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Frequency, Temperature and Salinity Variation of the Permittivity of Seawater

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With the emergence of unmanned marine robots, underwater communication systems have received much attention in recent years. To successfully develop radio wave based communication solutions, it is essential to understand properties of electromagnetic wave transmission in seawater. These properties are determined by the frequency variation of the permittivity of seawater. Existing models for the permittivity of saline water are empirical ones that best fit experimental data. We propose a physically realistic model, similar to the one used in plasma physics, for the variation of the dielectric constant of water with varying frequencies and salinities. Our model is in excellent agreement with existing empirical fits for frequencies between 1 and 256 GHz. We use this model to study the propagation of electromagnetic waves in seawater. We explain that large propagation distances would be possible at MHz frequencies if the conductivity of seawater decreases at small field strengths due to the hydrogen bonding of water molecules. However, we were unable to experimentally verify any reduction in the conductivity of seawater
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Frequency, temperature and salinity variation of the
permittivity of Seawater
Ram Somaraju and Jochen Trumpf
Abstract
With the emergence of unmanned marine robots, underwater communication systems have received much
attention in recent years. To successfully develop radio wave based communication solutions, it is essential to
understand properties of electromagnetic wave transmission in seawater. These properties are determined by the
frequency variation of the permittivity of seawater. Existing models for the permittivity of saline water are empirical
ones that best fit experimental data. We propose a physically realistic model, similar to the one used in plasma
physics, for the variation of the dielectric constant of water with varying frequencies and salinities. Our model is
in excellent agreement with existing empirical fits for frequencies between 1 and 256 GHz. We use this model to
study the propagation of electromagnetic waves in seawater. We explain that large propagation distances would be
possible at MHz frequencies if the conductivity of seawater decreases at small field strengths due to the hydrogen
bonding of water molecules. However, we were unable to experimentally verify any reduction in the conductivity
of seawater.
Index Terms
Permittivity, Underwater radio propagation, Electromagnetic propagation in plasma media, Attenuation
Ram Somaraju and Jochen Trumpf are with the Department of Information Engineering, Research School of Information Sciences and
Engineering, Building 115, The Australian National University, Canberra, ACT 0200, Australia and National ICT Australia Limited, Locked
Bag 8001, Canberra, ACT 2601, Australia.
National ICT Australia Limited is funded by the Australian Government’s Department of Communications, Information Technology and
the Arts and the Australian Research Council through Backing Australias Ability and the ICT Centre of Excellence Program.
1
Frequency, temperature and salinity variation of the
permittivity of Seawater
I. I
NTRODUCTION
E
XISTING systems for underwater communication
largely depend on acoustic technologies. However,
acoustic communication is riddled with problems in-
cluding time-varying multipath propagation and large
latencies. Therefore, Al-Shammaa et. al. claim that radio
communication is a viable alternative [1]. To understand
the properties of radio wave propagation in seawater it is
essential to know the frequency variation of seawater’s
relative permittivity because the rate of attenuation of
plane electromagnetic waves is a function only of the
relative permittivity of the medium.
However, existing models used for the permittivity of
seawater are empirical ones that best fit experimental
data and are not based on a sound physical model.
We propose a model for the permittivity of seawater
that is similar to the one used for ionic plasmas. In
the following, we briefly review the general theory of
polarization of dielectrics including Debye’s theory of
molecular relaxation. This is followed by a description
of models that are currently being used to determine the
permittivity of fresh and sea water. We then explain our
model, and continue with an evaluation of the model
including implications on electromagnetic wave propa-
gation. As will be shown in section VI the predictions of
this model disagree with the results of Al-Shammaa et.
al. [1] but agree with the measurements we calculated.
II. P
OLARIZATION
Any dielectric substance placed in an electric field
undergoes polarization, which involves the appearance
of bound charges on the surface of the dielectric. Polar-
ization is defined as the dipole moment per unit volume
and it may be divided into two categories: induced and
orientation polarization [2], [3].
Debye investigates the contribution of these two forms
of polarizations to the relative permittivity of a dielectric
substance [4]. In section IV we will introduce atomic
polarization leading to a theory more appropriate for
seawater.
A. Debye’s Theory
In the book Polar Molecules [4], Debye explains the
relationship between relative permittivity and the fre-
quency of electromagnetic waves in a dielectric medium.
Debye assumes that the molecules are free and do not
interact with each other [2], [3] and that the polarization
of the dielectric consists of induced and orientation
components.
