Part A | 8.3
180 Part A Fundamentals
while the time-harmonic vector wave equations become
r
2 E D r r M i C
1
"
rq e C i!!J i
C i!!!E !
2
"E ;
(8.27)
r
2 H D Dr r J i C M i C
1
rq m
C i!"M i C i!!!H !
2
"H :
(8.28)
Although J i and q e can represent either real or virtual
sources, M i and q m are only virtual. In a source free
region, J i D q e D 0 and M i D q m D 0, so that (8.27)
and (8.28) reduce to the homogenous form of the wave
equation
r
2 E
2 E D 0 ;
(8.29)
r
2 H
2 H D 0 ;
(8.30)
where
2
D i!!! !
2
" :
(8.31)
Equations (8.29) and (8.30) describe electromagnetic
waves traveling in a lossy medium with a propagation
constant given by .
The last equations needed to completely define electromagnetic fields in the ocean are boundary conditions.
Boundary conditions describe how the fields respond
at an interface formed between two media of different constitutive properties, such as at the sea surface
and seafloor. In this case, air is nonconducting with
a permittivity and permeability the same as free space,
while seawater is conductive with a high permittivity.
Although the seafloor is conductive, its effective conductivity is one to four orders of magnitude less than
the ocean’s; depending on the depth the fields propagate below the bottom.
The generalized boundary conditions are well
known. Their detailed derivation from Maxwell’s equations can be found in [8.2]. Let the unit normal at the
interface between two different media be given by O
n and
pointing into the second region. Let the fields in media
1 and 2 be given by E 1 , H 1 , D 1 , B 1 and E 2 , H 2 , D 2 ,
B 2 , respectively. Then the boundary conditions at the
interface between them are
O n .E 2 E 1 / D M s ;
(8.32)
O
n .H 2 H 1 / D J s ;
(8.33)
O
n .D 2 D 1 / D q es ;
(8.34)
O
n .B 2 B 1 / D q ms ;
(8.35)
where M s and J s are the linear magnetic and electric
surface currents along the boundary, and q es and q ms
are the linear surface charge densities on the boundary. Although M s and q ms are virtual boundary sources,
they do arise in equivalent source representations of the
fields within a closed region.
8.3 Plane Wave Propagation
The homogenous wave (8.29) and (8.30) can be solved
in any of the separable coordinate systems. Their solution in rectangular coordinates will result in the expressions for plane waves. Using the method of separation
of variables, the solution to (8.29) and (8.30) for a wave
traveling along the positive z-axis is
E D E 0 e
z
;
(8.36)
H D H 0 e
z
;
(8.37)
where E 0 and H 0 are the amplitudes of the electric and
magnetic fields. The propagation constant , defined by
(8.31), can be written as
D
p
i!!.. C i!"/ ;
(8.38a)
D ˛ C iˇ ;
(8.38b)
and ˛ is the attenuation constant in Np=m, and ˇ is the
phase constant in rad=m. If the waves were traveling
in the negative z direction, then z in (8.36) and (8.37)
would be replaced by z. Solutions to the homogenous
wave equation in cylindrical and spherical coordinates
are derived in [8.2, pp. 12–18].
The electric and magnetic fields in (8.36) and (8.37)
are linked through Maxwell’s equations. For a sourcefree region, (8.24) can be written as
E D
1
C i!"
r r H :
(8.39)
If the magnetic field vector H in (8.39) has only an x
component, given by H x , taking its curl and substituting
it in (8.36) gives
E y D
C i!"
H x :
(8.40)
Substituting (8.38a) into (8.40) and reducing gives
E y D DÁH x ;
(8.41)
where
Á D
r
i!!
C i!"
:
(8.42)
180 Part A Fundamentals
while the time-harmonic vector wave equations become
r
2 E D r r M i C
1
"
rq e C i!!J i
C i!!!E !
2
"E ;
(8.27)
r
2 H D Dr r J i C M i C
1
rq m
C i!"M i C i!!!H !
2
"H :
(8.28)
Although J i and q e can represent either real or virtual
sources, M i and q m are only virtual. In a source free
region, J i D q e D 0 and M i D q m D 0, so that (8.27)
and (8.28) reduce to the homogenous form of the wave
equation
r
2 E
2 E D 0 ;
(8.29)
r
2 H
2 H D 0 ;
(8.30)
where
2
D i!!! !
2
" :
(8.31)
Equations (8.29) and (8.30) describe electromagnetic
waves traveling in a lossy medium with a propagation
constant given by .
The last equations needed to completely define electromagnetic fields in the ocean are boundary conditions.
Boundary conditions describe how the fields respond
at an interface formed between two media of different constitutive properties, such as at the sea surface
and seafloor. In this case, air is nonconducting with
a permittivity and permeability the same as free space,
while seawater is conductive with a high permittivity.
Although the seafloor is conductive, its effective conductivity is one to four orders of magnitude less than
the ocean’s; depending on the depth the fields propagate below the bottom.
The generalized boundary conditions are well
known. Their detailed derivation from Maxwell’s equations can be found in [8.2]. Let the unit normal at the
interface between two different media be given by O
n and
pointing into the second region. Let the fields in media
1 and 2 be given by E 1 , H 1 , D 1 , B 1 and E 2 , H 2 , D 2 ,
B 2 , respectively. Then the boundary conditions at the
interface between them are
O n .E 2 E 1 / D M s ;
(8.32)
O
n .H 2 H 1 / D J s ;
(8.33)
O
n .D 2 D 1 / D q es ;
(8.34)
O
n .B 2 B 1 / D q ms ;
(8.35)
where M s and J s are the linear magnetic and electric
surface currents along the boundary, and q es and q ms
are the linear surface charge densities on the boundary. Although M s and q ms are virtual boundary sources,
they do arise in equivalent source representations of the
fields within a closed region.
8.3 Plane Wave Propagation
The homogenous wave (8.29) and (8.30) can be solved
in any of the separable coordinate systems. Their solution in rectangular coordinates will result in the expressions for plane waves. Using the method of separation
of variables, the solution to (8.29) and (8.30) for a wave
traveling along the positive z-axis is
E D E 0 e
z
;
(8.36)
H D H 0 e
z
;
(8.37)
where E 0 and H 0 are the amplitudes of the electric and
magnetic fields. The propagation constant , defined by
(8.31), can be written as
D
p
i!!.. C i!"/ ;
(8.38a)
D ˛ C iˇ ;
(8.38b)
and ˛ is the attenuation constant in Np=m, and ˇ is the
phase constant in rad=m. If the waves were traveling
in the negative z direction, then z in (8.36) and (8.37)
would be replaced by z. Solutions to the homogenous
wave equation in cylindrical and spherical coordinates
are derived in [8.2, pp. 12–18].
The electric and magnetic fields in (8.36) and (8.37)
are linked through Maxwell’s equations. For a sourcefree region, (8.24) can be written as
E D
1
C i!"
r r H :
(8.39)
If the magnetic field vector H in (8.39) has only an x
component, given by H x , taking its curl and substituting
it in (8.36) gives
E y D
C i!"
H x :
(8.40)
Substituting (8.38a) into (8.40) and reducing gives
E y D DÁH x ;
(8.41)
where
Á D
r
i!!
C i!"
:
(8.42)