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Some electromagnetic wave functions for propagation in stratified media
Journal of Atmospheric andTerrestrial Pliysics,I@&%, Vol. 26, pp. 336to 340. Pergamon PressLtd. PrintedinNorthern Ireland
Some electromagnetic wave functions for propagation in stratified media R. N. GOULD and R. BURMAN Physics Department,
Victoris (Received
University
of Wellington,
New Zealand
15 Pu’ovember 1963)
Ab&&-The propagation of el~troma~eti~ waves in pbmar or spherically stratif%ed isotropic media is described by certain well-known differential equations (BAAED, 1949 and 1958). Considerable use (e.g. BUDDEN, 1961%) has been made of exact solutions which have been obtained for special profiles of the refractive index. In this paper some further exact solutions are given for both plane and spherical geometry. 1. INTRODUCTION theory of electromagnetic fields in stratified media finds important applications to the problems of radio propagation in the ionosphere and the troposphere since, for many purposes, these regions may be regarded as stratified (BREMMER, 1949 and 1958; BUDDEN 196la and 1961b). This is particularly true at v.1.f. since local ionospheric irregularities are usually much smaller than the wavelength. The case of spherically stratified media is also relevant to scattering problems (TAI, 1963). For the propagation of electroma~etic waves in planar or spherically stratified isotropic media the general field consists of a superposition of two component parts, which are known as fields of the “electric” and “magnetic” types. They can be represented by electric and magnetic Hertz vectors respectively, these vectors being perpendicular to the stratifications. In the plane case, taking the z-axis to be perpendicular to the stratifications, the z-wise variation of the Hertz vector is given by a function f(z) which satisfies the equation (BREMMER, 1958) THE
(1) where, for fields of magnetic type J&r = k(z), and for fields of electric type
I& = k@(z)- k(x)
(2)
z3{l/L(z)}.
In these equations k(x) is the product of the refractive index and the free space plane wave propagation constant, and y is a separation constant giving the propagation constant of the waves. The other equations, arising from the process of separation and giving the variation of the Hertz vector in planes of constant z, can be solved in simple terms in either Cartesian or cylindrical polar co-ordinates. In the spherical case, introducing spherical polar co-ordinates with the radial 335
co-ordinatc r measured ~erpendiclllar to the strati~~atioi~s, the r-wise variation of the Hertz vector is given by a(~),$ where u- satisfies the equation (BREXXER, 1949 and 195X) (4) and k,, is given by equations (2) and (3) on replacing z by r. The problem thus reduces t,o solving equations (1) and (4). In the case of high frequencies the ionosphere may be regarded as a slowly varying medium and approximate soiut.ions to the differential equations can be obtained by the WKB method. However at r.1.f. and e.1.f. the WKB solutions are not applicable, since the ionosphere may vary significantly within a wavelength. IIence at these frequencies it is of particular interest to have exact solut.ions to the differe~~tial equations for various special profiles of the refractive index. discussions of differential equations having the form of equations (I) and (4), namely y” -t f(x)?/ = 0 have been given by KAMXE (1948)and by RICHARDS (1959). Fields of magnatic type in planar stratified media are most amenable to treatment and various solutions exist for particular profiles of the refractive index (BREKHOVSKIKH, 1960; BU~DEX, 1961a; EPSTEIN, 1930). For fields of electric type in a planar stratified medium the differential equation is seen to be more complicated; nevertheless exact solutions can be obtained (BURMAN and GOULD, 1963). The case of propagation in a spherically stratified medium appears to have been given less attention ; however for a profile of the form k(r) = (a + h/r -l- ~/7~)"~ it has been mentioned by BREMMER (1949), that equation (4) for fields of the magnetic type can be solved in terms of Whittaker functions. A discussion of the differential equations governing the electromagnetic fields in a spherically stratified medium has recently been given by TAT (1963). In t.hix paper some further exact solutions are given for the cases discussed above. Spherically s0ratificd media arc considered in Section 2 and planar stratified media in Fiection 3. 2. S~~HERI~~AI.I.~ STRATIE'IIW MEDIA First? a profile having the form k(r) --- 1’~ $ br2 + c/r2 ,
(5)
where (1, b, and c arc in(l~~en(lent of r, is considered for fields of the magnetic type. ~~ransforn~ing to variables q and v(g) according to the equations g = 9.2 u(rf) = r1j2 u(r),
and
(6)
equat,ion (4) bocomca d2v ;i;r12
I +
This may be compared
-$
[bg2 + ag -t
with Whittaker’s
{c +
1, - n(n + 1)p.J = 0.
confluent hypergeometric
a2X2 + 433~x.X + (1 -
4m2)]W = 0,
equation
Some electromagneticwave functions for propagation in stratified media
which has linearly independent solutions W,,,(CCE) and W_,,,( Whittaker functions (MAGNUS and OBERHETTINGER,1949). Thus $.-l/2 w -,,J
U(T) = ~-1/2~~*~(,~~~ r2), are linearly independent
-ax)
known
-il/br2),
337
as
(9)
solutions, where p = -&L/41/6 m =$y/,(r&
and
-t- 1) + $ -c.