When a static electric field E is applied to a dielectric,
the induced component of polarization is assumed to
have no inertia and almost instantaneously attains a value
of P
i
= ǫ
0
(ǫ
1)E. However, the orientation polar-
ization rises exponentially to reach a maximum value of
P
o
= ǫ
0
(ǫ
s
1)E ǫ
0
(ǫ
1)E at t = . Therefore,
for a static electric field E, the total polarization reaches
a maximum value of P = ǫ
0
(ǫ
s
1)E. The time-constant
τ of the exponentially increasing orientation polarization
is called the relaxation time. Also, ǫ
s
and ǫ
are the
static and infinite frequency relative permittivities of
the dielectric and ǫ
0
is the permittivity of free space.
It depends on the temperature of the substance and
is independent of the nature of the electric field and
the time of application of the field. Note that other
parameters such as viscosity and pressure influence ǫ
r
.
But, this variation is not studied in the paper. Based on
these assumptions, it can easily be shown (see [2], [3])
that the frequency dependance of the relative permittivity
ǫ
r
may be written as
ǫ
r
(ω) = ǫ
+
ǫ
s
ǫ
1 + jωτ
(1)
Here, ω is the angular frequency of oscillation of the
electric field. It should be noted that the terms in the
above equation are functions of the temperature T of the
substance and therefore it is more appropriate to write
equation (1) as
ǫ
r
(ω, T ) = ǫ
(T ) +
ǫ
s
(T ) ǫ
(T )
1 + jωτ(T)
(2)
1) Dielectric properties of real molecules: Debye’s
model is based on the assumption that there are no
intermolecular interactions and this simple model does
not accurately predict the permittivity of real dielectrics.
Several dielectrics may be better modelled using the
Cole-Cole model [5] which states that the relative per-
mittivity is given by
ǫ
r
(ω, T ) = ǫ
(T ) +
ǫ
s
(T ) ǫ
(T )
1 + (jωτ(T))
1h
(3)
2
where 0 h 1. This is an empirical model and does
not really have a physical basis. There is however an
interpretation of the Cole-Cole model as the result of
a distribution of relaxation times rather than a single
relaxation time. Several fits have been proposed for
the permittivity of both sea and fresh water based on
both Debye and Cole-Cole models as explained in the
following section.
III. E
XISTING MODELS FOR SEA AND FRESH WATER
PERMITTIVITY
An extensive set of experimental measurements [6]–
[9] are available for the permittivity of fresh water. The
dielectric properties of fresh water may be modelled us-
ing equ. 1 for frequencies up to 100 GHz [10]. However,
for higher frequencies a double-Debye model is found to
be more appropriate. The double-Debye model is based
on the assumption that there exists a second polarization
process with a different relaxation time and is given by
the equation
ǫ
r
(ω, T ) = ǫ
(T ) +
ǫ
s
(T ) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
(4)
The widely used equations of Liebe et. al. [10] are based
on such a double-Debye model. Liebe et. al. [10] claim
that their model may be used for frequencies up to 1THz
and may be extended up to 30THz by the inclusion of
two Lorentzian terms. Double-Debye fits for fresh water
are found in several other papers including Stogryn et.
al. [11] and Meissner et. al. [12].
Until recently, comprehensive models based on exten-
sive experimental measurements were not freely avail-
able for seawater. Descriptions used for seawater until
the early 1990s consisted of works of Stogryn [13]
and Klein [14]. However, in the last decade and a
half several single and double Debye type models were
developed by Ellison et. al. [15], Stogryn et. al. [11] and
Meissner et. al. [12]. The double-Debye model used by
Meissner et. al. [12] and Stogryn et. al. [11] is similar
to the fresh water model with the addition of the effect
of conductivity on the dielectric constant and may be
written as
ǫ
r
(ω, T, S) = ǫ
(T, S) +
ǫ
s
(T, S) ǫ
1
(T, S)
1 + jωτ
1
(T, S)
+
ǫ
1
(T, S) ǫ
(T, S)
1 + jωτ
2
(T, S)
+ j
σ(T, S)
ǫ
o
ω
(5)
Here, S is the salinity of seawater in parts per thousand
(ppt). Ellison et. al. [15] use a single Debye model to
fit to experimental data. All these models evaluate the
functional dependance of the terms in equation (5) on the
salinity and temperature by fitting polynomial, rational or
exponential functions to experimental data. For example,
Meissner et. al. [12] use the fit
ǫ
s
= ǫ
s
(T, 0) · exp(b
0
S + b
1
S
2
+ b
2
T S) (6)
for the static relative permittivity of seawater. Here,
ǫ
s
(T, 0) is the static relative permittivity of fresh water
and the constants b
i
are evaluated by fitting the best curve
to experimental data. By the author’s own admission,
there is no physical basis for the model (equation (6))
used. In addition to using the dielectric model of fresh
water Ellison et. al. [15], Stogryn et. al. [11] and
Meissner et. al. [12] respectively use 30, 13 and 12
parameters that are determined from experimental data
to predict the variation of all the terms in equation (5)
with temperature and salinity. In contrast, our model
is not only physically realistic but also uses only two
additional parameters to describe the dielectric behavior
of seawater.
