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Submitted on 1 Jan 1979
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SECOND ORDER HARMONIC SURFACE WAVES GENERATED BY ONE FUNDAMENTAL WAVE
A. Shivarova, T. Stoychev
To cite this version:
A. Shivarova, T. Stoychev. SECOND ORDER HARMONIC SURFACE WAVES GENERATED BY ONE FUNDAMENTAL WAVE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-607-C7-608.
�10.1051/jphyscol:19797294�. �jpa-00219283�
JOURNAL DE PHYSIQUE ColZoque C7, s u p p l d m e n t au n07, Tome 4 0 , J u i Z Z e t 1979, paye C7- 607
SECOND ORDER HARMONIC SURFACE WAVES GENERATED BY ONE FUNDAMENTAL WAVE
A. Shivarova, T. Stoychev.
FacuZty o f P h y s i c s , S o f i a U n i v e r s i t y , 11 26 S o f i a , B u l g a r i a .
I n t h i s r e p o r t we d e s c r i b e e x p e r i m e n t a l = tan-'
sin@
+ ~ s i n g ) / ( C c o s O + ~ c o s ' q ) ] i n v e s t i g a t i o n s on p r o p a g a t i o n of second jj = tan-' [ ( c s i n o - B s i n ~ ) / ( C c o s o - ~ c o s y ) ] orden harmonic s u z f a c e waves i n a g a s d i s - B = [(2fa+,f2W)2+
( 2 k j k y )21
-1J2charg-e plasma. 2 -1/2
c
='[(
26,-[2a) + ( 2kW-k2*)] -
1The t h e o r e t i c a l e x p l a n a t i o n of our ex-
3
= t a n , ( ( 2 b+ b2*)
/ ( 2kw + k2W)]p e r i m e n k a l d a t a i s b a s e d on a model g i v e n 6 = tan-' ((26,
-
b u ) / ( 2 k u-
k2@)]by S a t o e t a1 [ I ] f o r n o n - l i n e a r mixing of I f t h e i n t e r v a l s I and 111 whose i n f l u e n c e d i s p e n s i v e waves, a p p l i e d i n t h e s p e c i a l
c a s e of p c o p a g a t i o n of one fundamental wa- v e of fnequency w, e x c i t e d a t z=0 i n a me- dium widh a weak n o n l i n e a s i t y . The optimum of t h e wave s t u d y by t h e methods used i n o u r e x p e r i m e n t s i s t h a t when t h e c o u p l e r i s n o t p l a c e d a t t h e plasma column e n d s . Thus, i n t h e t h e o r e t i c a l model [I] w.e i n - c l u d e t h e f a c t t h a t t h e fundamental wave of f r e q u e n c y 0 , phase v e l o c i t y vph=u/k, and s p a c e damping r a t e fu p r o p a g a t e s i n a d i r e c t i o n of z<O a s w e l l a s of z.70 ( P i g . 1 )
I c o u p l e r I I P ~ c Y - u P 111
I
-w,v ph
, ,b
m-P i g . 1
Because of t h e n o n - l i n e a r plasma f e a t u r e t h i s p r o p a g a t i n g wave a r o u s e a t any BE(-=,
+oo) ( F i g . 1) harmonic s i g n a l s of f r e q u e n c y 2w, which p r o p a g a t e i n b o t h d i r e c t i o n s (z$
B ) a c c o r d i n g t o t h e l i n e a r d i s p e r s i o n l a w w i t h phase v e l o c i t i e s vph=20/k2a and s p a c e damping r a t e s
kw.
