eighties, w hen air w as reg ard ed as an a lm o st p e rfe c t in su la to r an d
A N A L Y S I S OF T H E IO N O S P H E R E 457
e v ery in v e stig a to r w hen ta k in g a c c o u n t of e ith e r influence sim p ly
A N A L Y S I S OF T H E I O N O S P H E R E 459
i.e. m illions of cycles p er second) co rresponding to w av e-len g th s of 300, 30 a n d 3 m etres. H ere a lre a d y th e read er m eets th e fa c t th a t th e h e ig h t a t w hich such a signal is reflected, or th e q u estio n w h e th e r it shall be reflected a t all, d ep e n d s on th e freq u en cy of th e w aves and th e d e n sity of th e ions. F o r ev ery freq u en cy th e re is w h a t I shall call a “ m irro r-d e n s ity ” : signals are reflected as soon as th e y reach th e low est level in th e ionosphere w here t h a t m irro r-d en sity is a tta in e d . T h e higher th e frequency, th e higher th e m irro r-d en sity . T h e form ula
Fig. 1
will soon be d eriv ed a n d show n; b u t for th e m om ent, let us inquire how th e a ltitu d e of e ith e r lay er can be m easured, in term s of th e sim ple m odel of Fig. 1.
O ne w ay of m easuring th e a ltitu d e is v e ry obvious. Suppose th e o b serv er to go a know n d ista n c e aw ay from th e aerial, and m easure th ere th e angle w hich th e reflected w ave o r “ sky w a v e ” m akes w ith th e h o riz o n tal as it com es dow n to him . If th is can be done, th en clearly he can g e t th e a ltitu d e by th e sim plest trig o n o m etry — th e a ltitu d e of th e E -la y e r or th e F -layer, according to th e frequency
w h ich h e uses. I t can be d o n e, a n d i t w a s d o n e b y A p p le to n a n d
A X A L Y S T S OF T H E I O X O S P H E R E 461
“ m u ltip le e c h o e s” ; th e signal h a s tra v e le d tw o to five tim e s th e e n tire jo u rn e y from g ro u n d to ionosphere to g ro u n d ag ain , th e surface of th e e a r th being itse lf a good reflector. T h o se m a rk e d G a re d u e to th e signal sp read in g alo n g th e g ro u n d itself. If th e sen d er a n d th e receiver are p ra c tic a lly side b y tid e, as u su ally is th e case, th e kicks G o cc u r a t th e in s ta n ts of d e p a rtu re of th e signals. T h e reco rd is m oving la te ra lly w ith th e speed in tim a te d b y th e w a v y line b e n e a th , a n d acco rd in g ly th e d ista n c e alo n g i t from a G -kick to th e follow ing echo-kick is a m easu re of th e " d e la y o f th e echo.”
T h e d e lay o f th e echo is a n in d icatio n of th e a ltitu d e of th e m irro r w here i t w as reflected— th e la y e r E o r F. a s th e case m a y be. Signals of re la tiv e ly low freq u en cy b ein g reflected from E w hile th o se of m edium freq u en cy a re echoed a t F. one a d ju s ts th e freq u en cy according
r ; ______________ is :
.V A V W A m W .m 'A V .V v W A V W .V .V .V
y U : . ; i m
w w v/a w v.w A M A M M A M A M M
Cv'C _ £ 5 = £ = S E C C 'O
F it. 2— E choes. G, original signal: E-_. F-. echoes rem raing after a a n g le re
flection from and F respectively; F-- - - F*. echoes which have suffered tw o to five reflections a t /--Layer. A ppleton and Builder.
to th e la y e r w hich one w ishes to locate. If th e re should be n o t tw o b u t several lay ers of th e ionosphere, each h a v in g a g re a te r A -v alu e th a n th e o n e b e n e a th i t , one w ould lo cate th e m all w ith a p p ro p ria te frequencies. I f th e re is a co n tin u o u sly -risin g d istrib u tio n o f .V w ith h e ig h t in th e ionosphere, one m a y p lu m b i t b y v a ry in g th e frequency"
co n tin u o u sly . N ow w e a re a t th e p rinciple of th e e c h o -m eth o d : b u t before i t is used, th e re a re m a n y d e ta ils to d e a r up.
F ir s t a s to th e “ sig n als." a te rm w hich ( it m u s t h a v e been noticed rep laced th e te rm “ w a v e s ” in th e foregoing p a ra g rap h s. T hese signals a re w a v e -tra in s indeed, b u t n o t th e long co n tin u o u s uniform tra in s ta c itly assu m ed in th e d escrip tio n of th e o th e r tw o m ethods.
