JOURNAL OF RESEARCH of the National Bureau of Standards-D.

Radio Propagation
Vol. 67D, No. 1, January- February 1963

r Composition of Reflection and Transmission

Formulae
John Heading
C ontribution fro m the Unive rsity of Southampton, Southampton, England

(Received July 3 1, 1962)

An integral equation for t he eJect ri c fi eld in a continu ously strati fi ed io n ize ci m ed ium
is d eri ved ; t hi s is t hen m anipul ated to yield equatio ns fo r t h e refl ectio n and tran s mi ssion
coeffi cie nts, at t he sa me t ime bein g s usceptibl e to p h ys ical in terpretat ion . The equ at io ns
ar e solved b y s uccessive approxim at ion s, t he first term s being Fre~ne l -type coeffi cie nt s.
Va r io us appli cations of t h e r es ults ar e di sc ussed .

1. Introduction

In a r ecen t paper , W a i L [19 62] has followed a n icl ea of Bl'ekhovs kikh II960] a nd has ouLain eci
appro ximate for111 s £or t he r eflect ion coeffLcien Ls or VLF a nd ELF radio waves when t h e
ionosphere is no t sh arply bounded. In Lhe isotropic case, Lhe lll eLhod is to obta in a fir st order
nonlinear d ifferen tial equ atio n for R (z), a funct ion of t h e h eigh t z t haL r educes to t he r eflection
coefficient below t h e ioni zed layer. T h is equ ation is solved b.v s uccess ive app roxim ation s,
Lhe fLrst approximation b ein g t he familiar Fl'es nell'efi ec Lio ll coeHicient. Ne it he r BrekllOvskikh
in h is comprehensive text (p . 220) nor 'W ait discusses a ny p hys ical in Lel'preLaLion of the cor-
respondin g in tegral equaLion . H er e, we S llOW LhaL b y m ea ns o r a d ifl'ere n L, yet u l LimaLely
equiv alent, formul ation of the problem , t he r es ul t in g equation s a r e s uscep Lible to ph:ysical
in terpreta tion . T h e t heory is applicable to all a ngles of incide nce a nd to a ni sotropi c iono-
sph er es. T h e inLel'p retfL Lion is made possible by t he recenL inves t iga tion s of vVes LcoLL [1962<1 ] in
wh ich h e h as derived various exp ress ion s y ield ing t he effect or p a rtic ular region s of (h e
ionosph e re on Lbe r efl ected wave .

2 . Isotropic Model
V erti cal prOpltglttion in an isotropic ionosphcre , j nclud in g Lh e efrecL of Lh e coll is ion fr e-
q u c ncy, is govcrn ed by th e diifer c n tilll equaLio ll

wh er e the z-axis is v ertical , E d enotes a particuln,r horizontfLlly polarized compon en t of th e

electr ic fi el d , k= w/c the time j'fLctor eiw t being s uppressed , ./~= Ne2 / tomw2, a nd Z = v/w. W e s hall
write l -X/(l -?Z )= n 2 , wh er e n is th e refrac t ive index .
W e con s ider t h e following configuration for the ion ized m edium . For z< a t h er e exisLs
free sp ace for which n= 1 ; 1'01' z> b t her e exists homogen eous medium ex tendin g to infini ty
for which ./y =./ ~j , Z = Z" n = n j. For a<z<b t he prop erties or the medi llm ar e function s of z,
vary ing from n = 1 to n = nj .
Let r d eno te l1,n in termediate lleigh t in th e range a::;r::;b. W e mul tiply the equation

(1 )

by e-ikn j (Z- , ) a nd in Legntte from r to b. To this, we a dd th e result obt},in ed by multiply ing

by eik( Z- , ) and i n teg nt t in g from a to r. We have, upon in tegrating by p ar ts t wice in succession ,
6S
 I:
l bE " e- ikn j(z-ll dz= E' e- ikn j(z- r) + i k n jlbE' e- iknj (z-Il dz

I:
= (E' +iknjE) e- ikn j(z-r) -Fni i bE e-ikn j(z- Il dz . (2)
Similarly,
(3)

