Secondary Undulations of Oceanic Tides. 107 
(i.) Rectangular bay of constant depth. 
Let I and h be the length and the depth respectively of a 
rectangular bay of constant depth ; then the period T of the 
free oscillation of the bay, which has its node at the mouth 
and its loop at the end, will be given by the formula 
, T -c% w 
provided the correction due to the mouth be neglected. 
This correction may be approximately found in the following 
way. 
Take the origin of the rectangular coordinates at the 
middle point on the mouth of the bay ; #-axis in the direction 
of length, positive inwards, and ^/-axis upwards. Assume 
the vertical displacement 77 inside the bay to be given by 
. 7T.V 2irt 
T)=asm -gjcos-™-. 
If we neglect the vertical acceleration, we have 
where f? is the horizontal displacement ; hence 
and 
l 4/ irx . 2irt 
f=-«-cos-sm-^-. 
If b be the breadth of the bay, the kinetic and potential 
energies inside the bay are given by 
K.E. = ipkb$?dx and P.E. = ±gbp$i?dx ; 
assume also the^ kinetic energy outside the bay to be 
Phb 2 pt; 2 , where f is the value of f at x=0, p the density of 
water, and P the dimension of a number. Neglect 'the 
potential energy outside the bay, which is very small, and 
write down the condition of the constancy of energy. From 
the two relations which are obtained" by putting t = 
and t=-in this equality, we get the expression for the 
period of oscillation : — 