IV. M
ODEL OF SEAWATER PERMITTIVITY
Seawater has several dissolved salts and is therefore
a good conductor. However, increased conduction is
not the only phenomenon that occurs when salts are
dissolved in water. The ions are hydrated to varying
extents (see [16]–[18]).
The hydration number is defined as the number of
water molecules in the immediate vicinity of the ion. It is
based on the dynamical behavior of the water molecules
in solution that move with the ion as a unit [17]. This
should be distinguished from the coordination number
of the ion which is the number of molecules in the
immediate neighborhood of the ion. The coordination
number depends on the distance of the water molecules
from the ion [17]. It is bigger than the hydration number
and includes all the molecules that are hydrogen bonded
to the molecules in the immediate vicinity of the ion.
The model we propose here assumes that the Debye
model of Stogryn et. al. [11] is adequate for fresh
water. However, we develop a physically realistic model
for the variation in the permittivity of seawater with
varying salinities and temperatures. There are three basic
differences between sea and fresh water that need to be
considered in order to develop this model.
The conductivity of water increases with the ad-
dition of ions and the increase in conductivity is
approximately proportional to the number of ions.
The extent of polarization due to the displacement
of bound charges (i.e. induced and orientation po-
larization) in seawater depends on its salinity due
3
to the presence of ions. Therefore, ǫ
s
, ǫ
and τ are
functions of seawater’s salinity.
The static relative permittivity ǫ
s
, of seawater re-
duces because all the water molecules that are
in the vicinity of an ion orient themselves with
respect to the ion. We assume that these molecules
do not contribute to the orientation polarization of
seawater. We further assume that the number of
water molecules that orient themselves about the
dissolved ions is directly proportional to the number
of ions. Hence, we would expect ǫ
s
to decrease
linearly with increasing salinity. This assumption is
in accordance with the model of Ellison et. al. [15]
and furthermore seems reasonable based on the
physical intuition given above.
The effect of the ions on the induced polarizability
is difficult to analyze. Firstly, each ion will have a
different absorption spectrum in the infrared region
and will contribute different amounts to induced
polarization. Further, the ions will affect the mag-
nitude of induced polarization of water molecules.
However, if the concentration of ions is small, these
effects may be ignored.
Also, the time constant τ should not be affected
by the addition of ions. This is because τ is based
on the inertial properties of orientation polarization
and we are assuming that the water molecules that
are oriented about the ions do not contribute to the
orientation polarization. Also, the inertial forces on
the water molecules that are not near the ions should
not be effected significantly by the presence of ions.
In addition to induced and orientation polarization,
there exists a third kind of polarization in seawater.
Non-uniform distribution of free ions in the water
will result in atomic polarization, P
f
. The contri-
bution of P
f
to polarization has to be taken into
account in calculating the relative permittivity.
A. Polarization of seawater
The model we propose here is based on the one used
for gaseous plasmas which is composed of positive and
negative ions, electrons and also neutral atoms [19].
The total polarization of seawater, P may be written
as P = P
b
+ P
f
. Here, P
b
is the polarization due to
the displacement of bound charges in water molecules
(i.e. induced and orientation polarization) and P
f
is due
to the displacement of ions inside water (i.e. atomic
polarization). We can write P
b
= ǫ
0
χE, where χ = ǫ
b
1
and
ǫ
b
(ω, T, S) = ǫ
(T ) +
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
+
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
(7)
This is similar to equation (4) that is used for fresh
water with the small but significant additional term
α(T )S in accordance with the assumption that the
static relative permittivity of seawater decreases linearly
with increasing salinity. The remaining terms in this
equation are assumed to be the same as the one used
by Stogryn et. al. [11] to model fresh water.
