- 13 - 281 



APPENDIX 



Proof triat equations (l) and (5) in the text are equivalent. 



We assume that the velocity u(t) is such that it can be expanded as a sum or integral of 

 decreasing exponential functions of the time. We apply to equation (5) the operator 



. t . f ^ — «hich has the effect of changimj p into the function which takes the 



ZTTi'^^QoaJ-rn sjr- 



value zero for t <^ and p~" for t > 0. This result follows by completing the contour by a 



semicircle below the real axis for t < and above the real axis, for t > 0. In either case, 



the integral round the semicircle vanishes asa)~*<" by Jordan's Lemma and the result follows. 



The pressure on the piston due to a velocity U, which can be expanded as a sum or integral If such 



functions in the form U » X *p P~" is given Dy applying the same operator to the right hand side 



of equation (5). The integrals that result are very similar to those considered by Gallop (see 



Watson's Sessel Functions page 421) and are best evaluated by expression of the Bessel Functions 



as integrals of Poisson's type and reversing the order of integration. Equation (5) thus gives 



us, for the reaction force due to such a velocity U. the expression:- 



P =^ -^oIIL^ /- AcJ^ ^.J^ sin^^p-^'^''^-^ ,e) dc. A(l) 



" in i -00 tj- i n y / 



For t < the contour integrals both vanish. For t > the first term becomes as before 



-p C Tr R Za p~" , while for the second term we must put the contour above the origin only for 



those values of 6 for which ai t > 2 kR cos 6, i.e. t > ^ '^ ^os ^ ^^^j-g ^g f,a„g_ |f ^. j^ j^g 

 cr it ical value, 



? ^ ? nr rT 7 2n R cos d 



F = -p^C77R^u+2 p^ C 77 R^ A^ p-"' /^^ 5in^i9p C ad a{2) 



Changing the variable sn that t' - -r- cos 9, it follows that a(2) Is equivalent to equation (l) 



in the text. For t > -~ th'^ integral in equation a(2) Is from to ? and that in iquation (l) 



2R 

 from to Y" • 



The more general form of equation (l), when U is not zero for t < 0, is 



F = -Po-«^/:^(f)(^o^_^.- *(3) 



where i// f -— J = ^ - - {s\n 2 y * 2 y ) 



m 



where sin y = 



Ct' 

 2R 



Substituting u - u^ p in equation A(3), it can he verified that the equation reduces to 



Raylelgh's expression (5). Thus the equivalence of the two methods is established. 



