ON THE HYDRODYNAMICAL THEORY OF SEICHES. 615 
As (10) is simply the canonical form of the linear differential equation of the second 
order, a variety of cases can be devised in which the seiche problem can be solved in 
finite terms; and in any case where A(x) and b(x) are given slowly varying functions, 
approximate solutions can be found, with more or less labour. 
§ 21. In all the seiche problems considered in this paper we have at the ends of the 
lake either A(x)=0 or else £=0; thatisw=0. 
Moreover, the equation (7) may be regarded as the equation of motion of a vertical 
string vibrating in one plane, the ratio of whose tension to the longitudinal density is 
go(v), v being the distance of any point P of the string from one end when the whole is 
at rest. The variable u denotes the lateral displacement of the point P at time ¢; and, 
in view of the conditions w=0 at both ends of the lake, we may suppose both ends of 
the string fixed. We can then deduce the seiche displacements from the motion of the 
string by the equations 
€=u/A(x), f= —odu/ov. 
It will be observed that the nodes of the string correspond to ventral points of the 
seiche, and vice versa; and it appears that we could, by experimenting with a string 
loaded so that its density is inversely proportional to the product of the area and 
surface breadth of the cross section of a lake, roughly determine in the laboratory the 
periods and nodes of the pure seiches that might occur in the lake. 
It follows from Srurm’s Oscillation theorem™ that in any given lake seiches are 
possible which have 
ventral points and 
nodes respectively. 
In other words, pure seiches of all degrees of nodality are possible; and the most 
general seiche disturbance is a sum of such pure seiches with arbitrary amplitudes and 
phases. We regard the ends of the lake as ventral points because u always vanishes 
there, although in most cases the horizontal displacement does not vanish, as it should 
do at a ventral point properly so called. 
The identification of the seiche problem with the theory of a vibrating string is not 
only very instructive from the physical point of view, but is very helpful mathematically. 
For example, when we have worked out the periods and nodes of a seiche for any simple 
configuration approximately fitting a given lake, we can correct for the divergence of 
the actual lake from the assumed mathematical form by means of the beautiful method 
described by Lorp RayteicH in his Theory of Sound, vol. i. § 90. 
* See RayueEran’s Sownd (1877), vol. i. § 142. 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 25). 91 
