4 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 34, NO. 1, FEBRUARY 1992
Reflection of VLF Radio Waves at a Junction
in the Earth-Ionosphere Waveguide
James R. Wait, Life Fellow, IEEE
Abstract-The conditions for the neglect of reflection of VLF
modes incident on a junction between the daytime and nighttime f
earth-ionosphere waveguides are considered. It is shown that
reflections are indeed small but some interesting and possibly
significant exceptions may occur at highly oblique incidence. In
fact, total reflection may take place for night-to-day transmission
at angles within several degrees of grazing. The same analytical
model is also relevant to oblique transmission across a flat lying
T HE propagation of VLF (very low frequency) radio waves
in the earth-ionosphere waveguide is considered to be
well understood provided the conditions along the transmission
path are uniform -. However, when the path from J
transmitting to the receiving antennas crosses the day/night Fig. 1. Perspective of idealized junction between daytime and nighttime
terminator and/or a coast line, mode conversion will take place earth-ionosphere waveguides where the height changes from h to h’. The
producing an interference pattern which complicates field source vertical dipole and observer at P are on the same side of the terminator
(i.e., the junction).
strength and phase prediction. Considerable theoretical and
numerical studies of this phenomenon have been performed
e.g., -. Commonly the reflection of the waveguide j k , ro, j k , is the propagation constant of the mth-order
modes at the discontinuity is neglected in the analysis. If +
mode, ro = [z2 (y - y ~ ) ~ ] land ,G,(z) is the height
attention is focussed on the characteristics of the field beyond gain of the observer normalized to 1 at z = 0. A time factor
the junction, it appears such an assumption is reasonable. But exp(jwt) has been assumed where w is the angular frequency.
if one is interested in the reflected waves we need a more The coefficient A,, in (l),is the excitation factor of the mth
refined treatment. This is the subject of the present paper. mode. Without loss of generality in this study, we will set it
equal to 1.
11. FORMULATION To facilitate the analysis, we rewrite (1) in the following
equivalent form [ 11:
A n idealized model is shown in Fig. 1. The two parallel
plate waveguides have a junction in the plane y = 0. The +=
structure is uniform for all z and the bottom surface is at
z = 0 for all y. We can regard the upper boundary of the two
E:: = (1/2)~,(z) /’ u,lexp(-umly - yo[)
semiinfinite waveguides as sharp but at different heights h and
h as indicated in Fig. 1. Also to simplify the discussion, the
lower boundary is characterized by fixed surface impedances
where U , = (g2 - k:)”* depends on the mode number
2 and 2’. source of the field is a vertical electric dipole
m . Now we postulate that the resultant vertical electric field
located on and just above the lower boundary at (O,yo,O).
strength E,,, for an incident mode of order m , for the region
Such a source will launch waveguide modes and a typical
propagation mode of order m , will have the form [l]
y > 0, will have the form E,, = E;; +
Here E;; is the vertical electric field of the primary or
incident mode of order m in the region y > 0, KOis the zero- (3)
order modified Bessel or MacDonald function of argument
where Rn,, is a function of g which characterizes the con-
Manuscript received March 28, 1991; revised August 6, 1991.
The author is at 2210 East Waverly, Tucson AZ 85719.
version from, say an incident mode of order m , into reflected
IEEE Log Number 9104201. or scattered modes of order n. Here U, = (g2 - k;)l/’ in
0018-9375/92$03.00 0 1992 IEEE
WAIT: REFLECTION OF VLF RADIO WAVES 5
terms of the propagation constant j k , for modes of order n where
and G,, ( 2 ) are the corresponding height-gain functions.
For the region y < 0. we adopt the form A+‘ = z+’/ - o (2” is the surface impedance a t z = h’).
2Emz = J’ ~ p , exp(-umyo
m +v p y ) ~ p ( z ) 111. MODE MATCHING
It is useful to note that the integral representations, for
. exp(-jgz) dg (4)
E,,, as postulated by (3) and (4), are indeed solutions of
where T,,,, is a function of g which characterizes the con- the Helmholtz equation in the respective waveguide regions
version of a mode of order m into modes or order p and for y > 0 and y < 0. Also, it can be verified that the
ITzl( z ) are the corresponding height-gain functions. In (4), we appropriate surface impedance boundary conditions at z = 0
note that vP = ( g 2 - N i k i ) ’’’
in terms of the propagation
constant j N P k Pfor modes of order p in the waveguide region
and at z = h and h’ are satisfied. But now a fundamental
complication arises when we attempt a rigorous matching of
y < 0. We may interpret NP as an effective refractive the tangential fields across the junction plane y = 0 between
index for horizontal “rays” as they pass from one wave- the two waveguide regions. In fact what happens is that the
guide region to the other. Such will become evident in what TM (transverse magnetic) type field representation that we
follows. have postulated must be supplemented with a TE (transverse
The specific forms of the height-gain functions have not, as electric) type representation. In the present approximate treat-
yet, been specified. But, for example, if the lower boundary of ment we will ignore this TM-TE coupling that would vanish,
the waveguide of width h. for y > 0, had a surface impedance in any case, at normal incidence (i.e., no field variation in the
2. then we could write, for the height interval 0 < z < h. that y direction). Thus, in the present situation, we are scalarizing
the problem in analogy to acoustic waves. The appropriate
G m ( z )= [I + Rm exp(-2jkCmz)]/(l + &) (5) boundary conditions are the continuity of E , and H , (or
where aE,/ay) across the junction plane at y = 0. Thus, on working
with (3) and (4), we require that
Here, C, is a solution of the “vertical mode equation” to be and
discussed below. Also, we note that C i + 5 ; = 1.
