174 Long Waves in Canals and Standing Waves in Closed Basins 



It will be necessary to introduce a correction for a canal or bay with a wide 

 opening to the free ocean, in whose width b at the opening is an appreciable 

 fraction of the length of the canal /. This correction increases the period, 

 similar to the behaviour of vibrations of air in pipes which are open at one 

 end. As a matter of fact, in the oscillatory processes at the opening, the water 

 not only flows forth and back in the longitudinal direction of the canal, but 

 it is also drawn from the side and thrown back sideways. The investigations 

 made by Rayleigh (1897) show, as is to be expected, that the correction 

 is the greater, the wider the opening of the bay is in proportion to its length. 

 The corrected period of the free oscillation then is T{\ +e), if T is the period 

 computed according to Merian's formula (VI. 74). For e we find, if b is the 

 width and / the length of the canal closed on one side, 



b G .nb 

 S= xl\2- ln 4l 



in which y is Euler's constant 057721 5. Different ratios 

 for 1+e (Table 22). 



Table 22 



y\ 



(VI. 75) 

 / gives the values 



Width 

 Length 

 Correction factor 



1/2 



1/3 



1/4 



1/5 



1-371 1-297 1-240 1-203 1-176 



1/10 



1110 



1/20 

 1066 



It can be seen from these values that when the width of the bay equals 

 the length, the increase in the period is 37 % but is reduced to about 1 1 % 

 if the width is one-tenth of the length. Proudman (1925, p. 247) has pointed 

 out that this mouth correction only applies if the shore from which the canal 



Fig. 74. Computation of the correction for the opening. 



branches out is straight and if there is a uniform distribution of a current 

 transverse to the open end of the canal. If the coastline is funnel-shaped and 



