ON THE SEICHES OF LOCH EARN. 511 



Again, if we take T = T x as before, but t x = — , 



dk x = B/ = § &dp ; 



8tj = - A x 'jk x n x = f &dplk x n x ; 



and the maximum value of dk x is "529p, corresponding to v = 97 (mile/hour). 

 Lastly, for T = T l5 t x = — 



7T 

 "2 



8A-j = - B x = - |<J>3^ ; 



9tj = A x '/k x n x = - #©3/)/^?^ ; 



and the maximum value of 3^ is "719/9, corresponding to y = 28 (mile/hour). 



Alternative Method. 



13. If the disturbance of phase is not required, the following method, by means of 

 which I originally obtained some of the results given above, will furnish the disturbance 

 of the extreme amplitudes of the various seiche components due to a given disturbance 

 of pressure, to the same degree of approximation as Rayleigh's method. 



If K = % + Q$ denote the whole energy of the seiche motion, p the pressure at any 

 point of the water surface, and i\ the velocity of the water at that point in the 

 direction of the normal to the surface drawn towards the water, then the following 

 equation holds : * 



-p.. = uxpv v = a dwpi: v . 



J -a J -I 



It is easy to show that, for our purposes, the above equation may be written 



diopC ..... (41); 



for in so doing we neglect only quantities of the orders of k„ z /ha (<1/10 8 ) or 

 k?/a 2 ( <L/10 10 ), already negligible if we are to apply the theory of long waves. 



Suppose now the seiche motion to be analysed into uni-, bi-, tri-, .... nodal 

 components whose amplitudes at the ends of the lake are k x , k 2 , k s , . . , . 



Since these components are normal modes of motion for the parabolic lake, we may 

 calculate the total energies for each of these seiches separately and independently ; and 

 the sum of these energies will be K. Let these partial energies be K x , K 2 , K 3 , . . . . 



Taking the v- nodal seiche by itself, we have the equation 



/•■+i 

 ^=-a\dw P l (42). 



In the integral on the left-hand side of (42) we need pay no attention to any constant 



* See Lamb's Hydrodynamics, 3rd ed. (1906), p. 9. 

 TRANS. ROY. SOC. EDIN., VOL. XLVI. PART III. (NO. 20). 77 