I f we p u t a pick-up a t z(>0) t h e t o t a l s i g n a l of f r e q u e n c y 2 W a t t h i s p o i n t , A ~ ~ ( Z ,t,),
i s a r e s u l t of t h e i n t e r f e r e n c e of t h e waves g e n e r a t e a hy a l l v i r k u a l e x c i t e r s Z and r e a c h i n g t o t h e po- i n t i a a t t h e moment t.:~ ~ ~t ) oc ( M 5 e x ~ ( - ( ~ ~ z ) , s i n ( kZoz
-
2wt -A) ( 1 )-
K e x p ( - 2 b z ) s i n ( 2 k w z-
2 ~ t-13)
where
2 1 / 2 FI = [ ( ~ c o s o +Bcoslp)
+
(CsinOsing) ]
2 1/2 N = [(CCOSO - ~ c o s ~ ) ~
+
( ~ s i n o -Bsin'g)]
i s dekermined by B , z a r e i g n o r e d , t h e r e - s u l t ( 1 ) r e d u c e s t o t h a t i n [ I ] .
E x p r e s s i o n ( 1 ) shows t h a t two t y p e s of harmonic waves of e q u a l f r e q u e n c i e s 2 0 p r o p a g a t e s i m u l t , a n e o u s l y . The f i r s t one s a t i s f i e s t h e l i n e a r d i s p e r s i o n r e l a t i o n ( 2 ) k t = k 2 ~ ' = 6 2 ~
and f o r t h e second one t h e c o n d i t i o n s ( 3 ) k" = 2km,
t'
=2tw
a r e v a l i d . The i n t e r v a l I1 ( F i g . 1 ) con- t r i b u t e s t o t h e a m p l i t u d e of b o t h t y p e s of waves, whereas t h e c o n t r i b u t . i o n of I ia- c r e a s e s t h e a m p l i t u d e of t h e f i r s t wave and t h e c o n t r i b u t i o n of I11 d e c r e a s e s t h e a m p l i t u d e of t h e second wave, s o t h a t M>N.
F o r t h e h i g h - f r e q u e n c y a z i m u t h a l l y s i m - m e t r i c s u r f a c e waves we have k2m>2k,,
tZw>
26,,,. We e n s u r e e x p e r i m e n t a l l y t h e p r e s e n c e of i n t e r v a l I ( F i g . 1 ) and we o b s e r v e , i n a c c o r d a n c e w i t h e x p r e s s i o n ( 1 )
,
d i f f e r e n t s p a t i a l r e g i o n s ( c o n s e c u t i v e l y a l o n g the plasma column) of a dominant p r o p a g a t i o n of e a c h t y p e of harmonic waves. .The pnopa- g a t , i o n b e g i n s c l o s e t o t h e c o u p l e r w i t h a p r e d o m i n a t i n g of t h e f i r s t ( s h o r t , e r ) wave which, however, i s more a t t e n u a t e d . Next t o t h i - s r e g i o n a n i n t e r m e d i a t e r e g i o n f o l - lows where t h e two harmonic waves pr0pag.a- t e w i t h comparable a m p l i t u d e s . When t h e c o n & i b u t i o n of t h e f i r s t . wave g o e s down, o n l y t h e p r o p a g a t i o n of t h e second ( l o n g - e r ) wave which h a s a s m a l l e r a m p l i t u d e a t z=O b u t i s l e s s a t t e n u a t . e d , r e m a i n s a l o n g t h e plasma column. T h i s a l l o w s u s t o ob-Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797294
t a i n d i r e c t l y experimenQ1 d a t a b o t h f o r t h e wavenumbers and t h e space damping r a - t e s by r e l a t . i v e l y p r e c i s e methods f o r wave
i n v e s t i g a t i o n s such a s t h e methods of time -space diagrams and of phase s h i f t measu- r ements [2 ,3]
.
!Uhe experiments a r e performed on a h o t - cathod neon g a s d i s c h a r g e plasma a t low p r e s s u r e .
T y p i c a l i n s t a n c e s of our e x p e r i m e n t a l d a t a f o r t h e p r o p a g a t i o n of t h e two t y p e s of s u r f a c e harmonic waves ( g e n e r a t e d by one fundamental wave of frequency
u )
oh- t a i n e d by phase s h i f t measurements, a r e p r e s e n t e d i n T a h l . 1.