W"e now consider a wave incident below the ionosphere. \iVe therefore restrict ourselves
to tha t solu tion of (1) for which

z= O being the phase-reference level Jor both coefficients Rand T. Th en

and

The sum of the two integrals (2) and (3) is now given by

H enco, when the final term in eq (1 ) is taken in to account, we ob tain

ik (1+ nj)EW - 2ike -ikr +F rr(1-n2)Ee ik (Z-ll d z +F.Jre(nI-n2)Ee-iknj(z - ll dz= 0,

.Ja

(4)

This constitutes a suitable in tegral equation for the field at any point .I in t he medium ;
the integration process is extended over those portions of the ionized medium where the r e-
fractive index differs from unity or from n t . In particular, if fr ee space exists above Z= b
(that is, if n = 1), the in tegral equation becom es
j

It sho uld b e pointed out that the right-hand sides of both eqs (4) and (5) are independent
of a, provided that a is any point in t he free space below the ionized region.

3. Reflection and Transmission Formulae

In eq (4), let .\= a, in which case
66
Then

reducing t.o

( 6)

In particular, if nj= 1 (that is, if Xj = O), this reduces to

1 .
R=2 tk
[b1-X ·Z E( z)e - 'k-dz
'. . (7)
• a t

It is obvious that the limits may be replaced by - 00 and + 00 respectively, since the integr and
vanishes in these extended regions.
Similarly, we place 1;= b in eq (4), in which case

Hence,

reducing to

(8)

If free space exists above z = b, this result simplifi es to

T= l +~ ilc .f13z E(z )eikZdz . (9)

It should be pointed out that results (6), (7), (8), and (9) are independent of a and b provided
that a and b lie in the respective homogeneous r egions b elow and above the ioni zed m edium.

4 . Varying Region Extending to Infinity

If there exists no homogeneous medium. of refractive index nl above z= b, then the varying
medium will extend to infinity. In particular, the medium becomes infinitely overdense
there if X --'7 OO as Z --'7 OO. Under these circumstances, we multiply eq (1) by eik (z-n and in-
tegrate from a to I; as before, but now we multiply eq (1) by e- ik(Z-n and integrate from I; to
ex:>. The same analysis as b efore leads to the consideration of

(E' + ilcE) e- ik(z- n 100

•

It should be observed that t his factor does not converge at the upper limit if a homogeneous
m edium extends to infinity . It was for this reason that previously we employed the factor
e-ikn1(z- n to secure a result that vanished when 8= b. For convergence in the present case,
we r eq uire E' +ikE--'70 as Z--'7 OO . This limit certainly exists if the region becom es infinitely
overden se, since the allowed W.K.B.J. solution for E then becomes exponentially small in
magni tude as Z--'7 OO. We obtain the r esult

E(!;) = e - ikl +~ ik ( I ~ E(z)eik(Z-ll dz +~ ik ( OO ~ E( z)e-ik (Z- lld z .

2 .J a 1 - tZ 2 .J 1 1- t Z
67
 We could have multiplied by e-ikn1 (z-n in the second integral, where nl is arbitrary (Rlnl > 0),
but this would have been an irrelevant procedure in this case since nl would have no definite
meaning in the ionized medium.
The formula for R becomes

_1 'k ( '" X E() - ikZd

R -2 (10)
~ J a 1- iZ z e z,

where a occurs below the ionization. No formula for T exists in this case.

5. Physical Interpretation
There are three distinct contributions to the field Em in eq (4). These are:
(i) 2e- ikr / (1 + nl);
(ii) elements of the form 1 tnl . 1 ~iz E (z) oz eik(Z- r) originating below the height L

