3 MARCO CONTEXTUAL.
3.1 PROGRAMA EDUCACIÓN PARA LA PARTICIPACIÓN Y LA CONVIVENCIA CIUDADANA-EDUPAR
p la n e
Eemembering t h a t and P a re fu n c tio n s o f p , v/e fin d t h a t
rem ains f i n i t e a s p —^ 0 , b u t t h a t a t p - - b / vn th e re i s a s in g u l a r i t y v/Mch le a d s to th e c o n tr ib u tio n to y ( L p ) - S t/w i - 1 3 v 1: p * J L (4*14) ^ 2 T 1 + 6 lAL I
F o r a l i g h t f i b r e B /vn i s a freq u en cy o r decay c o n s ta n t much M g h er th a n any we a re i n te r e s te d
in,
and so v/e ta k e th e l i m i t a s :L t Y » ( l t ) « 0 f o r a l l • ( • > 0 . (4 .1 5 )
'B/vk“^oo
T his c o n tr ib u tio n i s th e r e f o r e n eg lected #
The rem ain in g s i n g u l a r i t i e s a re g iv en by th e ro o ts o f
tk i^L B 1 « (4*16)
Let
p. K when |) B oc^ , so t h a tf o r B / |
(4.1 7)
1
1
Let , ___ Then (2f,16 ) becomes "gi j W T (e -I- L&) in th e l im i t t h a t B /v n —5> <^ , i . e . f o r a l i g h t f i b r e . T his e q u a tio n h as s o lu tio n swith 6 = 0
g iv e n byta n §” B ^ (4 # 20)
iv \T r
I n th e e x p erim e n tal c o n d itio n s
I
, so t h a t th e s o lu tio n sare
fV\T
S' i TT (-) , Z'lT (-) , l~) , ••.)
( 4 . 2 1 )
6 s= 0
S in ce in o u r p r e s e n t
approximation
^ Fh » th e s e s i n g u l a r i t i e s o f Y g iv e n o n - o s c illa to r y ,exponentially decaying
c o n tr ib u tio n s to y (ij't) w ith decay c o n s ta n tsI s no
(45)
For v ery la r g e v a lu e s o f S th e approxim ation j
B /
1lo n g e r v a lid and so th e c o n tr ib u tio n s to a re n o t p u ie ly exponential# I n th e extrem e o p p o s ite l i m i t , where B /m ^ oT eq u atio n (4 *16) becomes h ' which h a s s o lu tio n s tu T J
T
(4.23) 7f
(4.24)Tfith a larg'o in te g e r# Those alm o st undamped, o s c i l l a t o r y o on t ii- b u tio n s to y a re n e v e rth e le s s n o t p h y s ic a lly Im p o rta n t, b ecau se t h e i r a c tu a l am p litu d e i s v ery sm all#
I n o rd e r to fin d s i n g u l a r i t i e s o f w ith n o n -zero v a lu e s o f C 9 e q u a tio n (4*19) i s s e p a ra te d in to r e a l and im ag in ary p a r t s , g iv in g th e sim u ltan eo u s e q u a tio n s
C t i t j l t a d A") ^ +
Ud t
t a d s ' M T i tan (T J j - t d c ) r f j *î‘Î
tan'^(T M T5 r
( d (Th' I d _< r + <r^) (4.25)(
4.
