110 Messrs. K. Honda, T. Terada, and D. Isitani on the 
(iii.) Dumbbell-shaped bay. 
The above formula does not hold when a portion of the 
lake is very much contracted. In this case, then, we may 
treat the problem in a quite different way. 
When two basins communicate with each other by a 
narrow canal, the mode of oscillation of the longest period 
takes place when the levels of the two basins rise and fall 
alternately. If the breadth of the canal be very narrow 
compared with the dimensions of the two basins, we may 
assume that the rise and fall of the level are uniform for 
each basin, and that in the canal the level is invariable, the 
motion of the water being chiefly horizontal. Then, denoting 
the areas of the basins by S and S', the breadth, the depth, 
and the length of the canal by b, A, and I respectively, the 
displacement of water in the canal in its direction by f, and 
the vertical displacement of the surface of S and S' by 97 
and 7]' respectively, the potential and the kinetic energy are 
given by 
P.E.^I (S^-f SV 2 ), and K.E. = ^f 2 . 
Again, the correction to the kinetic energy on each end of 
the canal is nearly 
^(i-7-io g ?>, 
in which \ is the wave-length, if the basins be infinitely 
wide, and may be considered nearly equal to four times the 
length of the basin in the direction of oscillation. 
Since St?= — SV and S?7 = ^£, 
we obtain in the usual manner for the period of oscillation 
Special interest is due to the case when one of the basins 
becomes infinitely large ; in which case the problem reduces 
itself to that of a bay communicating with the open sea 
through a narrow neck. Taking A, = \' = 4L, where L is the 
length of the bay measured along the probable direction of 
propagation of waves, we obtain from the above equation, 
•vst+a— -a} 
(5) 