(iii) elements of the form 1 tnl C 3Z-1 ~z) E (z) oze- ikn1 (Z- rJ originating above the
height \.
The vector sum of all these waves yields the total electric field at the height r. (i) is a
kind of transmitted wave complete with the appropriate Fresnel transmission coefficient.
(ii) represents a plane wave propagated vertically upwards as in free space (according to the
factor e- ikr ), originating from the elementary layer of thickness oZ situated at z< 5' All such
layers below 5" yield waves of similar forms . (iii) represents a plane wave propagated vertically
downwards as in a homogeneous medium of refractive index nl (according to t h e factor e iknll) ,
originating from the elementary layer of thickness oZ situated at z> 5". All such layers above
5" yield waves of similar forms.
The work of Westcott [1962a] shows th at this is t he correct interpretation of the elements
involved. Each individual layer of thickness oz in an ionized medium gives rise to reradiated
waves, their originating strength depending on the exact valu e of the electric field within the
lay er oz. In Westcott's series of papers he considered these reradiated waves as propagated
in free space, thereby excluding the possibility of a homogeneous medium existing above z= b
of refractive index other than unity. His formulae would not converge if such a medium
existed there. We have modified the theory to allow for this, at the same time providing
the reflection and transmission formulae derived in section 3. It should be observed that
our method has not proved the interpretation placed upon the elements involved; to accomplish
this would r equire the more elaborate analysis of Westcott, considering from first principles
the Hertz vectors involved in the reradiation processes.

6. Approximation Methods
In order to calculate the reflection coefficient (6), it is necessary to know E( z) throughout
the medium beforehand. If E( z) were known, R would also be known, implying that there
would be no need to evaluate R from form ula (6), which after all is but an identity. But an
iterative method of solution may be adopted, using approximate solutions derived from eq (4).
A series of approximations may form ally be written down based on successive substitution
into eq (4). We write

- ~ (l ~ E () ik(z-rJ d +~ ( b(~_~)E () -iknl (z- rJl

E()5" -l
n + nl J a 1- iZ Zen-Iz l + nl J r 1- iZ 1- iZI Ze Gz. n- l

68

L_
 -I
Now I

and if throughout the range a to b

then

It follows that t he series for E converges rapidly provided

that is, provided the layer is sufficiently thin reb-a) small enough] 01' provided M is small
enough (the ionization density small throughout).
If frce space exists above z= b, the formulae simplify. The successive approximations are

·1 r r
Ein (t ) -_12 UC X E () ik(z- rJl +1 ·1 f ~ .X E n 1 () - ikl'- rld
J a l - iZ " n _ 1 Z e GZ 2~/C r l - iZ f _ Z e Z

=~ ik.r 1 ~Z E n - 1 (z) e-iklz-rl d z .

If M denotes the maximum of X I I l - iZ I, the series convcrges rapidly if

HkM(b- a) « 1.
If X contains a smaIl constant multiplier a, evidently this process yields E (t) as a power-series
cxpansion in powers of a, since E n(l) cx a n.
The meaning of the individual tcrms in the scries becomes clear. E1(1) rcprcsents the
reradiated field if the cause of thc rcradiation process at each level is given by Eo(t) and not
E (t ) . Similarly, E 2 (t ) represents thc rcradiated field due to the field El Ct) only acting on the
free electrons, and so on.
Substituting the successive terms of ECt) into eq (6) for B , we obtain the corrcspondin g
scrics for B. The first two terms of (6) become

If nl = l above z= b, this reduces to

(11)

from eq (7). Under these circumstances, eq (9) yields for the first term in T,

This value of B arises from the contributions to the reflected wave produced by the incident
fi eld e- ikz being regarded as the sole cause of the reradiation process at each level. It corre-
sponds, in fact, to the use of the Born approximation.
For this case, we have from eq (5 ),
658514- 63- 6 69
yielding the second term in the series for R:

R -! 'k (a X( z) -ikZ l . ! 'k ( a X(t) -ikt -iklt-z i lt

2-2'/, Jal - iZ( z ) e G Z 2'/, Jal - iZ(t) e e G .

With no collisions, we may write this development of R in the form

If \ve place
v=iE' /n1kE,
then differentiation and the use of eq (1) show immediately that

dv
-- _ - -ik (2 2 2)
n - nv
dz nl I,
where v= 1 when z> b and
e-ikZ_ R eikZ
v nl(e ikz+ ReikZ)

when z<a. This is the equation derived by Wait [1962] and which was solved by successive
approximations. The corresponding integral equation would be

