ON THE SEICHES OF LOCH EARN. 



375 



It follows from the classical researches of Osborne Reynolds, " On the Motion of 

 Water and on the Law of Resistance in Parallel Channels," * that, if the velocity in the 

 access tube does not exceed w e = 2008v/b, where v is the viscosity of water at the 

 temperature 0°C, viz., v = '0178/(1 + -03370+ -OOO2210 2 ), then the resistance in the tube 

 will be proportional to the mean velocity of flow, and we can apply the well-known 

 formula of Poseuille to calculate the difference of pressure at the two ends of the tube 

 for a given velocity of flow. 



For a seiche of recorded amplitude B, and period T, the maximum velocity of rise or 

 fall of level is 2ttB/T, which gives a velocity of w b = 27rBa 2 /b 2 T in the access tube. 

 Now, in Loch Earn the range recorded rarely exceeded 4 cm. ; so that B = 2. We may 

 take a= 15 cm., which was about the diameter of the small wells used with the index 

 limnographs. Taking the temperature of the water to be 17°C, so that v= -0109, we 

 get, for the uninodal seiche of Earn (T = 870 sec ) : — 



w e = 21-89/6, iv b = 3-250 /b\ 



Hence the following table for the three diameters b which were actually used : — 



b 



w. 



Wl 



1-27 

 •63 



•47 



17-23 

 34-74 

 46-36 



2-02 



8-19 



14-71 



The actual maximum is in all cases well under the critical velocity.! 

 Since the motion is very slow, we may treat it as steady, and then we have at any 

 time t, by Poseuille's Law : — 



a 2 dx ■wb i g(x - y) 



4 dt 128W 



Hence, taking v= -0109, g = 982, and putting 



x = gb*/32vla 2 = 281 3&V« 2 



we get 



■ft- -xk*-v) 



(!)■ 



(2), 

 (3). 



Let us now suppose that the lake level is subject to a harmonic denivellation of 

 amplitude A and period T ; then, if n = 2-tt/T, and d be the mean depth at the opening 

 of the access tube, we shall have y = A sin nt + d ; and, therefore, 



dx 



~j7+X x = X-A sin nt + \d 



(4). 



* Phil. Trans. Roy. Soc. Lond., 1883. 



t It should be noticed that this is not the critical velocity (u c = l3000v/d) at which turbulent motion begins. The 

 critical velocity v c at which the resistance begins to be proportional to (velocity)", where n is a number varying from, 

 say, l - 72 to 2'00, is again different, viz., v c =l"325w c . 