T h o se m e th o d s a re a d a p te d to tra in s o f indefinite len g th , b u t n o t th e ech o -m eth o d , w hich fo r a n obvious reason req u ires w a v e-train s of lim ited le n g th — a n d th e m o re lim ited, th e b e tte r.
N ow , a lim ited w a v e -tra in is e q u iv a le n t to an in fin ity of infinitely
A N A L Y S I S OF T H E IO N O S P H E R E 463
In v acu o , th erefo re , th e re is a c u rre n t-d e n s ity (1/4-71-) tim e s th e
A N A L Y S I S OF T H E IO N O S P H E R E 465
3 N otice incidentally th a t owing to th e curvature of th e earth and its overhanging ionosphere, th e angle of incidence can never rise to 90°; it follows th at w aves of frequency beyond a certain value (ordinarily around 30 me.) never suffer total reflection.
a m o u n ts to say in g t h a t th e p hase-speed is in fin ite w hen N = N c, im a g in a ry w hen N > N c. H ow ever, th e c o n cep t of p h ase-sp eed is of such a q u a lity of a b stra c tn e ss, t h a t even th ese s ta te m e n ts im p ly n o th in g a b su rd in th e physical situ a tio n . T h e signal-speed itself re m ain s safely finite a n d real.
T h e signal-speed is s tric tly indefinite, since th e signal d is to rts itself as it proceeds. H ow ever, th e p ra c tic e is to id e n tify it w ith th e g roup- speed v, w hich, as I in tim a te d (page 462), is th e speed of th e b e a ts form ed b y tw o superposed w a v e -tra in s differing infin itesim ally in w av e-len g th , each such b e a t b ein g a v e ry special ty p e of signal.
T h e fo rm u la is,
v = u — X(du/dX) = u j ^ 1 — ~ ^ ^ • (6) I t is difficult to visualize or d eriv e w ith o u t a d ia g ra m ,4 b u t th e d e riv a tio n m a y be su m m arized as follows. Im ag in e tw o su p erp o sed w ave- tr a in s of p hase-speeds u a n d u + du, w a v e -len g th s X a n d X + d \ \ consider tw o co n secu tiv e w av e-crests A , A ' of one a n d tw o co n secu tiv e w av e-crests B , B ' of th e o th e r; tra n sp o se te m p o ra rily to a fram e of reference in w hich th e fo rm er w a v e -tra in is s ta tio n a ry . A t a c ertain place an d tim e A a n d A ' will coincide, an d th e m ax im u m of one of th e b e a ts will be rig h t th ere. L e t th e tim e dX/du elap se; w hen it h a s elapsed, th e c rests B an d B ' will be coinciding a n d th e m ax im u m of th e b e a t will h av e m ov ed on b y one e n tire w av e-len g th . T h e b e a t th e re fo re tra v e ls w ith speed Xdu/dX in th e te m p o ra ry a n d w ith speed u — X(du/dX) in th e original fram e of reference (th e m in u s sign is e v id e n t w hen th e reasoning is gone th ro u g h in d e ta il).
C o m bining (6) w ith (2) one finds:
v = c2/m; (7)
th e g re a te r th e p h ase-speed, th e slow er th e signal ! R e la tiv is ts will be pleased to observe t h a t acco rd in g to th is fo rm u la, th e signal n ev er a tta in s a n y speed g re a te r th a n c; s tu d e n ts of q u a n tu m m ech an ics m a y be m isled b y its superficial resem b lan ce to a fo rm u la re la tin g phase-speed to gro u p -sp eed for de B roglie w aves, w ith w hich it has n o th in g to do. S tu d e n ts of th e ionosphere sh o u ld rem em b er its ap p ro x im a tiv e c h a ra c te r. A lm o st all t h a t needs to be know n for th e purp o ses of th is artic le is, t h a t as a signal clim bs in to th e ionosphere it goes m ore a n d m ore slow ly, th e n e a re r N a p p ro a c h e s to t h a t v a lu e N c w here th e signal finds its ceiling.