(13)

no doub t simpler than our eq (5), but certainly less comprehensive in physical content. M ore-
over, our eq (6) yields R directly as a series expansion, but Wait has shown how eq (13) may
be solved approximately, yielding R as a series expansion only indirectly through the expansion
of v.
7. Examples
As the first example, we may consider the model for which free space exists above and
below the slab with bounding surfaces z= O and z= h; homogeneous medium exists in thp, slab
such that n 2 = 1-X. It ma~- easily be shown that the reflection coefficient is given by
. (n 2 - 1) (1 - e2iknh)
R = (n+ 1) 2e2iknh_ (n - ] )2'

In order to apply the second-order reflection formula (12), X is assumed to be small within
the slab. EA,])anding R to the second order in X, we easily obtain
R = iX(l- e -2ikh) + iX2(1- e -zikh-2ikh e -Zikh). (14)
We may also use eq (12). Since X is constant throughout t h e range of integration, we
obtain

These integrals are trivial to evaluate, the result without any approximation being equal to (14).
Secondly, we may consider a symmetrical Epstein [1930] profile, tending to free-space
conditions above and below the layer. Let

ae~ Z
X = (e 8z + 1)2'

70
 The work of Epstein (or of Budden [1961]) shows that the reflection coefficient has the valu e

R = r(- 2ik/{3) r(N + 2ik/(3) r(1 - N +2ik/(3)

r(2ik/{3)r(N)r(1 - N) ,

where N is given by N (N - 1) + k 2a/{32= 0. Since a is assumed to be small, we take N = Pa/{32

approximately, in order to calculate the first term in the development of R in terms of a. To
this first order, we may neglect the N in the two gamma functions occurring in the numerator.
Jn the denominator 'we write

r(N)r(l - N) = 7r/sin (7rN) ' . 1/N

sin ce N is small. Tben to the first order
R= r( - 2ik/(3) r (1 + 2ik/(3)N
7rN

(15)
{32 sinh (27rk/{3)

This result should also be recovered by the direct applicaLion of formu la (11 ). This yields

This may be evaluated by considerin g the correspondin g complex contour integral Laken
around the rectangle with vertices ( ± L, 0), ( ± L , 2i7r/(3) , the contour enclosing one double
pole at z= i7r/{3. The result tur ns out to be identical with (15).

8. Integral Identities
The refl ection formul ae (7 ) and (10) ma~T be used to obtain certain interesting inflllite
integrals, when the valu es of R , X , and E are known from other considerations.
Our first illu stration involves the exact solution of a tractable model. Consider the
exponential profile X =e az/k2 with no collision frequency. The work of Budden [1961] shows
that
r (- 2i lc/a) -4ikla
R r (2ilc/a) a ,

E (z)= H~\~/ a (2i eazI 2/a) .

~ - i 7r lr(2ik/a)( ei7r/2/a )- 2ikla
1
Direct substitution into eq (10) and simplification yield
1
I
\
Finally, let
v= 2lc/a, t = 2eazI2 /a,

where v is real and positive and arg t = O along t he positive real axis. The integra.l reduces to
the result containing the single parameter v:

- V7r- lr (- iv)e7rvI221 -iv= r oo tI -i. H g) (it)dt .

.J o
Our second illustration concerns an approxim ate solu tion of eq (1). Let n 2 vary
monotonically from 1 when Z = - ro to - ro when z = + ro, in such a way that E tends to the
71
form e-ikz+ Reikz for large negative z. Evidently n 2 possesses a zero at some real value of z ;
we shall suppose that this transition point occurs at the origin O. Provided that the valu e of
kis lal\ge e'nough (that is, at the higher frequencies), many authors (see for example [Heading,
1962, page 29]) have shown that an approximate solution for E along the whole real h eight
axis is given by
E =Cn- 1/2C.f1Ul z} /6 Ai [Gei1r/2kl znd zy /] (16)

where Ai denotes the standard Airy integral and C is a constant. This result is a generalization
of the simpler W.K.B.J. solutions

n- I / 2 exp ( ±ikiZndz}

solutions that cease to be valid near the transition point z= O, and that require connection
formulae to effect a suitable connection across z= O. Equation (16) , however, suffers from
none of th ese disadvantages ; this solu tion if' valid even at the transition point.
To be specific and consistent, let
arg n=O for z< O,
arg n =-}f7r for z>O,
arg z= O for z> O,
arg Z= 7r for z<' ().
, Vhen z> O, we use the standard asymptotic expression
J
Since n 2 is negative, we must write n=e- i1r /2m, where m > 0; hence