(10)
For a profile of the form k(r) =
-A--
(11)
a + br ’
the second derivative of l/k(r) vanishes. Therefore ice, = k(r) for fields of both electric and magnetic types. On changing the independent variable from r to X, where it: = ---?~,/a, equation (4) becomes, for fields of both types, d2u -+ dx2
1
1 x2(x -
- b2
1)2
The hypergeometric
n(n + 1) x2 + 2n(n +
n(n + 1) u = 0.
d2Y
CEY &
in normal form by changing the dependent y
=
Ix/-W
.
11 _ xy+P-Y+w2
+
q$y
1 -
d%
where
+
i
i12.
4x2
1 -
-+ 4(x -
A=y--1,
p2
. gx) .
A2+p2--2-1
I)2 +
= 0,
variable from y to s where
Thus (KAMKE, 1948)
d>
(12)
equation
x(x - If @ + Ku + p + 1)x - y] is obtained
1
1)x -
4x(x -
1 f s=.
1)
v=M:
Iu=~+B-Y,
(14
-
B*
(15) (Ifi)
Comparison of equations (12) and (15) shows that ~(~~ may be expressed in terms of the hypergeometric function with h2 = 1 + 4n(n + I), $
and
=
1 -
4/b2
v2 = 1 + 4&
(17) + 1) -
4/P.
1
3. PLANAR STATIONED MEDIA
For a medium having a variation
of refractive
index given by
(18) equation (1) becomes, for fields of the electric type
(19)
R. N.
338
GOULD and
R. BURMAN
and for fields of the magnetic type
where q = az + b. Comparison
of equations f(z)
(21)
(B), (19) and (20) shows that
= K,?J2Y(~
+
bla)l,
Kv,J?$(z
+ b/a)1
(22)
are the required solutions, where
P = WW,
(23)
and m is zero for the electric type field and 4 for the magnetic type field. For a variation of refractive index defined by k(z) = c set (az + b), equation
(24)
(1) becomes, for fields of the electric type dz2 + [a2 + c2 -
y2 + c2 tan2 (ax + b)]f = 0.
For fields of the magnetic type equation zG< + [c2 Transforming
(25)
(1) becomes
y2 + c2 tan2 (az + b)]f = 0.
(26)
equations (25) and (26) by writing x = tanh (i(az + b)},
(27)
gives
for electric type fields and
t
(I-~2)~-2~~+[-$+~]~=0 for magnetic type fields. ated equation (1 -
Equations
x2) 2
-
J
(28) may be compared
2x2
+ [I+
the solutions of which are I’,,@(x), Q”(x). f(z) where and ,u = &d/1 fields.
= P/[tanh
+ 1) -
- y2/a2 for electric
with Legendre’s
&-&
= 0,
associ-
(29)
Thus
{i(az + b)}],
Y = &{ -1
(28)
f
d/1 -
&/[tanh
(;(a~
+
b)}l,
4c2/a2}
type fields and p = -&/a
(30) (31)
for magnetic
type
Some electroma~metic
w&xx functions
for propagtitiorl in xtwtifitvf media
339
In the case of a homogeneous medium ke, is independent of z for fields of both types and the solutions of equation (1) may be expressed in terms of exponential functions. For magnetic type fields the homogeneous model is the only one giving rise to constant values of k,.ff. For electric type fields this is not however true: k,, is independent of z providing k(z) satisfies the equation
k2(z) - k(z)
l I = a, g2(,i,,
(32)
where a is a constant. Equation (32) does not contain the independent variable explicitly and can be solved by standard methods (RICHARDS, 1959). The solution is seen to be ’ = For the particular
s
41/k)
d/b -
a/k2 -
(33)
2 log k *
case where a = 0 = b it can be shown that z=C+il/T2erf*i;,
(34)
where C is a constant. For a variation of refractive index obeying equation (34) k,, is zero and the fields obey the exponential law e+. Finally for a medium having a linear variation of refractive index the treatment already given for electric type fields (BURMAN and GOULD, 1963) can be extended to the case of magnetic type fields. Thus, if k(z) = a -f- bz,
(35)
putting and
q = (a + bz)2 V(P) = (a + bz)l’~(~),
(36)
I
it can be shown that for magnetic type fields where
f(z) = (a + bz)-“2WtP,~,4[fi(a p = iy2/4b. 4.
