631 Prof. T. H. Havelock on the Decay of Oscillation 
The method of formation of O(x) is clear from the equa- 
tions for a finite number of terms; multiply the left-hand 
side of (12) by y(z) and a factor to make the value unity 
for « zero. From (11) and (12), we have 
yee | 24(5 a oh e <r) we (13) 
oh\2v  4u 0 Ow? 
Writing the roots of (12) as «= —vd/h? and collecting 
the results from (6), (12), and (13), the first step in the 
solution of (3) gives 
PY 6 4p? (* Q2e-vAX(t-7)/A? 
Sa 2d = as |, eee) 
ah }o 
(14) 
where the summation extends over the positive roots of 
A tanA=2ph/o=k. Pate rf: OLS) 
Following the method of reduction for this type of equa- 
tion*, integrate (14) with respect to ¢ from O to @, using 
Dirichlet’s formula to transform the order of integration 
of the last term. Since the initial value of dy/dt is zero, this 
leads to 
dy(@) _ App h 6 & ea vA2(0—t)/h2 
i a JO™ aa Kee ee (16) 
Integrate (16), in the same manner, with respect to 0 
from 0 to T; finally, for convenience, replace T by ¢ and 
t by 7, respectively, in the result. Then we obtain 
(3 op’h  App?h3 7 vAMt=7)/h2 
t ea oth >; aaa” Mr S0h, 
y( +f vo Ti a eRe OA ee a. (17) 
The solution of (17) can be completed by means of (6), 
(8), and (11). The new exponents are yiven by 
op*h 4up?h® 1 ih 
z) of “AI ERLEO|@roye) | oS) 
Resolving the summation into one of simple partial frac- 
tions and using the properties of the roots of (15), this 
equation can be reduced to 
2pitaz h 
a? + coth = + p=0. so o (9) 
* Volterra, ‘Tecons sur les Equations Intégrales,’ p. 140. 
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