4 Cf. this journal, 9, 173 (1930), or m y Introduction to Contem porary P h ysic s, 2nd edition, p. 147.
A N A L Y S I S OF T H E IO N O S P H E R E 467
th e le a st d ep en d in g on th e d a sh ed p a r ts of th e N (z ) cu rv e of F ig. 3A o r in d ic a tin g a n y th in g w h a te v e r a b o u t th o se p a r ts ex c e p t t h a t th e y do n o t rise ab o v e th e o rd in a te Ne
-N ow if th e signal an d th e echo tra v e le d fro a n d to w ith th e speed c, th e d e la y T of th e echo m u ltip lie d b y \ c w ould be th e h e ig h t of th e ceiling. T h is, how ever, is n o t th e case, since th e signal-speed d ep en d s on N . W e m u s t th ere fo re d e n o te th e p ro d u c t \ c T b y a n o th e r sym bol h', a n d m ak e a n in q u iry in to th e p ro b a b le d ep en d e n ce of h ' on / , ta k in g in to a c c o u n t o u r v ag u e know ledge as to th e d ep e n d en ce of signal-speed on N .
A B
Fig. 3— A . T he “ curve of inference” : conjectural dependence of number N of electrons per unit volum e on true altitu de h. B . T he “ curve of d a ta ” : dependence of virtual altitu de h' of ceiling (one-half th e delay of the echo, m ultiplied b y c) on frequency / .
I t is easily seen t h a t h' m u s t be g re a te r (or a t le a st no less) th a n h , a n d t h a t th e excess of h' over h m u s t be larger, th e fa r th e r th e signal tra v e ls th ro u g h regions w here N is a lm o st b u t n o t q u ite eq u al to N c.
T h e cu rv e m u s t th erefo re lie ab o v e th e (h , f ) cu rv e, a n d fa rth e s t abo v e it in th e im m e d iate neig h b o rh o o d of th e g a p on b o th sides.
T h e re will still be an E -b ra n c h a n d a n E -b ran ch , b u t th e u p tu rn s to w a rd th e rig h t-h a n d ends of th e se b ra n c h e s will be ex ag g e rated , an d an u p tu rn ru n n in g to th e left will be in tro d u c e d in to th e le ft-h a n d end of th e E -b ran ch . I t is conceivable t h a t th ese u p tu rn s m a y becom e so large, t h a t th e ( h ',f ) cu rv e will a p p e a r to show a p e a k w h ere th e (h, f ) c u rv e w ould show a gap.
W ith th e re m a rk t h a t h' is know n as “ v irtu a l a ltitu d e ,” “ v irtu a l h e ig h t,” “ e q u iv a le n t h e ig h t,” o r “ effective h e ig h t,” I tu r n now to exam ples of th e c h a ra c te ristic cu rv e s of th e ionosphere.
A N A L Y S I S OF T H E I O N O S P H E R E 469
FREQUENCY ( f ) IN MEGACYCLES PER SECOND
Fig. 4— Exam ple of characteristic or fh ', f ) curv es, showing gap between E-branch and E-branch, and crinkle in E-branch indicating presence of Ei-layer. Duplication of curve due to earth’s m agnetic field (page 479). (Appleton.)
FREQUENCY IN KILOCYCLES PER SECOND
F ig. 5— Another exam ple of ( h ',f) curve, showing gap and crinkle.
• (Schafer and Goodall.)
F ig u res 4 an d 5 show tw o exam ples of th ese curves, from d a ta o b ta in e d w hile th e sun w as high in th e sky. A c tu a lly th e re a re tw o cu rv es in each of th e figures; th e a p p e a ra n ce is t h a t of a single cu rv e, re p e a te d w ith a sidew ise sh ift. I m en tio n t h a t th is re p e titio n is d u e to th e e a r th ’s m a g n etic field, b u t a sk th e re a d e r to ignore for th e p re se n t th e rig h t-h a n d cu rv e an d fix his a tte n tio n on th e le ft-h a n d one.
H ere he will see th e E -b ra n c h , th e gap, an d th e F -b ra n c h . T h e
TIM E IN M INUTES
Fig. 6— Characteristic (h ', f ) curves obtained with th e m ulti-frequency a p p a ra tu s;
sun high in sky, F\ crinkle apparent. (Carnegie In stitution of W ashington.)
u p tu rn s to rig h t an d left of th e gap are s trik in g on Fig. 5, in significant in F ig. 4. T h e F -b ran ch is deform ed b y an en o rm o u s h u m p or crinkle.
T h is is supposed to co rresp o n d to a second g ap , th e u p tu rn s on rig h t a n d le ft being so p ro n o u n ce d as to give a p e rfe c t sem blance of a p e ak ; indeed one sees in F ig. 5 how re ad ily th e g ap b etw een E a n d F m ig h t h av e been d raw n as a peak . C u rv e s of th is so rt a re th erefo re ta k e n as evidence for th re e lay e rs in th e ionosphere, d e n o te d b y E a n d Fi an d F2. S om etim es th e re a re signs of a fo u rth , ly in g b etw een E a n d F 2, an d d e n o te d b y M o r E 2.
