Part A | 8.7
190 Part A Fundamentals
This no-propagation approximation for air is sometimes called the quasi-static condition, and should
not be confused with the quasi-DC case discussed
previously.
With the solutions to f 1 and f 2 in hand, (8.125) and
(8.126) can be placed in (8.120) through (8.122) to
compute the electric and magnetic fields above and below the sea surface. For some special cases, the field
equations can be solved in the analytical form; but in
general, the Sommerfield integrals must be evaluated
numerically. Unfortunately, the integrands are oscillatory and slowly damped, increasing the difficulty in
their numerical integration. Special techniques have
been developed to efficiently evaluate Sommerfieldtype integrals, most of which have been reviewed
in [8.4].
Derivation of the more complicated equations for
a VED in a shallow ocean, which includes the interface
at the seafloor, follows the same procedure as shown
above for the deep water analysis. In the shallow water
case, a third Hertz vector is needed to describe propagation within the sea bottom, and another term must
be included in (8.126) to account for reflections off the
seafloor. Satisfying the boundary conditions at both the
seafloor and sea surface produce four equations with
four unknowns, this time f 1 through f 4 . The Sommerfeld
integral formulations for a VED in a three layer media
can be found in [8.5], and applied to the shallow-ocean
problem using the appropriate constitutive parameters.
The complementarity problem of a VMD in a deep
or shallow ocean is solved in an analogous manner using a magnetic Hertz vector in place of the electric
vector. The in-air fields from a time-harmonic VMD
submerged in shallow water have been reported in [8.6],
while the in-water fields from an airborne VMD are
found in [8.7]. Because of the presence of the ocean
surface and/or bottom interface, the field solutions for
an HED require both horizontal and vertical electric
Hertz vectors to satisfy the boundary conditions [8.8].
Although this complicates the algebra by doubling
the number of boundary equations and unknowns, the
Sommerfeld formulation for the in-water electric and
magnetic fields from a submerged HED in shallow water have been derived, with the solution given in [8.9].
Similarly, computing the fields for an HMD in the ocean
requires both the horizontal and vertical magnetic Hertz
vectors. The equations for the in-air electric and magnetic fields from submerged VED, HED, and HMD
sources have been complied in [8.10], while the inwater fields from these three sources located above the
ocean are summarized in [8.11]. The referenced shallow water equations can be reduced to the deep-ocean
case by setting the bottom conductivity equal to that of
the water.
Although the electromagnetic fields from dipole
sources submerged in a deep ocean have been formulated in the generalized form of Sommerfeld integrals,
it is very convenient to have algebraic equations readily accessible for obtaining quick estimates; even if
they are only valid for special cases. The algebraic inwater and in-air electric and magnetic field equations
for VED, HED, VMD, and HMD sources submerged
in a deep ocean will be presented in tabular form for
a few special cases. These tabularized equations have
been found to be useful as a fast and easy reference.
The coordinate system and geometry for the tabulated equations are again shown in Fig. 8.10. All
sources will be located at a depth h, with the VED and
VMD aligned in the positive z direction, while the HED
and HMD are oriented along the positive x-axis. The
planner sea surface interface is at z D 0. The magnetic
and electric dipole source strengths will be designated
as m and p, respectively.
The magnetic and electric fields for static magnetic
and electric dipoles in an unbounded ocean were given
previously in spherical coordinates. At times, it is more
convenient to apply the formulas in the Cartesian system. For easy reference, the Cartesian field equations
for a static VMD and HMD source in an unbounded
ocean (no sea surface) are listed in Table 8.1. Although
static magnetic sources produce no electric fields, static
VED and HED dipoles create both magnetic and electric fields as given in Table 8.2 for an unbounded
ocean.
If the sea surface interface is now reintroduced at
z D 0 (Fig. 8.10), the magnetic field equations for the
static VMD and HMD sources in Table 8.1 remain
unchanged. These equations are valid whether the observation point is in water, air, or sea bottom. The
reason is that all three media are nonmagnetic with
a permeability equal to that of free-space.
The jump in conductivity at the ocean’s surface
does, however, affect both the electric and magnetic
fields from submerged electric dipoles. Because the air
is nonconducting, the electric current in the conducting
seawater will stay confined within it and will not flow
across the surface interface. The sea surface is equivalent to an ideal reflector of static electric current. In
addition, the discontinuity in the electric current across
the surface interface produces additional components of
magnetic fields.
Mathematically, the effects of the sea surface on
the in-water static electric fields is equivalent to placing images of the VED and HED directly above the
water at a distance equal to the depth of the sources.
