Long Waves in a Rectangular Trough. 163 



fitted down the middle of the trough dividing it into two. 

 For a fresh surface, h=5 cms., on removing this partition k 

 diminished approximately 10 per cent. 



§ 4. It is interesting to note the manner in which the 

 velocity falls off at the hottom of the trough. 



Z = real part {cosh kh — cosh k(h — z)}, 

 = cos ^J^h cosh ^/Ih-coa yf£ (h-z) cosh y/^ v Qi-z). 



Taking h = 5 cms. we find that at different depths Z is 

 roughly proportional to the following numbers, the value at 

 the surface being taken as unity : — 



z 



Z 



0-0 mm. 



•000 



0-5 „ 



•129 



i-o „ 



•520 



1-5 „ 



•732 



2-0 „ 



•811 



3*0 „ 



1-03 



4'0 „ 



1-04 



5-0 „ 



1-02 



6-0 „ 



1-01 



7-0 „ 



1-00 



Thus the velocity reaches its surface value 3 mm. from the 

 bottom. 



§ 5. The case of a circular sheet with symmetry about the 

 centre can be treated in the same way as the rectangular 

 trough. 



If u denote the radial velocity the equation of continuitv 

 is 



h-fM^ (23 > 



The equation of motion is 





Eliminating p by means of (1) and (23) we get 



B- 





2* 



M 2 



(25; 