So g re a t is th e in te re s t in cu rv es like these, a n d so m u ch do th e y v a ry from tim e to tim e a n d from place to place, t h a t la te ly th e re h a v e been m ore th a n a score of s ta tio n s o v er th e w orld engaged in m ak in g th em . A t som e of th e se th e tra c in g of th e cu rv es is speeded u p a n d m ad e in cessan t b y a re m a rk a b le m ach in e d ev elo p ed a t th e
B u reau of S ta n d a rd s a n d th e C arnegie In s titu tio n . A u to m a tic a lly sending o u t th e signals ten tim es in a second, an d changing th e fre q u en cy b y (on th e average) 1600 cycles betw een each signal a n d th e n e x t w hile th e p h o to g rap h ic film is m oved a tin y b it from left to rig h t, th is “ m u lti-fre q u en cy a p p a r a tu s ” traces th e cu rv e o v er th e
A N A L Y S I S OF T H E I O N O S P H E R E 471
0.516 1 2 3 4 5 6 7 8
FR EQ U E N C Y IN MEGACYCLES PER SECO ND
Fig. 7— Characteristic (h ' , f ) curves obtained with the multi-frequency apparatus;
sun low in sky, F i crinkle missing. (Carnegie Institution of W ashington)
freq u en cy -ran g e betw een 0.516 me. a n d 16 me. in fifteen m inutes, a n d th e n goes rig h t b a c k an d does i t over an d over again. Figures 6, 7 an d 8 show ind iv id u al curves th u s a u to m a tic a lly ta k en , an d Fig. 15 a sequence of th em sp an n in g several ho u rs of th e day.
In Fig. 6 are curves w ith crinkles in th e F -b ran ch , sim ilar to those of Figs. 4 an d 5. In Fig. 7, how ever, th e crinkle is m issing, an d th e
F -b ra n c h sw eeps sm o o th ly a n d slow ly u p w ard from its co m m en cem en t.
(T h e forking signifies t h a t h ere are tw o sim ilar cu rv es lying side by side as in th e p rev io u s figures, b u t o v e rlap p in g so m u c h t h a t o v er a large p a r t of th e ir course th e y a re n o t d istin c t.) M a n y o b se rv a tio n s h av e co n cu rred in show ing t h a t th e crinkle is p re se n t o n ly w hen th e
J I I I_______________I___ I I________ L _________ I_____ I 1____ I_________ I________ I___ I
0 .5 I 2 3 4 5 6 7 8
FR EQ UENC Y IN M EG AC Y C LES P E R S E C O N D Fig. 8— M ultip le echoes. (Carnegie In stitution of W ashington.)
sun is high in th e sk y (w ith in som e 40° of th e z e n ith )— th erefo re a b s e n t b y n ig h t a n d a t th e b eg in n in g an d end of d a y , a n d indeed a b s e n t all d a y in w in te r w here th e la titu d e is high. T h is is o u r first exam ple of th e d ep en d en ce of th e ionosphere on su n lig h t, a v ery im p o rta n t featu re .
A N A L Y S I S OF T H E IO N O S P H E R E 473
E A S T E R N S T A N D A R D T IM E
Fig. 10— Dependence of critical frequencies on season and hour of day.
(Sm ith, Gilliland and Kirby.)
13,000
1 2,000
11,000
10,000
9000
80 0 0
7000
6000
oz oo
UJ if)
crUJ Û.
if)ILI
_Jo
>-oo
-J
X.
z oz
UJ=>
o
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5000
4 0 0 0
3000
2 000
1000 0
A N A L Y S I S OF T H E IO N O S P H E R E 475
cu rv es w ith th e su n h a v in g been p re se n te d as if for its ow n sake.
A N A L Y S I S OF T H E IO N O S P H E R E 477 beginning to stu d y the distribution of A in the horizontal plane, e.g. b y using obliquely-sent as well as vertically-em itted signals.
N ow w e turn briefly to a difficulty affecting n o t the relation between h' and h, b u t the relation between X c a n d /p r e s e n te d as equation (5). (A~, s) curves. I will risk a few statem ents about the curves them selves.