+ bzJ2/bl )
(37)
CONCLUSIOK
In this paper several exact solutions have been given for the propagation of electromagnetic waves in planar and spherically stratified media. These could be of value in the theoretical treatment of low frequency radio propagation, as well as in other fields. It is very probable that these solutions arc by no means exhaustive, nevertheless they provide a basis for the treatment of a wider variety of ionospheric models than hitherto. The applications of solutions in closed form can, in practice, be extended in two ways. In the first place it is possible to obtain exact solutions in which the variation of refractive index is completely determined by the propagation constant y. In such cases the model cannot be changed by free choice of arbitrary constant and hence these results are of limited value; for this reason examples have not been given here. Secondly it proves to be possible to obtain exact solutions of equations
R. N. GOULD
340
and R. BURMAN
which differ only slightly from equations (I) and (4) derived from given variations of the refractive index. Such solutions thus enable an a~c~ate, though approxima~, treatment of the model to be given, at least for restricted ranges of the frequency. REFERENCES BREKHOVSKIKHL.M.
1960
BREMMERH.
1949
BREMMERH.
1958
BUDDENK.G.
1961a
BUDDENK.G.
196lb
BCRXAN R. andGouI,~ R.N. EPSTEIN P. 8. KAMKEE.
1963 1930 1948
MAGNUS
1949
W. ~~~OBERHE~INUERF.
RICHARDS P. I.
1959
TAI G. T.
1963
Waves in Layered Media. Academic Press, London and New York. !l’ewestrial Radio Wavea. Elsevier Publ. Co., Amsterdam and New York. Handbueh der PhyGk, 16. SpringerVerlag Berlin. Radio Waves in the lonos~~e~e. Cambridge University Press. The Wave-guide Mode Theory of Wave Pro~agat~~~. Logos Press, London. J. &wx@+. Few. Phys. 25,643. Proc. Nat. Acad, Sei. (U.S.A.), 16, 627. D.~~ereentialgleichungen, Liiaungsmeth.oden und Ltisungen. Ghelsea Publ. Co., New York. Special Functions of Mathematical Physics. Chelsea Publ. Go., New York. Manual of Mathemutical Ph,ysics. Pergamon Press, London. J. Res. N.R.S. 67D, 199.
Some electromagnetic wave functions for propagation in stratified media R. N. GOULD and R. BURMAN Physics Department,
Victoris (Received
University
of Wellington,
New Zealand
15 Pu’ovember 1963)
Ab&&-The propagation of el~troma~eti~ waves in pbmar or spherically stratif%ed isotropic media is described by certain well-known differential equations (BAAED, 1949 and 1958). Considerable use (e.g. BUDDEN, 1961%) has been made of exact solutions which have been obtained for special profiles of the refractive index. In this paper some further exact solutions are given for both plane and spherical geometry. 1. INTRODUCTION theory of electromagnetic fields in stratified media finds important applications to the problems of radio propagation in the ionosphere and the troposphere since, for many purposes, these regions may be regarded as stratified (BREMMER, 1949 and 1958; BUDDEN 196la and 1961b). This is particularly true at v.1.f. since local ionospheric irregularities are usually much smaller than the wavelength. The case of spherically stratified media is also relevant to scattering problems (TAI, 1963). For the propagation of electroma~etic waves in planar or spherically stratified isotropic media the general field consists of a superposition of two component parts, which are known as fields of the “electric” and “magnetic” types. They can be represented by electric and magnetic Hertz vectors respectively, these vectors being perpendicular to the stratifications. In the plane case, taking the z-axis to be perpendicular to the stratifications, the z-wise variation of the Hertz vector is given by a function f(z) which satisfies the equation (BREMMER, 1958) THE
(1) where, for fields of magnetic type J&r = k(z), and for fields of electric type
I& = k@(z)- k(x)
(2)
z3{l/L(z)}.
