WAVE RESISTANCE OF A CYLINDER 331 
with 
{es (GAY) =P) 
ey HO) Oe Sh. 
pe ans (18) 
wca 1 2, AVP 
ary Peon ial 
f 0 0 
Senin (eV) (t-7)_g-in (et V)(t-7M bg 2a S 5/2 dy 
Taking the real part of (16) and integrating with respect to 7, we 
obtain for the wave resistance 
ongped if “Bes k(e - oye sin k(¢ + Mh. Miivte. (GO) 
Gao) Coe 
Putting x = x,u* = gu*/c* this becomes 
0 
{oo} 
R= tngox iat | [Selene Ot Dh + ene | e Bury sdy (20) 
0 
with d=x,ct, B = 2x of. 
For suitable values of the parameters a@,8 the integrals in (20) may 
be computed by direct quadrature, or from convergent and asymptotic 
expansions which may readily be deduced. In particular, the limiting 
value as ¢ becomes infinite follows directly from the first term in the 
integrand and is 
R = 42? gpx ate ?* of (21) 
the wave resistance for uniform motion. The next approximation for 
t large is of order ¢7!/? and can be obtained from the same integral by 
the method of stationary phase. This gives, as t > ~ 
204 
R +4 Baio! ap uf 
7 gpk 4 2a4e + + T9PK a Kot 
J eat’ fein (Teg ct-7) 22) 
Thus ultimately the resistance oscillates about the steady value, 
the amplitude of the oscillations diminishing slowly with the distance 
travelled and the period being roughly 4A), corresponding to the per- 
turbation of the wave motion given in (12). 
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