For the region y < 0, in the interval 0 < z < h’, we have umGm(z)- unRn,,Gn(z) = ~(
. U ~ T ~ , ~ H(14) Z ) .
H p ( z )= [l RL e x p ( - 2 j k C k z ) ] / ( l +I?;) (7)
the interval 0 > z > h when h > h and over the interval
0 > z > h when h > h‘.
Rb = (CL - A’)/(CL +A’) For the surface impedance conditions adopted above, it is
not difficult to show that the following orthogonality condi-
where tions hold
A’ = Z’/fio (2’ the surface impedance at z = 0).
For the sharply bounded waveguide model adopted, the
G,,(z)G,(z) dz = 0 if m #n (15)
explicit form of the mode equation, for the region y > 0, is
exp(-2jkCmh) = e x p ( - 2 j n ~ =)1 (9)
where H m ( z ) H P ( zdz = 0
) if m # p. (16)
R L = (Cm - A’) / (C, + A’) (10)
where Now to deal with (13), we multiply both sides by H p t ( z ) and
integrate over z from 0 to h’. Then, on utilizing (16), we
- ’ = Z+/fio (2’ is the surface inipedance a t z = h ) .
Similarly, for the region y < 0. the mode equation is
RbR;’exp(-2jkCLh’) = exp(-2jpr) = 1 (11)
Rp” = ( A - A+’)/(CL + A + ’ )
6 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 34, NO. 1, FEBRUARY 1992
Here Pp is a normalizing integral having a value of the order
of h/2 for the daytime earth-ionosphere waveguide. In any
case, it can be computed easily by employing (7). The upper
limit h’ of the coupling integral Cm,p,as given by (19), has
Fig. 2. Plan view of the incident mode of order ni and the reflected mode
been replaced by h on the assumption that G,(z) (or E,%) of order n .
can be assigned the value zero in the interval h < z < h’.
This prescription is applicable when 0 < (h’ - h ) << h.
In a similar manner as above, we reduce (14) to If the contrast between the waveguide regions is small such
that IC1,2I2 and IC2,1I2<< 1, we see that (26) and (27) are
~mcm,p unRn,mCn,p = ~ l p ~ p . m p p , (20) simplified to
Then, on eliminating Tp., from (17) and (20), we obtain the
single set of algebraic equations and
At this stage, if we assumed the coupling coefficient Cn,pwas
Under these same conditions, 1111 - 1121 << 1, in which
zero for n # m. we would have the simple result
case, the mode conversion reflection coefficient R2.1 has a
Rm., (U, - %)/(U, + 1)
1,. (22) relatively small amplitude.
In this case, an incident mode of order m reflects only as Iv, sTATroNARY H~~~~~~~~
a mode of order m. To within the same approximation, the
transmission mode, for y < 0. is also of order m. The we now return to the representation given by (3).
transmission coefficient is then given approximately by A stationary phase approximation (see the Appendix) can be
obtained readily if Rn,,(g) is a slowly varying function of
T,,,, Z 1 + R,,,, 2um/(um+ U,). (23) g in the region where the exponential factor has a stationary
point. The latter occurs when
To deal effectively with (21), we truncate the series over d(u,yo + u n y + j g z ) / d g = 0. (30)
n by a finite number of terms, say n = 1,2,3,....q Then .
correspondingly, we let p range from p = 1,2,3, . . . , q . The Such a is
resultant system can then be solved for Rn,m for any fixed
value of the incident mode number m.
g = gs = ICm sin ,
O, = IC, sin On
In keeping with the approximate tenor of our analysis and where and are constrained by
to provide physical insight, we choose q = 2 and m = 1. Then
the (severely) truncated system is given by yo tan ,
8 + y tan On = z. (32)
(‘1~1 + v l ) c 1 1R1 1 + ( ~ +2 ? / 1 ) c 2 1 = ( ~ - ~ i ) C 1 , 1(24)
The relevant geometry is shown in Fig. 2. Then, for y > 0.