The c h a r a c t e r i s t i c s of &he f i r s t ( k t , ( ) and second ( k t ' , i t ) t y p e s of harmonic waves a r e compared w i t h t h e c o r r e s p o n d i n g c h a r a c t e r i s t i c s of t h e s u r f a c e waves e x c i t e d d i r e c t l y a t t h e dou- b l e f r e q u e n c y 2w(kZw, ) and a t t h e fun- damental f r e q u e n c y w (k,,&)
r e s p e c t i v e - l y ;ape
i s t h e e l e c t r o n plasma frequency.I n t h e columns denoted a s &L1 and nL" t h e end poinlm of t h e r e g i o n s of dominant wave p r o p a g a t i o n a r e g i v e n ( Z = O i s t h e c o u p l e r 5 o s i t i o n ) .
Tabl. 1
We e s k a b l i s h t h a t a t lower f r e q u e n c i e s w o n l y t h e f i r s t type of harmonic waves ( k'-jk2w,
dMkcJ)
p r o p a g a t e s a l o n g t h e who- l e l e n g t h (L=55cm) of t h e plasma column.Wim
t h e i n c r e a s e of t h e frequency t h e d i - s t a n c e over which we observe a predominant p r o p a g a t i o n of t h i s wave d e c r e a s e s . A%h i g h e r f r e q u e n c i e s t h e f i r s t t y p e of h a r - monic waves a t t e n u a t e s c o n s i d e r a b l y over a s h o r t e r U s t a n c e t h a n t h e plasma column l e n g t h and we observe a l s o a r e g i o n of a dominanf p r o p a g a t i o n of t h e second t y p e of harmonic wave ( kt'&2k,, , f 2 2 @ )
.
The wave-numbers measured i n t h e int.ermediate r e - g i o n change w i t h t h e d i s t a n c e b u t t h e y m c of t h e o r d e r of ( k 1 + k t t ) / 2 which i s tiha wa- uenumber of t h e r e s u l t i n g wave o b t a i n e d h.y t h e i n t e r f e r e n c e of t h e t.wo waves w i t h wa- venumbers k 1 and ktt and e q u a l a m p l i t u d e s .
Wim
ihurther i n c r e a s e 02 t h e frequency t h e r e g i o n of dominant p s o p a g a t i o n of t h e f i r s t wave and t h e i n t e ~ m e d i a t e r e g i o n va- n i s h c o n s e c u t i v e l y and t h e r e g i o n of p ~ o - pag-ation of second wave b e g i n s t o s h o r t e n , T h i s p i c t u r e of haxmonic wave p r o p a g a t i o n i s a s s o c i a t e d t o t h e @owing a t t . e n u a t i o n o f t h e waves w i t h t h e frequency i n c r e a s e . We suppose t h a t t h e d i s c r e p a n c i e s 3rom eqs ( 2 , 3 ] obtained above t h e e x p e r i m e n t a l ex- r o r a r e due t o t h e dependences of t h e wave c h a r a c t e r i s t i c s on t h e e x c i t i n g s i g n a l am- p l i t u d e . The s p a t i a l s e p a r a t i o n of t h e r e - g i o n s o b t a i n e d e x p e r i m e n t a l l y i s more a t z o n g l y t h a n t h e t h e o r y ( f o r m u l a ( ti),) p r e d i c t . The c a s e of t h e c o n s e c u t i ~ e ob- s e r v a t i o n of t h e t h r e e r e g i o n s i s i l l u s - t s a t e d i n F i g . 2a, whene t h e t i n e - s p a c e diagram of a harmonic wave i s shown. The,diagrams of t h e s u r f a c e waves e x c i t e d d i - x e c t l y a t t h e fundaneniial and double f s e - q u e n c i e s a r e g i v e n for: comparison i n P i g .
F i g , 2 R e f e r e n c e s
1
.
S a t o N. ,
Mark E. and Popa G. ( 1976) Plasma Phys. l8, 897.2 . Shiuarova A . , Stoychev T . and Rwseva.
S. (1975) J.Phys.D: Appl.Phys.
3,
383.3. B a l t o v a K . D . , Shivarova A . P. asnd Sto- ychev T . T . (1,978) P h y s . L e t t .