1) Evaluation of P
f
: We make the following assump-
tions in deriving a model for the variation of atomic
polarization with frequency.
Seawater is composed of water and several dis-
solved ion types, indexed by i, with mass m
i
and
charge q
i
. m
i
is the total mass of the i
th
type of ion
and all the water molecules in the hydration shell
of this type of ion.
The drift velocity of the water molecules is zero and
the drift velocities of all other ions are measured
with respect to water.
The density of ions is small and so collisions
between ions may be ignored and only collisions
between neutral water molecules and ions are sig-
nificant.
If collisions are ignored, the rate of change of the drift
velocity of the i
th
type of ion v
i
may be written as
N
i
m
i
µ
v
i
t
+ v
i
· v
i
= N
i
q
i
E + N
i
q
i
v × B
+ N
i
m
i
g p
i
(8)
where N
i
is the number of ions of type i per unit
volume, p
i
= N
i
kT
i
is the pressure and N
i
m
i
g gives the
gravitational force. Though not shown explicitly for ease
of notation, both N
i
and m
i
are functions of temperature
and salinity. For wavelengths that are large compared to
atomic dimensions, the pressure gradient and the non-
linear v
i
· v
i
terms may be ignored [19]. Furthermore,
gravitational force is small compared to the force due to
the electric field and can therefore be ignored. Collisions
are incorporated into equation (8) by adding a damping
term that is proportional to v
i
and an effective collision
rate ω
eff
i
and we get,
N
i
m
i
µ
v
i
t
= N
i
q
i
E + N
i
q
i
v
i
× B
N
i
m
i
ω
eff
i
v
i
(9)
If we define the drift displacement r
i
of the i
th
ion by
v
i
=
r
i
t
(10)
4
then we can write
P
f
=
X
i
N
i
q
i
r
i
(11)
Also, equation (8) may be re-written in terms of drift
displacement as
N
i
m
i
2
r
i
t
2
= N
i
q
i
E + N
i
q
i
r
i
t
× B
N
i
m
i
ω
eff
i
r
i
t
(12)
For waves with exponential dependance of the form
exp{j(kr ωt)}, this equation may be written as
(jω)
2
µ
i
r
i
= q
i
(E + jωr
i
× B) (13)
where µ
i
= m
i
{1 + j(ω
eff
i
)}. Substituting equa-
tion (13) into (11), and ignoring the contribution of the
magnetic field, which tends to be small compared to that
of the electric field in non-magnetic materials, we get
P
f
=
X
i
N
i
q
2
i
µ
i
ω
2
E (14)
B. The relative permittivity
Now, the total displacement field D = ǫ
0
E+P
f
+P
b
.
Using equations (7) and (14) we get
ǫ
r
(ω, T, S) = ǫ
(T ) +
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
X
i
N
i
q
2
i
ǫ
0
µ
i
ω
2
(15)
Substituting the value of µ
i
from equation (13) we get
ǫ
r
(ω, T, S) = ǫ
(T ) +
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
X
i
c
i
ǫ
0
ω
2
(1 + jω
eff
i
)
(16)
where c
i
=
N
i
q
2
i
m
i
.
It is however difficult to calculate and measure ω
eff
i
for the individual ions. Therefore, we assume that the
effective collision rate is the same for all the ions and is
equal to ω
eff
. We can then rewrite equation (16) as
ǫ
r
(ω, T, S) = ǫ
(T ) +
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
c(T, S)
ǫ
0
ω
2
(1 + jω
eff
(T, S))
(17)
where c(T, S) =
P
i
c
i
. Both c(T, S) and ω(T, S) are
functions of both Salinity and temperature because the
number N
i
of ions of type i in solution and the mass
m
i
of i
th
type of ion along with its hydration shell are
functions of temperature and salinity.
1) Evaluation of α(T ), c(T, S) and ω
eff
(T, S):
α(T ) · S is equal to the fraction of water molecules
that are oriented towards ions in solution. Let the con-
centration of the i
th
type ion in water be ρ
i
parts per
thousand. Now, some fraction, say β
i
, of these ions will
be dissociated and these are the only ions that contribute
towards the reduction of the static permittivity of water.