we have E,, = E + EE, where
(U1 + ‘@)C1 1 + (UZ + v1)cz z = ( U 1
2R1 - Vi)ci,2. (25)
= [ ~ / ( 2 . ~ k ) [x
+ (Y ~ o ) ~ ] - ~ / ’ G , ( z )
Thus ‘ exP[-Jhn[J2 + (Y - Yo)2]1/2]
R11 = is the asymptotic approximation of the incident waveguide
(PLI - vl)C1 1(u2+ U * ) C ~ (ul - 7 1 2 ) c 1 , 2 ( U 2 + v l ) c 2
,~ - mode of order m , and where the scattered modes of order n
( U 1 + W)C1 l ( U 2 + W)C2 2 - ( U 1 + 112)C1,2(U2 + l)l)C2,1
(26) = [~/(2~1c,)]~/~
[s, + en cos2 O m / cos2 8,]-1/2
. Gn(z)Rn,m(kmsin ),6 ep-(,,
R2 1 2Ui(Vi - as
vz)Ci,iCl~/Denom. in (26). (27) (34)
WAIT. REFLECTION OF VLF RADIO WAVES
In the special case where mode conversion is neglected (i.e.,
R,,, = 0 for n # m), we see that (34) reduces to
E:= = [7r/(2jkm)] / 2 [X2
1 + (y + ~ o ) ~ ] - ~ ’ ~ G “ ( z )
. R,,,(k, sin 8,) exp[-jk,[x2 + (y + YO)^]^'^]
where t a n 0, = x / ( y yo). Now of course the angles of
incidence and reflection are equal. But, in general, this is not
true for n # m when mode conversion is present.
DISCUSSION THE REFLECTION
The reflection coefficient R,,,, as a function of 8,, is
0 1 2 3 4 5
given approximately by (22). Dropping the subscript m, we GRAZING ANGLt
write this in equivalent form as (NORMALIZED)
R = [sin 9 - ( N 2 - cos2 9)1 / 2 ] Fig. 3. Reflection in decibels for a mode of order m being reflected into
mode of order m for small grazing angles at the junction.
. [sin 9 + (N’ - cos2 X I J ) ” ~ ] - ~ (36)
where 9 = (7r/2)-8, is grazing angle and N = k L / k , is an
The integral appearing in (3) is
effective refractive index. Now if q2 << 1 and IN - 1 <<
1, we see that (36) is well approximated by
R E [ ( 9 / Q C ) ( 9 / 9 c ) 21]1/2]
uA’Rn,m(g) e x ~ [ - j $ ( g ) Idg (AI)
- 1]1/2]-1 (37) where
where 6,= 2(1 - N ) . When N is real and < 1, (RI = 1 4(g) = (IC; - g2) 1’2yo + (k: - g2)‘l2y (A2)
for 9 < 9,. the other hand, if N is real and > 1, R is
On is the complex “phase” function. The derivatives are
conveniently rewritten as
4’(d = - g y o / ( G - g2) gY/(k: - g 2 Y 2 (A31
4”(g) = 4 yo@; - 92)3/2 - k; y/(k: - g2) .
where @o = 2 ( N - 1). In this case, -R is real and < 1. (A41
The two cases considered above correspond to propagation
from nighttime to daytime (i.e., h’ < h ) and from daytime to The stationary phase point is when 4’(g) = 0 yielding
nighttime (i.e., h’ > h), respectively. In fact, for a localized g = gs = k , sin 8, = k , sin 8, ( 4
sharply bounded ionosphere model with hDAy = 70 km and
~ ~ I G H T90 km, it turns out [l] that IN - 1
= 1 at where
20 kHz. Thus, for this example, Qz
= 9; S 2 x or yotan 8, + y t a n 8, =x (A6)
9, = 90 E 2 + degrees.
The reflection loss in dB can be defined as 20 log,, IRI. which is a constraint on the complex angles 19, and 19, as is
This quantity is plotted in Fig. 3 as a function of the normal- clear from the construction in Fig. 2.
ized grazing angle which is 9/9, night-to-day transmis-
for The corresponding phase function is given by
sion and 9/90 day-to-night transmission. In the example
given above, the lower scale in Fig. 3 would run from 0’ to 4s = 4(gs) = k , yo cos8, + k , y cost), + k,xsin 8,
1 2 + O but there is a caveat here in that for angles less than (A71
1 / 2 degree or so, the stationary phase evaluation of the field
where, of course, the latter term could also be written
integral is questionable. In any case, it is certainly evident IR(
becomes very small as the grazing angle exceeds 5’ or so. In
k,xsin8,. Noting that $s = k,s, +
kntn we see that, as
indicated from the geometry in Fig. 2, 4s is the total complex
fact at normal incidence, I I E +IN - 1 2 0.5 x lop3 (or
phase of the reflected ray path from mode m into mode n.
