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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))

1−h

(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

/ω)}. 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

/ω)

(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)/ω)

(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)/ω

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 Ω 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.