The number of such ions N
i
in 1 Kg of solution is given
by
N
i
=
β
i
ρ
i
ν
i
moles (18)
Here, ν
i
is the atomic mass of the i
th
type ion. If the
coordination number of this ion is k
i
, then the total
number of water molecules that are oriented about the
i
th
type of ion is N
i
k
i
. Therefore the fraction of water
molecules, α(T ) · S that are oriented about all the ions
in solution is given by
α(T ) · S =
X
i
N
i
k
i
1/0.018
=
X
i
0.018β
i
ρ
i
k
i
ν
i
(19)
Note that we used the fact that the molecular mass of
water is 18 g/mole in deriving the last equation.
To evaluate c(T, S), we need to know the hydration
number and not the coordination number of an ion. Let
the hydration number of the i
th
type of ion be h
i
and
A
v
= 6.023e23 be Avogadro’s number. Then the mass
of the i
th
type of ion and all the water molecules in its
hydration shell is m
i
= (ν
i
+ 0.018h
i
)/A
v
. Using this
and equation (18) we get
c(S, T ) =
X
i
c
i
=
X
i
N
i
q
2
i
m
i
=
X
i
A
2
v
β
i
ρ
i
q
2
i
ν
i
(ν
i
+ 0.018h
i
)
(20)
Finally, because the ionic conductivity of water
σ(T, S) c(T, S)
eff
(T, S) we can calculate the
ratio of ω
eff
(T, S) to c(T, S) using the well established
values of ionic conductivity of water.
However, the hydration number, coordination number
and percentage of dissociation are difficult to ascertain
accurately from experimentation. For instance, the ex-
perimental values for the hydration number of Na+ ions
varies from 4-8 [16]. Therefore it was decided that α(T )
and c(T, S) should be determined using experimentally
5
Ion Type Density (ppt)
Cl
19.135
Na
+
10.76
SO
2
4
2.712
Mg
2+
1.294
TABLE I
DENSITY AND MOLALITIES OF COMPONENTS OF SEAWATER WITH
SALINITY S = 35 PPT
measured values of the permittivity and then the resulting
value for the hydration and coordination numbers be
compared to existing predictions.
V. RESULTS
It is difficult to experimentally measure the permittiv-
ity of seawater for a varying range of frequencies because
it is extremely lossy. Therefore, it was decided that
the validity of the model be ascertained by generating
pseudo-data from the empirical fits to experimental data
mentioned in section III. Figure 1 compares the real and
imaginary parts of permittivity of seawater of our model
with the fits of Stogryn et. al. [11], Meissner et. al. [12],
Ellison et. al. [15] and Wentz et. al. [20] for various
values of salinity and temperature.
It is clear from these figures that our model is in ex-
cellent agreement with these fits for frequencies between
1 and 256 GHz. The maximum deviation of our model
from the fits of Stogryn et. al. [11] and Meissner et.
al. [12], is 7.6% and 6.9% respectively in the frequency
range 1-256GHz. The fits of Stogryn et. al. [11] and
Meissner et. al. [12] differ by as much as 11.5% in the
same frequency range. Further verification of the model
comes from the fact that the values of α(T ) and c(T, S)
obtained by fitting best curves to pseudo-data are the
right order of magnitude. We get,
α(T ) = 0.00314 ppt
1
(21)
c(T, S) 1 × 10
12
· S C
2
/Kg (22)
It should be noted that no temperature dependance was
detected for both c(T, S) and α(T). To check that
these values are as expected, we need to know the
composition of seawater. Table I shows the densities
of four components of seawater that have the highest
concentrations [21].
We note that the only significant elements are Sodium
and Chlorine. Because we are only interested in the order
of magnitude of the hydration numbers we assume that
seawater only consists of these two components and from
the calculated value of α we get
β(k
Cl
+ k
Na
) = 11.3 (23)
Here, β is the degree of dissociation of NaCl and
k
Cl
and k
Na
are the coordination numbers of Cl and
Na respectively. These numbers are as expected [17].
Similarly it can be shown that c(T, S) is of the right
order of magnitude.
However, more experimental results over a wider
range of frequencies are required to completely validate
the model and get an accurate value for c(T, S). Such
experiments are particularly important for frequencies
greater than a few hundred gigahertz. High frequency
measurements are needed to accurately compute c(T, S)
because ω
eff
is of the order of 1THz and we require
measurements in this frequency range to accurately de-
termine c(T, S).
