2 the initial distance of the surface 
‘particle in question from one end, all measured in feet, and g=322:¢ denotes the 
time, and A, and 7 are arbitrary constants (amplitude and epoch), 
lt follows that in a lake of uniform depth the period of the uninodal seiche is 
-2l/,/(gd); and the periods of the uninodal, binodal, trinodal, etc, seiches are 
proportional to the terms of the harmonic series 1,4,4,.... Also the uninode 
* See Airy, art, “Tides and Waves,” §§ 187 et seqg., Encyclopedia Metropolitana, 1848. 
