Long Waves in Canals and Standing Waves in Closed Basins 161 



of very complete theoretical investigations. The literature on the seiches 

 theories is very extensive and, therefore, we can only refer here to the most 

 important works. The theoretical considerations only apply for oblong basins, 

 i.e. such basins where there are no components of the horizontal motion 

 in a direction perpendicular to the "Talweg" so that the motion of the water 

 is always parallel to the .v-axis, i.e. along the "Talweg" of the lake (see 

 Stenij, 1932). 



The equation of motion has the same form as it had previously [see 

 equation (VI. 5)], whereas the equation of continuity [see equation (VI. 27)] 

 becomes 



» = ~W)^ sm ' (VL47) 



if b(x) and S(x) are respectively the variable width at the surface and the 

 variable cross-section of the lake at the point x of the "Talweg". The boundary 

 conditions, neglecting the friction, are 



£ = for x = and x = I . 



(i) The theory of Chrystal (1904, p. 328; 1905a, p. 599; 19056, p. 637). 

 If we introduce into the equations (VI. 28) and (VI. 47) two new variables 



and if we put 

 we obtain 



u=S{x)l; and v = / b{x)dx (VI. 48) 



S(x)b(x) =o(v), (VI.49) 



8 2 u , .8 2 u , 8u 



For a lake of a constant width b, but of a variable depth h(x) we obtain, 

 if we put u = h{x)£, 



8 2 u . , .8 2 u , 8u f r _ . 



_= ? /,(.v)- and *--£. (VI. 51) 



The equations (VI. 50) and (VI. 51) are identical if we replace x with the 

 variable v. Chrystal proceeds as follows: a great number of cross-sections 

 close to each other are drawn. The width of these cross-sections at the 

 surface is b x ... b 2 ... b„. The area of each section is called S t ... S 2 ... S n . The 

 surface of the lake from section o to section i is designated v t . Chrystal draws 

 a curve which has as abscissa v t and as ordinate the product a t = ^.5,-. This 

 curve is called by Chrystal, the "Normal curve". A comprison of (VI. 50) and 

 (VI. 51) shows that the oscillations of an arbitrarily shaped lake [b(x), S(x)] 

 are the same as those of a rectangular basin with variable cross-section in the 

 x-axis, provided the "normal curve" is the "Talweg" of the lake. The form of 



n 