Also, at smaller frequencies seawater behaves as a
conductor and this can easily be seen by taking the low
frequency limit of equation (17). Then the permittivity
reduces to
ǫ
r
(T, S) =
c(T, S)
jǫ
0
ωω
eff
(T )
=
σ(T, S)
jǫ
0
ω
(24)
This is what one would expect for a good conductor.
In fact the low frequency limit of the empirical equa-
tion (5) is identical to the one used in this model. It is
therefore safe to assume that the model is valid for low
frequencies.
VI. I
MPLICATIONS FOR THE PROPAGATION OF
ELECTROMAGNETIC WAVES
Our model for the permittivity of seawater assumes
that it is independent of the applied electric field strength
and is only a function of the temperature and salinity of
seawater and the frequency of the electromagnetic wave.
Therefore the rate of attenuation of an electromagnetic
wave in seawater, which depends only on its permittivity,
is not a function of the distance from a transmitting an-
tenna. This is in accordance with the classical literature:
the articles [22]–[26] indicate that no such change in
attenuation occurs as the distance from the transmitting
antenna increases even if the antennas are insulated [26],
[27]. Therefore, we expect the range in seawater to be
comparable to the skin depth, which is of the order of 0.3
m at a frequency 1 MHz if we assume the conductivity
of seawater to be 4 s/m [28].
However, Al-Shammaa et. al. claim that radio com-
munication over a distance of 100m is possible at MHz
frequencies in seawater [1]. Al-Shammaa et. al. [1]
further claim that as the distance from the transmitting
antenna increases, the rate of attenuation of electromag-
netic waves reduces greatly. In fact, figures 13 and 14
in [1] indicate that there is minimal attenuation once the
distance from the transmitting antenna increases beyond
6
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 15°C, S = 35 ppt)
(a)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℑ{ǫ}
ℑ{ǫ} vs log
2
(f) (T = 15°C, S = 35 ppt)
(b)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 25°C, S = 35 ppt)
(c)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
90
100
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℑ{ǫ}
ℑ{ǫ} vs log
2
(f) (T = 25°C, S = 35 ppt)
(d)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 15°C, S = 25 ppt)
(e)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
ℑ{ǫ}
log
2
f (GHz)
ℑ{ǫ} vs log
2
(f) (T = 15°C, S = 25 ppt)
(f)
7
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 25°C, S = 25 ppt)
(g)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
ℑ{ǫ}
log
2
f (GHz)
ℑ{ǫ} vs log
2
(f) (T = 25°C, S = 25 ppt)
(h)
Fig. 1. Real and Imaginary parts of permittivity as a function of frequency
3-4 meters. Al-Shammaa et. al. [1] explain this reduction
in attenuation with increased distance by claiming that
the conduction current losses may be ignored once the
distance from the transmitting antenna becomes large.
We believe that such a change in attenuation of
electromagnetic waves in seawater could only occur
if seawater behaves differently at small electric field
strengths and hence at large propagation distances from
a transmitting antenna. If the conductivity of seawater
decreases at small electric field strengths, then, as the
distance from the transmitting antenna increases, the
amplitude of the transmitted electromagnetic wave would
reduce and therefore we would see a reduced rate of
attenuation. One possible explanation as to why seawater
might be a poor conductor at small field strengths is as
follows.
A positive and a negative ion may be bonded to each
other through water molecules that are hydrogen bonded
to each other (See figure 2). These bonds, if they do
exist, will be extremely weak and easy to break apart.
Therefore at high electric field strengths, with forces
acting in opposite directions on positive and negative
ions, these bonds might be broken apart and we would
get free positive and negative ions. However, for small
electric field strengths, there would be no free ions to
conduct and therefore the conductivity might decrease
drastically. This would particularly be the case at higher
frequencies because with alternating fields, the time
available to break these weak bonds would be shorter.
It is well known that the conductivity of seawater is
not constant above a certain frequency. Gabillard et.
al. [25] show that if the conductivity was constant for
+
_
Two water molecules hydrogen
bonded to each other
Hydrogen
Bond
Ions and their
hydration shells
Fig. 2. Ions bonded to each other through hydrogen bonded water
molecules
high frequencies then yellow light would only be visible
up to 29 cm underwater. But we can see a yellow lamp
much further than 29 cm in seawater.