26)
Qliese e q u atio n s a r e unchanged by th e sim u ltan eo u s r e v e r s a l o f th e sig n s o f £ and S” , a s we ex p ect s in c e i t i s o n ly (c -f th a t ha;
(a)
iS 4ÎTC1
3 iTC( 2ijT (fjL)- halfplane ITT - T T TC l m ( p ) A pure e x pon e n t i a l decay p - p lon e Re( p) further roots l i g h t l y d a m p e d oscillationFigure 4 . 2 . S i n g u l a r i t i e s of Y ( l ^p ) |n (a) t h e - j - p l a n e of (pi) (b) the p-p lone
(k6)
p hy s ic a l s ig n if ic a n c e . They a re a ls o unchanged i f th e sig n o f o n ly one o f C o r <r i s re v e rs e d . Hence i f (T J i s a so lu tion, so i s
r £ . M
6
\ , and we need lo o k o n ly f o r p o s i t iv e v a lu e s . W ith •“ —« T 'O^
1IP M T
th e r e i s a r o o t w ith C 0 and o n e a r th e m iddle o f th e i n te r v a l (Oj, ^ , Good a n a ly tic a l approxim ation i s cumbersome, b u t i t can be
t r-z
v e r if ie d t h a t th e f i r s t ap p ro x im atio n t ~ 0 le a d s to an o s o il^ la to r y s o lu tio n w ith p e rio d tim es t h a t o f a r i g i d sim p le pendulum o f th e same le n g th , A b e t t e r appro x im atio n can be o b ta in e d n u m e ric a lly by an i t e r a t i o n p ro ce d u re.
There appears to be no f u r t h e r ro o t w ith t o f s im ila r m agnitude b u t d if f e r e n t ^ , and no ro o t w ith sm a lle r b u t n o n -zero £ • I f £ i s
l a r g e , c i , and so by e q u a tio n (U»26)
However, f u r t h e r approxim ation on t h i s b a s is le a d s to th e same ro o t a s th a t o b ta in e d from
^
, The s i n g u l a r i t i e s o fY
p )
v/hich have been found f o r a l i g h t f i b r e a re mapped in th e complex h a lf-p la n e o f in f ig u r e 4 * 2 (a ), In f ig u r e if<,2(b) th e y a re mapped in the complex j) -p la n e, where im ag in ary p a r ts le a d to o s c i ll a t o r y c o n tr i b u tio n s to ^ .rea l p a r ts to e x p o n e n tia l c o n tr ib u tio n s . F or a f i b r e o f sm all but f i n i t e mass th e s e r i e s of s i n g u l a r i t i e s along th e n eg a tiv e r e a l a x is o f p b e g in s to diverge from th a t a x is n e a r th e p a r t i c u l a r s i n g u la r ity a t = - B /< n ( s e e (l* 1 4 )) and to wheel back(47)
sy m m etrica lly tow ards th e p o s itiv e and n e g a tiv e im ag in ary a x is a s j|>| in creases*
The o n ly im p o rtan t o s c i l l a t o r y con trib u tion s eoino from th e tv/o ro o ts n e a r th e o rig in ^ "which have eq u al, n e g a tiv e r e a l p a r ts , end eq u al and o p p o s ite Im aginary p a r t s . T og eth er th e y th e r e f o re g iv e to y
a con'fcribution w hich i s a damped s in e wave, p ro b a b ly p h a se -sh ifte d w ith re s p e c t to th e zero o f tim e .
(b) The n u m erica l ev a lu a tion o f th e so lu tion
I h i s p h y s ic a lly i n t e r e s t i n g , os c illa tor y c o n tr ib u tio n i s e v a lu a te d a s follows* liquations (4*23) and (4 ,2 6 ) a re added and s u b tra c te d to g iv e th e e q u a tio n s t ^ t ton^ (T) 4- f ta n (T (^1 - 'tlv 1 + t ta n ^ 5" M T 1 ^ R,;. tan^’*f') (T ta n 1 -i- tsua^ (T
MT
-■ T )(
4.
28)
When & I? / iWT h as been c a lc u la te d as a fu n ctio n o f tem p era tu re f o r th e g iv en e x p erim en tal c o n d itio n s , f i r s t e s tim a te s and ^ a re su b s ti-- tu te d i n th e l e f t hand s id e s o f (4 .2 ? ) and (4 .2 8 ) to g iv e t S )
^T- ^ t ^ J
and — r e s p e c tiv e ly .
Z
and can th e n be found and th e4 T ^ ^
whole p ro c e ss re p e a te d . T h is i s co n tin u ed u n t i l and a t t a i n s u f f i c i e n t l y s ta b le v a lu e s . The r e s u ltin g c o n trib u tio n t o j/ i s g iv e n by
( L , t ) cC exp ^ - X ( exp ± % T
[ J