Adiabatic radiative acceleration of a piston in a uniform cylindrical waveguide having an arbitrary singly connected cro
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RADIATIVE
IN A UNIFORM
HAVING CROSS
AN
ARBITRARY
ACCELERATION CYLINDRICAL SINGLY
OF
A
WAVEGUIDE
CONNECTED
SECTION L.
S.
Terekhov
UDC 538.569:531
1. Study of the r a d i a t i v e a c c e l e r a t i o n of p l a s m a is c u r r e n t l y in its initial stages and in fact has cons i s t e d e n t i r e l y of the study of p l a s m a s in an intense h i g h - f r e q u e n c y field [1]. However, since it is hoped that the difficulties involved in r a d i a t i v e a c c e l e r a t i o n of p l a s m a in a waveguide will be o v e r c o m e , it s e e m s worthwhile to analyze r a d i a t i v e a c c e l e r a t i o n f r o m the p u r e l y energetic point of view, and, neglecting l o s s e s , to t r a c e the t r a n s f o r m a t i o n of e l e c t r o m a g n e t i c e n e r g y in the f o r m of standing waves filling the closed waveguide cavity into the m e c h a n i c a l e n e r g y a s s o c i a t e d with the t r a n s l a t i o n a l motion of the f r e e end of a w a v e guide cavity; in this p r o c e d u r e the moving end of the cavity is identified with the a c c e l e r a t e d p l a s m a . 2. We a s s u m e that a waveguide cavity with p e r f e c t l y conducting and absolutely rigid walls is f o r m e d f r o m a section of a uniform waveguide having an a r b i t r a r y singly connected c r o s s s e c t i o n s (Fig. 1). One end Sm can m o v e f r e e l y along the waveguide axis x, t h e r e b y f o r m i n g a cavity of v a r i a b l e length X. Some oscillation mode, f o r example, an H mode, is excited in the cavity at f r e q u e n c y v. We set the c r o s s - s e c tional a r e a S equal to unity, so that the p r e s s u r e p and the f o r c e acting on the piston a r e equal to each other and so that the length X is equal to the cavity volume. The piston begins to move at t i m e t 0. We r e s t r a i n the piston motion by the adiabatic condition, i . e . , we r e q u i r e that the state of the " w o r k ing m e d i u m , " in this c a s e the standing e l e c t r o m a g n e t i c wave, differ at any instant only infinitesimally f r o m the equilibrium state, so that the group velocity Vgr at which the e l e c t r o m a g n e t i c wave p r o p a g a t e s in the waveguide m u s t be much l a r g e r than the piston velocity X: A p a r t i c u l a r f o r m of the mechanical t h e o r e m on adiabatic invariance [21 is the action t h e o r e m , according to which the action UT (where U is the e n e r g y of the oscillation, which occurs o v e r a period T) is invariant with r e s p e c t to adiabatic d e f o r m a t i o n [3] or, equivalently, we have U. ~ = c0nst, (2) where X is the f r e e - s p a c e wavelength. The second group of conditions which we adopt as initial data follow f r o m the f a m i l i a r r e l a t i o n s for standing waves in a cavity. The e n e r g y U in the cavity and the p r e s s u r e p on piston Sm a v e r a g e d o v e r period T for an H wave can be written in t e r m s of integrals of the t r a n s v e r s e (Hs) and longitudinal
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