The electric fields produced by these images are listed
in Table 8.3 [8.12]. To get the total in-water static
electric fields for the deep ocean case, the VED and
190 Part A Fundamentals
This no-propagation approximation for air is sometimes called the quasi-static condition, and should
not be confused with the quasi-DC case discussed
previously.
With the solutions to f 1 and f 2 in hand, (8.125) and
(8.126) can be placed in (8.120) through (8.122) to
compute the electric and magnetic fields above and below the sea surface. For some special cases, the field
equations can be solved in the analytical form; but in
general, the Sommerfield integrals must be evaluated
numerically. Unfortunately, the integrands are oscillatory and slowly damped, increasing the difficulty in
their numerical integration. Special techniques have
been developed to efficiently evaluate Sommerfieldtype integrals, most of which have been reviewed
in [8.4].
Derivation of the more complicated equations for
a VED in a shallow ocean, which includes the interface
at the seafloor, follows the same procedure as shown
above for the deep water analysis. In the shallow water
case, a third Hertz vector is needed to describe propagation within the sea bottom, and another term must
be included in (8.126) to account for reflections off the
seafloor. Satisfying the boundary conditions at both the
seafloor and sea surface produce four equations with
four unknowns, this time f 1 through f 4 . The Sommerfeld
integral formulations for a VED in a three layer media
can be found in [8.5], and applied to the shallow-ocean
problem using the appropriate constitutive parameters.
The complementarity problem of a VMD in a deep
or shallow ocean is solved in an analogous manner using a magnetic Hertz vector in place of the electric
vector. The in-air fields from a time-harmonic VMD
submerged in shallow water have been reported in [8.6],
while the in-water fields from an airborne VMD are
found in [8.7]. Because of the presence of the ocean
surface and/or bottom interface, the field solutions for
an HED require both horizontal and vertical electric
Hertz vectors to satisfy the boundary conditions [8.8].
Although this complicates the algebra by doubling
the number of boundary equations and unknowns, the
Sommerfeld formulation for the in-water electric and
magnetic fields from a submerged HED in shallow water have been derived, with the solution given in [8.9].
Similarly, computing the fields for an HMD in the ocean
requires both the horizontal and vertical magnetic Hertz
vectors. The equations for the in-air electric and magnetic fields from submerged VED, HED, and HMD
sources have been complied in [8.10], while the inwater fields from these three sources located above the
ocean are summarized in [8.11]. The referenced shallow water equations can be reduced to the deep-ocean
case by setting the bottom conductivity equal to that of
the water.
Although the electromagnetic fields from dipole
sources submerged in a deep ocean have been formulated in the generalized form of Sommerfeld integrals,
it is very convenient to have algebraic equations readily accessible for obtaining quick estimates; even if
they are only valid for special cases. The algebraic inwater and in-air electric and magnetic field equations
for VED, HED, VMD, and HMD sources submerged
in a deep ocean will be presented in tabular form for
a few special cases. These tabularized equations have
been found to be useful as a fast and easy reference.
The coordinate system and geometry for the tabulated equations are again shown in Fig. 8.10. All
sources will be located at a depth h, with the VED and
VMD aligned in the positive z direction, while the HED
and HMD are oriented along the positive x-axis. The
planner sea surface interface is at z D 0. The magnetic
and electric dipole source strengths will be designated
as m and p, respectively.
The magnetic and electric fields for static magnetic
and electric dipoles in an unbounded ocean were given
previously in spherical coordinates. At times, it is more
convenient to apply the formulas in the Cartesian system. For easy reference, the Cartesian field equations
for a static VMD and HMD source in an unbounded
ocean (no sea surface) are listed in Table 8.1. Although
static magnetic sources produce no electric fields, static
VED and HED dipoles create both magnetic and electric fields as given in Table 8.2 for an unbounded
ocean.
If the sea surface interface is now reintroduced at
z D 0 (Fig. 8.10), the magnetic field equations for the
static VMD and HMD sources in Table 8.1 remain
unchanged. These equations are valid whether the observation point is in water, air, or sea bottom. The
reason is that all three media are nonmagnetic with
a permeability equal to that of free-space.
The jump in conductivity at the ocean’s surface
does, however, affect both the electric and magnetic
fields from submerged electric dipoles. Because the air
is nonconducting, the electric current in the conducting
seawater will stay confined within it and will not flow
across the surface interface. The sea surface is equivalent to an ideal reflector of static electric current. In
addition, the discontinuity in the electric current across
the surface interface produces additional components of
magnetic fields.
Mathematically, the effects of the sea surface on
the in-water static electric fields is equivalent to placing images of the VED and HED directly above the
water at a distance equal to the depth of the sources.
The electric fields produced by these images are listed
in Table 8.3 [8.12]. To get the total in-water static
electric fields for the deep ocean case, the VED and