In these equations k(x) is the product of the refractive index and the free space plane wave propagation constant, and y is a separation constant giving the propagation constant of the waves. The other equations, arising from the process of separation and giving the variation of the Hertz vector in planes of constant z, can be solved in simple terms in either Cartesian or cylindrical polar co-ordinates. In the spherical case, introducing spherical polar co-ordinates with the radial 335
co-ordinatc r measured ~erpendiclllar to the strati~~atioi~s, the r-wise variation of the Hertz vector is given by a(~),$ where u- satisfies the equation (BREXXER, 1949 and 195X) (4) and k,, is given by equations (2) and (3) on replacing z by r. The problem thus reduces t,o solving equations (1) and (4). In the case of high frequencies the ionosphere may be regarded as a slowly varying medium and approximate soiut.ions to the differential equations can be obtained by the WKB method. However at r.1.f. and e.1.f. the WKB solutions are not applicable, since the ionosphere may vary significantly within a wavelength. IIence at these frequencies it is of particular interest to have exact solut.ions to the differe~~tial equations for various special profiles of the refractive index. discussions of differential equations having the form of equations (I) and (4), namely y” -t f(x)?/ = 0 have been given by KAMXE (1948)and by RICHARDS (1959). Fields of magnatic type in planar stratified media are most amenable to treatment and various solutions exist for particular profiles of the refractive index (BREKHOVSKIKH, 1960; BU~DEX, 1961a; EPSTEIN, 1930). For fields of electric type in a planar stratified medium the differential equation is seen to be more complicated; nevertheless exact solutions can be obtained (BURMAN and GOULD, 1963). The case of propagation in a spherically stratified medium appears to have been given less attention ; however for a profile of the form k(r) = (a + h/r -l- ~/7~)"~ it has been mentioned by BREMMER (1949), that equation (4) for fields of the magnetic type can be solved in terms of Whittaker functions. A discussion of the differential equations governing the electromagnetic fields in a spherically stratified medium has recently been given by TAT (1963). In t.hix paper some further exact solutions are given for the cases discussed above. Spherically s0ratificd media arc considered in Section 2 and planar stratified media in Fiection 3. 2. S~~HERI~~AI.I.~ STRATIE'IIW MEDIA First? a profile having the form k(r) --- 1’~ $ br2 + c/r2 ,
(5)
where (1, b, and c arc in(l~~en(lent of r, is considered for fields of the magnetic type. ~~ransforn~ing to variables q and v(g) according to the equations g = 9.2 u(rf) = r1j2 u(r),
and
(6)
equat,ion (4) bocomca d2v ;i;r12
I +
This may be compared
-$
[bg2 + ag -t
with Whittaker’s
{c +
1, - n(n + 1)p.J = 0.
confluent hypergeometric
a2X2 + 433~x.X + (1 -
4m2)]W = 0,
equation
Some electromagneticwave functions for propagation in stratified media
which has linearly independent solutions W,,,(CCE) and W_,,,( Whittaker functions (MAGNUS and OBERHETTINGER,1949). Thus $.-l/2 w -,,J
U(T) = ~-1/2~~*~(,~~~ r2), are linearly independent
-ax)
known
-il/br2),
337
as
(9)
solutions, where p = -&L/41/6 m =$y/,(r&
and
-t- 1) + $ -c.
(10)
For a profile of the form k(r) =
-A--
(11)
a + br ’
the second derivative of l/k(r) vanishes. Therefore ice, = k(r) for fields of both electric and magnetic types. On changing the independent variable from r to X, where it: = ---?~,/a, equation (4) becomes, for fields of both types, d2u -+ dx2
1
1 x2(x -
- b2
1)2
The hypergeometric
n(n + 1) x2 + 2n(n +
n(n + 1) u = 0.
d2Y
CEY &
in normal form by changing the dependent y
=
Ix/-W
.
11 _ xy+P-Y+w2
+
q$y
1 -
d%
where
+
i
i12.
4x2
1 -
-+ 4(x -
A=y--1,
p2
. gx) .
A2+p2--2-1
I)2 +
= 0,
variable from y to s where
Thus (KAMKE, 1948)
d>
(12)
equation
x(x - If @ + Ku + p + 1)x - y] is obtained
1
1)x -
4x(x -
1 f s=.
1)
v=M:
Iu=~+B-Y,
(14
-
B*
(15) (Ifi)
Comparison of equations (12) and (15) shows that ~(~~ may be expressed in terms of the hypergeometric function with h2 = 1 + 4n(n + I), $
and
=
1 -
4/b2
v2 = 1 + 4&
(17) + 1) -
4/P.
1
3. PLANAR STATIONED MEDIA
For a medium having a variation
of refractive
index given by
(18) equation (1) becomes, for fields of the electric type
(19)
R. N.
338
GOULD and
R. BURMAN
and for fields of the magnetic type
where q = az + b. Comparison
of equations f(z)
(21)
(B), (19) and (20) shows that
= K,?J2Y(~
+
bla)l,
Kv,J?$(z
+ b/a)1
(22)
are the required solutions, where
P = WW,
(23)
and m is zero for the electric type field and 4 for the magnetic type field. For a variation of refractive index defined by k(z) = c set (az + b), equation
(24)
(1) becomes, for fields of the electric type dz2 + [a2 + c2 -
y2 + c2 tan2 (ax + b)]f = 0.