1 . (3 (r Z )
"'ZC7r - 1/2 e'1r/6 Zk ) -1/6 m - 1/2 exp -kjomclz,

an exponentially evanescent solution as z-,;>co. This solution therefore satisfies the boundary
condition there.
When z<O, let z=e i1r t, where t > O, yielding Ii'

f 'ndt
=Cn-1/2ei1r/6 ( Jo rrndt)2/ 3J.
)1/6Ai [ - (3zkJo
We now quote the standard asymptotic e)..rpression

yielding

72
 In order to introduce the in cident and r eflec ted waves, we note that if rl is large and
positive, then

if o
n(- r)dr= i f!
0
n(- r)dr+ i-fI
z
dr,

since n = l b elow Z= - I l ' H ence

rfn(-
Jo r~
g-) dg-=-Z- rl + Jo f [n(- g-) - l]dg-,
n(- r)dg-=- z+ Jo
oo

where the upp er limit is replaced by DO since the integrand mu st vanish at t hese additional low
heigh ts. It follows t h at as n -71,

l.

In order that t h e in cident field should be or uni L amplitude, wc choose

] . (3
('- I = 2" 7r- 1 /2e '~/6 2 k ) - 1/6csp (. ~kJ o
1 .)
r oo [n ( - r)- 1]d r-4 7r~ '

It follow s t hat
R = i esp ( - 2ik.Fo oo [n (- r) - I]d .\}
X = 1- n 2 ,

l / n - 1/2
rznd z)1/6At.[(3
(Jo . rzo ndz )J2/3
2 ei7r/2lcJ
r E 12" . (3
7r- 1 / 2e' ~/ 6
2 'C
) 1/6exp ( . 1 .)'
r oo [n ( - n - l]cl .\-4~
~IcJ o

Final sub stit ution in Lo cq (10) yields after cross multipli caLion

1 (3
27r- 1/2e5irr/ 12 2lc ) - 1/6 exp (-.~lcJro oo [n( - g-)-- l ]dr
.)

=2 1. f oo O - n
~lc -00
.
2)e-'kZn- 1/2 rzndz )116 At.
( Jo [(3
2 e'.7r Ic Jrz
/2
o nd z
)2/3J dz.
This formula embraces t h e buildup of t he r eflection coefficient from every la:ver of a slowly
varying medium of the type postulated.

9 . Reflection and Transmission Formulae for Anisotropic Ionospheres

The relevant equations leading to the analysis of t his section are given, for examplo, by
Budden [1961 , chapter III] . The differential equation for the electric fi eld E within an amso-
tropic ionosphere is
(17)

where the 3 X 3 susceptibility matrix M is given by

r -U2+ l2P inYU+ lmP im yu+ lnpl

M"U(U~p) l -inYU+ lm p -U2+ m2p il Y~~ mnY2 J '
-imYU+ lnP -ilYU+ mnP _U2 + n2y2

73
Here, X, Y, Z are the usual dimensionless parameters describing the ionosphere, U = I-iZ,
and - (l, m, n) are the direction cosines of the earth 's magnetic field . The vector ME is pro-
portional to the electric polarization vector.
I n evalu ating curl E, we choose t h e customary coordinate system in which o jox= -ik
sin 0, o joy= O, and ojo z is replaced by a prime. In component form, eq (1 7) may be written as

E~' +ik sin 8E;= - FEx- F( ME)x, "\

E~/-k2 sin 8Ey= - 1c2Ey- k (ME)y,
2 l 2 (18 )

ik sin 8E~-k2 sin 8E z= -FEz-F(ME )z. J

We now mult iply these three equations by e- ikz co s ° and integrate from a to b, namely
over the whole height range in which ionization occurs, assuming free space for z> b and z<a.
Noting that

( bEll e-ikz cos 0clz= (E' +ik cos OE )e - ikZ cos 0lb- k 2 cos 2 8 ( bE e-ikZ cos 0d z,
Ja a Ja
we obtain from eq (18 )

