632 Prof. T. H. Havelock on the Decay of Oscillation 

 In the previous notation, we have 



/ , has* . ov*x* . . hxi\ '."] 



t(^=\ co ^ + -^T sinh ^FJ' 



a . x / vi vixt\ . , fa* 2/*a , ft.** I" ' (20) 



0(#)=l r— i + —si Jsinh— H ^-cosh — > 



v \liX2 p l h) v* crpvi v\ J 



the formation o£ the latter being clearly indicated in the 

 reduction from (18) to (19). The coefficients of the solving 

 function can now be formed by (11). Finally, substituting 

 in (6) and carrying out the integration, we arrive at the 

 result 



_ 4:/jip 2 a ote at 



y ~ ah a 4 - {2/il<rh) [1 + 2ph/<r)a* + 2p 2 oc 2 + 6fip 2 a/ah +p* ' 



. . . (21) 

 the summation extending over the roots of (19). 



3. We may verify the result by other methods which are 

 available in this case. We choose Bromwich's method of 

 complex integration*, referring to his paper for the general 

 principles, and writing down the results briefly for the 

 present problem. 



Suppose the fluid velocity and the displacement of the 

 plane to be given by 



where u and 77 are functions of a, and the paths of integration 

 are in the plane of a complex variable a and enclose all the 

 poles of these functions. The differential equation of fluid 

 motion, 'dvj'dt = vd 2 vl'dx 2 , with the conditions u = for x=h 

 and u=-dr)jdt for x='Q, gives the solution, for x positive, 



^ sinh{a*(ft-dQ/p*} 



dt sinh(aM/v*j ' ' ' ^ 6) 



From the boundary condition (1), after introducing terms 

 due to the initial conditions y — a and dy/dt = for t = 0, and 

 using (23), we obtain 



au 2 7)+ ^-r-^coth— 1 +- <rp-7} — lax +■ -4- coth — )a. (24) 



Hence we have 



_ 1 C a{* + Q2p*il*v*) coth (hgtlviy^e^du. 

 y ~2iriJ a 2 i-(2^f/o-^)coth(A a W)fp 2 ' ' ^ ^ 

 Forming the residues of the integrand at the zeros of the 

 * T. J. I'A. Bromwich, Proc. Loud. Math. Soc. xv. p. 401 (1916). 



