ON THE SEICHES OF LOCH EARN. 499 



Finally, I must apologise to the Trustees of the Lake Survey and to those who 

 assisted in the Seiche Survey for the long delay in the completion of the report. This 

 delay has been due to the heavy pressure of unavoidable professional and public duties 

 that has fallen upon me during the three years that have passed since the pleasant 

 autumn when we worked together upon Loch Earn. 



PART V. 



MATHEMATICAL APPENDIX ON THE EFFECT OF PRESSURE DISTURB- 

 ANCES UPON THE SEICHES IN A UNIFORM PARABOLIC LAKE. 



Estimation of the Effect of Pressure Disturbances on the Seiches in a 

 Symmetric Parabolic Lake of Uniform Breadth. 



I. In what follows I shall use the method of Normal Co-ordinates introduced by 

 Lord Rayleigh,* to which reference was made in my memoir on the Hydrodynamical 

 Theory of Seiches, § 21. f 



With very slight and obvious modifications, the notation employed is the same as 

 in the memoir just referred to, and, to make the results approximately applicable to 

 Loch Earn, it may be supposed that the length, 2a, of the symmetric parabolic lake is 

 G miles, say 10 6 cm., and the maximum depth 270 feet, say 8000 cm. Unless the 

 contrary is indicated, C.G.S. units are used throughout. 



Then we have, if £ and £ be the horizontal and vertical displacement at time t of a 

 particle on the surface of the lake, 



(1 - w' 2 )£ = u = - %al( v cos v v {t - t„)Q„(w) .... (1), 

 £= + %k v cos n v {t - t„)Q'„(w) .... (2), 



where nj = ghv(y + 1 )/a 2 , iv = x/a ; lc v is the extreme amplitude of the y-nodal seiche 

 corresponding to x= +a, i.e. to w= + 1 ; and Q v (w) is a solution of the equation 



(l-iv*)Q"v(w) + v(v+l)Q v (zo) = .... (3), 



which vanishes when w- ±1, and is such that Q r „(l) = 1. 



It is convenient for our present purposes to use the forms of the Seiche Functions 

 for which c = v{y + 1), given by Dr Halm, J viz. — 



*M-S-.s£<" , - 1 >' • • • (4) ' 



< *M-K\£p- l r .... <5). 



* Theory of Sound, vol. i., § 87 (1877). 

 + Trans. R.S.E., vol. xli. (1905). 

 + Ibid., p. 660 (1905). 