For fields of the magnetic type equation zG< + [c2 Transforming
(25)
(1) becomes
y2 + c2 tan2 (az + b)]f = 0.
(26)
equations (25) and (26) by writing x = tanh (i(az + b)},
(27)
gives
for electric type fields and
t
(I-~2)~-2~~+[-$+~]~=0 for magnetic type fields. ated equation (1 -
Equations
x2) 2
-
J
(28) may be compared
2x2
+ [I+
the solutions of which are I’,,@(x), Q”(x). f(z) where and ,u = &d/1 fields.
= P/[tanh
+ 1) -
- y2/a2 for electric
with Legendre’s
&-&
= 0,
associ-
(29)
Thus
{i(az + b)}],
Y = &{ -1
(28)
f
d/1 -
&/[tanh
(;(a~
+
b)}l,
4c2/a2}
type fields and p = -&/a
(30) (31)
for magnetic
type
Some electroma~metic
w&xx functions
for propagtitiorl in xtwtifitvf media
339
In the case of a homogeneous medium ke, is independent of z for fields of both types and the solutions of equation (1) may be expressed in terms of exponential functions. For magnetic type fields the homogeneous model is the only one giving rise to constant values of k,.ff. For electric type fields this is not however true: k,, is independent of z providing k(z) satisfies the equation
k2(z) - k(z)
l I = a, g2(,i,,
(32)
where a is a constant. Equation (32) does not contain the independent variable explicitly and can be solved by standard methods (RICHARDS, 1959). The solution is seen to be ’ = For the particular
s
41/k)
d/b -
a/k2 -
(33)
2 log k *
case where a = 0 = b it can be shown that z=C+il/T2erf*i;,
(34)
where C is a constant. For a variation of refractive index obeying equation (34) k,, is zero and the fields obey the exponential law e+. Finally for a medium having a linear variation of refractive index the treatment already given for electric type fields (BURMAN and GOULD, 1963) can be extended to the case of magnetic type fields. Thus, if k(z) = a -f- bz,
(35)
putting and
q = (a + bz)2 V(P) = (a + bz)l’~(~),
(36)
I
it can be shown that for magnetic type fields where
f(z) = (a + bz)-“2WtP,~,4[fi(a p = iy2/4b. 4.
+ bzJ2/bl )
(37)
CONCLUSIOK
In this paper several exact solutions have been given for the propagation of electromagnetic waves in planar and spherically stratified media. These could be of value in the theoretical treatment of low frequency radio propagation, as well as in other fields. It is very probable that these solutions arc by no means exhaustive, nevertheless they provide a basis for the treatment of a wider variety of ionospheric models than hitherto. The applications of solutions in closed form can, in practice, be extended in two ways. In the first place it is possible to obtain exact solutions in which the variation of refractive index is completely determined by the propagation constant y. In such cases the model cannot be changed by free choice of arbitrary constant and hence these results are of limited value; for this reason examples have not been given here. Secondly it proves to be possible to obtain exact solutions of equations
R. N. GOULD
340
and R. BURMAN
which differ only slightly from equations (I) and (4) derived from given variations of the refractive index. Such solutions thus enable an a~c~ate, though approxima~, treatment of the model to be given, at least for restricted ranges of the frequency. REFERENCES BREKHOVSKIKHL.M.
1960
BREMMERH.
1949
BREMMERH.
1958
BUDDENK.G.
1961a
BUDDENK.G.
196lb
BCRXAN R. andGouI,~ R.N. EPSTEIN P. 8. KAMKEE.
1963 1930 1948
MAGNUS
1949
W. ~~~OBERHE~INUERF.
RICHARDS P. I.
1959
TAI G. T.
1963
Waves in Layered Media. Academic Press, London and New York. !l’ewestrial Radio Wavea. Elsevier Publ. Co., Amsterdam and New York. Handbueh der PhyGk, 16. SpringerVerlag Berlin. Radio Waves in the lonos~~e~e. Cambridge University Press. The Wave-guide Mode Theory of Wave Pro~agat~~~. Logos Press, London. J. &wx@+. Few. Phys. 25,643. Proc. Nat. Acad, Sei. (U.S.A.), 16, 627. D.~~ereentialgleichungen, Liiaungsmeth.oden und Ltisungen. Ghelsea Publ. Co., New York. Special Functions of Mathematical Physics. Chelsea Publ. Go., New York. Manual of Mathemutical Ph,ysics. Pergamon Press, London. J. Res. N.R.S. 67D, 199.