629 Prof. T. H. Havelock on the Decay of Oscillation 
the ordinary solution of a damped harmonic vibration requires 
modification when the initial conditions are taken into 
account, but no explicit solution of this nature seems to have 
been piven; in certain experimental refinements, the dis- 
turbance may be of some importance. In the following 
notes, I have worked out in detail first the simpler case of a 
plane oscillating between two fixed planes. The problem 
can be solved by various methods: by normal functions, or, 
more readily, by operational methods. I have chosen to use 
it as an example of a type of integral equation, for which 
reference may be made to a previous paper*. In this case 
the equation of motion is an integro-differential equation of 
Volterra’s type, and it can be solved by a repeated appli- 
cation of Whittaker’s method which was used in the simpler 
cases; the solution may be of interest apart from the parti- 
cular problem. ‘The results are then verified by using 
Bromwich’s method of complex integration. Finally, the 
solution is indicated for a sphere oscillating within a fixed 
outer sphere, and the results are discussed in connexion 
with the experiments to which reference has already been 
made. 
2. Suppose that a viscous liquid can move in laminar 
motion between two fixed planes z=+h. Let the plane of 
uz be a thin rigid barrier of mass o per unit area, and let it 
be acted on by an elastic force parallel to Oy such that, if the 
liquid were absent, the plane would vibrate with a natural 
period 27/p. Further, suppose the motion starts from rest 
with the plane displaced a distance a from its equilibrium 
position. The equation of motion of the plane is 
d*y Ov 
v2 ~2#(5,), tPU=O - jd) ob (UY) 
where v is the fluid velocity. 
Now if the plane of yz has a velocity V(t), the fluid 
velocity may be written in the form 
22.n1rv - nie (* lyr /he 
al oa e7 Rm 2vt[h? sin e iV irjennndr. 5 @) 
Taking the value of Qv/dx for c=0, integrating by parts 
and noting that in this problem V(0)=0, equation (1) gives 
Vy , 2p (‘ay Se naaetear 
ae dalam m Lartpiy=0. (3) 
* Supra, p. 620. 
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