(E~+ik cos 8Ey)e- ikZ cos 01"= -1c2 ( "(ME )v e - ikZ cos odz, (19)
a .J a

Multiplying the third of these equation s by tan 8 and adding it to the first , we obtain the
simpler equation

(E ;+ik SQC 8Ex + ilc Sill 8Ez)e- ikZ cos 01"= -F ( b[(ME)x+ Lan 8(ME )z]e - ikZ cos °dz. (20)
a Ja
When the incident field is horizontally polarized, let
.Ex= cos 8.LR neikZ cos 0,
E y=e-ikZ cos 0+ .L R.LCikz co s 0,

E z= sin 8.L Rne ikz cos ° '1

for z<a and for z>b, !
E x=cos 8.L T l e- ikZ cos 0,

E y=.L T .L e- ikz cos 0,

E z=- sin 8.L Tne-ikZ cos 0,

the x- and t-factors b eing suppressed. 'iVhen the incident fi eld is vertically polarized, let

E x= cos Oe- ikz cos o+cos 8nR neikZ cos 0,

E y= I R J.eikz co s 0,

E z=- sin 8e- ikz co s o+ sin 811H IICikZ co s °

for z<a and for z>b,
74
 Ex=cos 0UT l e- ikZ cos 0,
E y= I T 1- e- ikZ cos 0,
E z=- sin 0Il T l e- ikZ cos 0.

Forming an arbitrary lin ear combination of these two fields, we write

E x= B cos Oe-,kz co s 0+ (A1- R u+ 13 I R u) cos 8e 'kz cos 0, }
E y= A e-'kZ co s 0+ (A1- R .l. + 13 I H.de'kz co s 0, (2 1)
E z= - 13 sin 8e-,kz cos 0+ (A .l.R u+ B uR I ) sin Oe,kz cos °
for z<a and fo r z>b,
E x= (A 1-T I + 13II T I ) cos 8e- ikZ COB 0, I
~
E y= (A.l.T .l. + B uT .l. )e- ikZ cos 0, ~ (22)
I
I E z=- (A .l.T II+ B uT I ) sin 8e- ikZCOSO. J

Equation (19) now r educes to

1
- 2ile cos O(A .l.R .l. + 13nR .l.)=-lc 2 a(ME )ye - ikZ cos odz,

while (20) takes the form

- 2ilc(A 1-R II+ B I R I ) =- 1c2. f[(ME)x+ tan 8(ME )z]e- ikZCOB adz.

The physical interpretfltion of t he e integrals at oblique incidence is demonstrated b y

considerations given by W estcott, [1962 b, cJ a nd in further papers t o be published in this eries·
In particular, if t he ionized layer is weak so th at the fi eld in th e medium may b e r eplaced
b y the incident field as th e first approximation, we obtain

A .l.R 1- + 13II R .l. = -~ i lc sec 8 I" (1V12IB cos 8+1\1£22A -11123B sin 8) e- 2ikz cos od z,

A.l.RII+ BII RII = -~ ilc I" (Mll B cos 0+ 1I1I zA -.M I3B sin 0+1\1£3113 sin 8
+ 1V13zA tan 8- M 33B tan 0 sin O)e-Z fkICOS adz.
H ence, if Z is constan t, we obtain

I R .l. = -~ i lc sec 0 ( b(1\1121 cos O- M 23 sin 8)e - 2ikZcos 0dz

~ .J a
_~ ·le 0 (i11YU- lmY2) cos 0+ (i l YU + mnY2) sin 0 ( by -2ik, coso d
- 2 ~ sec U (U 2_ Y2) Ja.Ll..e z,

.L
Rn= -.!.2 i le Jfaa 01,([I ?+
.
M 32 tan 8)e - 2ikZ cOSOdz
_.!. . j (- inYU - lmY2)+ Lan 0(i lrU- mny2) ( " Y -2ik' COBO l
- 2 ~/C U (U2 _ y 2) .L .L1. e G z,

1I R II= -~ ileJ:b (l\111 cos 0- 11113 sin 0+1\131 sin 8- 1V133 tan 8 sin 8)e - 2ikZ cosod z

_.!. .j U2 cos 20 sec 8- (l2 cos 0- 112 tan 8 sin 8) Y 2- 2im sin oYU ( " Y - 2ikz COB 0d
- 2 ~/C U(U2 _ Y 2) Ja.L1. e z.