-66 dB). Nevertheless, the near grazing condition could occur Another needed quantity is
when the great circle transmission path tended to be parallel
to the day/night terminator or to a fairly straight coastline. 4; = 4”(gs) 1 - Y O / C O S ~ 8, - Y / C O S ~8,. (A8)
8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 34, NO. 1, FEBRUARY 1992
Then on remembering that ,s = y~/cosO, and 1, = REFERENCES
y/cosO, we have
J. K. Wait, Electromagnetic Waves in Stratified Media, 2nd. ed. New
(-4;)”’ = [s,/cos2 Om + !,/cos2 QnI1”. (A9)
York: Pergamon, 1970.
J. Galejs, Terrestrial Propagation of Long Electromagnetic Waves. New
York: Pergamon, 1972.
On neglecting higher order derivatives we see F. J. Kelly, “ELFArLFILF propagation and system design,” Naval Res.
Lab., Washington, D.C., Rep. 9028, 1987.
1 J. R. Wait, “Mode conversion and refraction effects in the earth-
4(9) % 4 s + 2 ( 9 - gs,”;. ionosphere for VLF radio waves,” J . Geophys. Res., vol. 73, no. 11,
pp. 3537-3548, June 1968.
Then R. A. Pappert and F. P. Snyder, “Some results of a mode conversion
program for VLF,” Radio Science, vol. 7, no. 10, pp. 913-923, Oct.
I,., 2 ( a j k , cos om)-’ exp[-j(k,s, 1972.
K. A. Pappert and D. G. Morftt, “Theoretical and experimental sun-
rise mode conversion results at VLF,” Radio Science, vol. IO, no. 5 ,
(All) pp. 537-546, May 1975.
K. Smith, “Approximate mode conversion coefficients in the earth-
-Cx ionosphere waveguide,” IEEE Trans. Antennas Propagat., vol. AP-23,
no. 11, pp. 868-870, Nov. 1975.
We now introduce a new variable z via T. Kikuchi, “Waveguide mode analyses of Omega VLF wave propagation
at 13.6 kHz, J . Atmospheric and Terrestrial Physics, vol. 48, no. 1,
( j / 2 y ( g - gs)(-4;)1/2 =j z (A12) pp. 15-23, Jan. 1986.
and note that JTzexp(-z)’dz = rl/’ yield
K. A. Pappert and J. A. Ferguson, “VLFILF mode-conversion model cal-
culations for air-to-air transmissions in the earth-ionosphere waveguide,”
Radio Science, vol. 21, no. 4, pp. 551-558, JulyiAug. 1986.
Inn.m [7r/(2jkm)]1’2[Sm+ (e, cos2 &&OS2 &)]-1/2
n,ksn,ep-(,, + k,t,].
(‘413) James R. Wait (SM’56-F’62-LF’91) received the
B.A. Sc., M.A.Sc., and Ph.D. degrees from the
A relatively trivial special case is when Rn,, = 1 and n = m University of Toronto, Toronto, Ont., Canada.
From 1955 to 1980, he was a member of the
from which scientific community in Boulder, CO. His positions
L., [7r/(2jk,)1~/~[s, + exp[-jk,(s, + e,] included: Senior Scientist in N O M , Professor Ad-
joint in electrical engineering at the University of
Colorado, Boulder, CO., Consultant to the Institute
(A141 for Telecommunications, and Fellow of the Co-
operative Institute for Research in Environmental
which is the appropriate asymptotic approximation for Sciences. In 1980 he become Professor of Electrical
K o [ j L ( s , +em)]. Engineering and of Geosciences at the University of Arizona, Tucson.
In obtaining the stationary phase approximation given by In 1988, he was appointed Regents Professor at the University of Arizona.
In 1989, he retired from the university to become a private consultant residing
(A14) via the change of variable in (A12) we have implicitly in Tucson, AZ. Dr. Wait has received numerous awards for his research in
ignored the lateral wave contribution which leads to terms electromagnetics and electrical geophysics including the: Balth van der Pol
having an inverse 312’s power distance dependence. Also, Gold Medal presented by UKSI in Helsinki in 1978, IEEE Centennial Medal
in 1984, IEEE Geoscience and Remote Sensing Achievement Award in 1985,
(A13) is not valid for near grazing angles where Om and On and the IEEE Antennas and Propagation Distinguished Achievement Award
are approaching 90”. in 1990. He is a Member of the (U.S.) National Academy of Engineering.