T he shape of the (A*, s) curs e, when the sun is low in the sky.' and
on a c e rta in d a y of e a rly sp rin g (1933) A p p leto n found 1 .2 - 105 a t th e
A N A L Y S I S OF T H E I O N O S P H E R E 479 A m ag n e tic field, th e e a r th ’s or a n y o th e r, should hav e no effect w h a tso e v e r on rad io w aves so long as th ese a re tra v e lin g in a ir com posed e n tire ly of n e u tra l m olecules. W hen, how ever, th e w aves are s e ttin g elec tro n s in to m o tio n , th e m oving electrons are affected b y th e field, w hich has a tw istin g actio n on th e ir p a th s. W e h av e seen a lre a d y t h a t th e m oving electro n s re act, so to speak, upo n th e w aves, raisin g th e w ave-speed thereof. B y a lte rin g th e m otions of th e electrons, th e m ag n etic field will influence a t second h a n d th e w aves them selves. B u t will th e re su lt be p erceptible? In view of th e fa c t
T IM E O F D AY
Fig. 12— Ceiling" for signal of fixed frequency shifting near sunrise from F to £ as the electron-density of the E-layer increases with increase of light. (Appleton.)
t h a t th e e a r th ’s m ag n e tic field is v e ry feeble b y com parison w ith th e fields betw een th e poles of o u r electro m ag n ets g re a t o r sm all, or even w ith th o se a ro u n d th e horseshoe m ag n ets w hich are p lay th in g s, one m ig h t well th in k th e influence n o t w o rth th e tro u b le of com puting.
B u t th o se w ho first u n d e rto o k to co m p u te it— N ichols an d Schelleng in A m erica, A pp leto n in d e p en d e n tly in E n g lan d , in th e w in ter of 1924-25— found it a serious influence, a n d v ery well w o rth th e trouble.
T h e problem is one of those w hich are n o t v e ry h a rd to s ta te , b u t can be v e ry te d io u s to solve except in special cases w hich m ay or m ay n o t be of p ra c tic a l im p o rtan ce. F o r th is problem it h ap p en s th a t tw o of th e special cases can be solved w ith relativ e ease, a n d one a t
le a st is realizab le in p ra c tic e . I c a n n o t v e n tu re to give th e th e o ry to strengthening the electron-current-density I e parallel to the electric vector. I t w ill be recalled (from pag e 464) t h a t i t is I e w hich for
A N A L Y S I S OF T H E I O N O S P H E R E 481
a fa irly close copy of th e o th e r, sh ifted a c e rta in d ista n c e alo n g th e
A N A L Y S I S OF T H E IO N O S P H E R E 483
V IR T U A L A L T IT U D E IN K IL O M E T E R S
B eing d iscovered how ever som e y e a rs before th e w ar, it w as a n d is
A X A L Y S I S OF T H E IO X O S P H E R r . 4S5
A ttem pting to supplv an answer to this question. I point o u t th at in such part of the theory as I have thus far given, there is n othing corresponding to absorption. This is because the electron-current
- je : :
-F ig . 15— lr e n d o f th e critical frequencies and o f th e echoed frequency as a ia d eo ct proceeds. B e t e e r .
I , in the former notation' is in quadrature w ith the electric vector in th e w aves, being thus a “ w a t t le s current-" T his in turn is because w e h ave assum ed each electron to be entirely free, oscillatin g in the
w av e-sw ep t a e th e r a s th o u g h th e re w ere n o th in g else in th e w orld
A N A L Y S I S OF T H E I O N O S P H E R E 487
ra d io v e ry fa r over th e e a rth . T h e re a d e r m a y th in k t h a t th is is n o t v e ry im p o rta n t: o u r a n c e sto rs lived w ith o u t rad io , w h y should we w o rry a b o u t lacking it? W ell, it is p ro b a b ly q u ite tru e t h a t if th e ionosphere w ere n o t o v e rh e a d , we should n o t be w o rry in g a b o u t th e lack of rad io . W e should in fa c t p ro b a b ly n o t be w o rry in g a b o u t a n y th in g a t all, for we should n o t be h ere to w o rry . T h e u ltra -v io le t lig h t of th e sun, p o u rin g dow n upon th e su rface of th e globe u n h in d e re d , w ould w o rk chan g es so severe on o rg an ism s as we know th e m t h a t life w ould h a v e to be v e ry d ifferen t, a n d p e rh a p s im possible. T h is leth al lig h t is like an en em y, w hich in a tta c k in g a c ity sp en d s itself in th ro w in g u p a b a rrie r a g a in st itself; a n d th e b a rrie r n o t o n ly k eeps th e enem y o u t, b u t is serviceable o th erw ise to th e dw ellers in th e city .