75
Moreover, we easily deduce the formulae for the isotropic case with Y = Q at oblique incidence,

R =~ ik sec eU- 1 { bX e- 2ikz cos adz

1- 1- 2 Ja '
I R I =~ ik cos 2e sec eU- li b X e-- 2 i k Z cos adz .

The transmission coefficients may be found by multiplying eq (18) by e ikz co s °and inte-
grating over the height range in which ionization occurs. We then multiply the third of the
resulting equations by tan e and subtract it from the first, yielding

(E~ - ik cos eEy) eikz cos 0lb= _p { b (ME )ye -ikz cos 0dz ,
a Ja
and
I, {b
(E~-ik sec eEx+ ik sin eEz)e ikZ co s aIa = -P J a [(ME )x+ tan e(ME)zle ikZ cos °dz. {

Inserting the values (21) and (22) of the field above and below the range of ionization ,
we obtain
1 {b
A 1-T 1- +B uT 1- = A -2 ik sec eJ a (M E)ye i k Z cos ° dz,

A1-Tu + Bu Tu = B-~ iki b [(ME)x+ tan e(ME )z]eikZ cos adz .

Finally, we deduce the values of the transmission coefficients to the first order, when the
field throughout the medium is replaced by the incident field:

I T1-= -~ ik sec e { b (Jl![21 cos e- lJ![23 sin e)dz,

~ Ja
1-Tu= -~ ikib (M 12 + tan eM32 )cl z,

I TD= l-~ ik { b (M1l cos e-1\113 sin e+ M 31sin (J - )YI33 tan (J sin e)dz .
.L
In the isotropic case, these formulae reduce to

1-T 1- = 1 +~ ik sec (JU - li bXd z,

u TlI= l + ~ ik cos 2(J sec eU- i

1
b
Xd z.
.1
c,

10. Comparison of the Two Methods

We should conclude with a note on the difference between the results of Wait [1962] and
of the method presented in this paper. Considering the isotropic model at vertical incidence,
the first-order reflection coefficient produced here is given by (11 ), namely

(23)

for zero collision frequency. On the other hand, the first approximation to the solution of
the integral eq (13) is obtained by taking V= 1, yielding
76
 Hence, when t=a,

or
(24)

when the left hand sid e is expanded by the binomial since IRI is small.
It is obvious that this result is only valid for a thin layer, in which case the phase factor
e- 2 i k z in the integrand of (23) is replaced by e- 2 i ka • H ence, for the homogeneous slab di scussed
in section 7, eq (24) yields (when a= O, b= h) ,

R=~ i k SohXdz=~ i khX.

The exact first term (14) reduces to t hi s only when kh is small. On the other hand, it should
be pointed out that 'IVait's expiLnsion for the homogeneous slab is or a different kind, valid
when Ikhyl(l - X) I« l.

11 . References

Brekhovskikh, L. lVI., Waves in layered media (Academic Press, London and I ew York, 1960).
Budden, K G., Radio waves in the ionosphere (Cambridge Univers ity Press, England, 1961).
Epstein , P . S., Refl ection of waves in an in homogeneous absorbing medium, Proc. Nat. Acad. Sci . (U.S.A.)
16, 627 (1930) .
H eadi ng, J. , An introduction to phase-integr al methods (M ethu en, London, 1962).
Wait, J . ll., On the propagation of VLF and ELF radio Wfl,ves when the ionosphere is not sharply bounded,
J . Resea rch NBS 66D (Radio Prop .) No.1, 53- 61 (J an.- Feb . 1962) .
Westcott, B. S., Ionospheric r efl ection processes for long radio-wavcs-I, J . Atmospheric and T errest. Phys.
24, 385- 399 (1962a) .
West cott, B. S., Ionospheric r efl ection processes for long radio-waves-II, J. Atmospheric and T erres t . Phys.
24, 619- 631 (1962b).
'Westcott, B. S., Ion ospher ic r efl ection processes for long radio-wav es-III, J. Atmospheric and T errest. Phys.
24,701- 71 3 (1962e) .

(Paper 67Dl- 244 )

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