It is hence conceivable that the rate of attenuation
decreases with decreasing field strength. It could well be
that such a reduction has not been detected previously
because one could not measure extremely small electric
field strengths until recently. However, with better mea-
suring equipment available now one might be able to
detect small field strengths.
We decided to experimentally verify if the conductiv-
ity of seawater changes by using the setup shown in
figure 3. We measured the amplitude of V 1 and V 2
using a lock-in amplifier to calculate the impedance
of salt water from the ratio V 1 and V 2. We used a
8
R1
B
A
V2
W
V1
Lock−in Amplifier
Fig. 3. Experimental setup to measure the conductivity of saltwater
Princeton Applied Research EG & G 5210 amplifier at a
frequency of 50 KHz and a Stanford Research Systems
SR844 amplifier at a frequency of 1 MHz. It was decided
that it was unnecessary to use a Wheatstone bridge
circuit because it is not essential that the impedance
be measured accurately. We were only interested in
measuring large changes in impedance as only this would
explain the large differences in the rates of attenuation.
As a control experiment, at 1 MHz the water cell was
also replaced by a 820 carbon resistor.
The electric field strength applied to the water cell
was reduced by increasing the resistance of the variable
resistor R1. All the components were shielded inside
grounded metal boxes to reduce the effects of external
noise. We detected no change in the conductivity of salt
water at 50 KHz. The smallest voltage applied to the
water cell was 600nV at 50 KHz. The cell is 5 cm
long and if we assume a uniform electric field then the
smallest field applied was 12µV /m.
However, at 1 MHz we initially detected a change in
the ratio of the voltages V 1 and V 2. Exactly the same
change in ratio was also present in the control experi-
ment, where we replaced the water cell with a resistor.
We concluded that this effect was due to capacitative and
inductive coupling between poorly shielded wires. After
shielding the wires properly we did not detect a change
in the ratios of the two voltages and hence no change in
the conductivity of seawater down to a voltage of 30 µV
at 1 MHz. If we assume a uniform electric field then the
smallest field strength applied was 1.5 mV /m.
We can conclude from this experiment that the con-
ductivity of seawater does not change for electric field
strengths as small as 12µV /m at a frequency of 50 KHz
or 1.5mV/m at a frequency of 1 MHz and hence the rate
of attenuation does not change for these field strengths.
We currently have no plausible explanation for the large
propagation range observed by Al-Shammaa et. al. [1].
VII. C
ONCLUSION
In this paper we derived a physically realistic model
for the frequency variation of the relative permittivity
of seawater for varying salinities and temperatures. The
model derived is in excellent agreement with existing
empirical fits to experimental data. Also, the model uses
only two parameters that need to be determined from
experimental data as opposed to more than 10 parameters
used by most empirical fits. Furthermore, the remaining
parameters in our model have a physical interpretation
and could hence theoretically be determined by indepen-
dent experiments. Moreover, because our model has a
physical foundation, we are confident that it is valid over
a wider parameter (frequency, temperature and salinity)
range and can be used for extrapolation in regions where
no experimental data is available.
This model however does not predict large propagation
distances for electromagnetic waves in seawater in the
frequency range of a few Megahertz as measured by
Al-Shammaa et. al. [1]. We believe that the only pos-
sible explanation for these large propagation distances is
that the conductivity of seawater changes at small field
strengths due to hydrogen bonding in water. However, we
measured no change in conductivity for electric fields as
small as 12µV /m and 1.5mV/m at frequencies of 50
KHz and 1 MHz respectively.
R
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Jochen Trumpf (M’04) received the Dipl.-
Math. and Dr. rer. nat. degrees in mathematics
from the University of W
¨
urzburg, Germany, in
1997 and 2002, respectively. He is working as
a Research Fellow for the Department of Infor-
mation Engineering in the Research School of
Information Sciences and Engineering at The
Australian National University, Canberra, Aus-
tralia, and is currently seconded to National
ICT Australia Ltd. His research interests include observer theory and
design, linear systems theory and optimisation problems in digital
communication.
Ram Somaraju received B.Sc. and B.E. de-
grees in Physics and Computer Systems en-
gineering from the University of Auckland,
New Zealand, in 2001 and 2003, respectively.
He is currently pursuing a Ph.D. degree in
the Department of Information Engineering in
the Research School of Information Sciences
and Engineering at The Australian National
University, Canberra, Australia. His research
interests include underwater radio communications, MIMO commu-
nication and uncertainty principles for communication channels.
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