Refracted Acoustic Communications in Wireless Underwater Sensor

Networks with Mobility

S. Blouin

, M. Barbeau

and E. Kranakis

Underwater Sensing, DRDC Atlantic Research Centre, Nova Scotia, Canada

School of Computer Science, Carleton University, Ottawa, Ontario, Canada

Keywords:

Routing, Refraction, Sensor Networks, Mobile Nodes.

Abstract:

Routing is a fundamental function of any wireless network. For battery-powered underwater sensors us-

ing acoustic waves, routing is even more challenging due to an ever changing and communication-opaque

ocean. Herein, we propose a shallow-water routing scheme that adapts to a unique physical phenomena of the

medium. In particular, the proposed routing exploits downward and upward refractions for underwater net-

works comprising mobile nodes. Solutions for node-to-node links are developed before extending the concept

to network routing. In particular, necessary and sufﬁcient conditions guaranteeing the existence of a single-hop

link through acoustic refraction are derived. Simulations convey results followed by concluding remarks.

1 INTRODUCTION

Routing is a fundamental function of any wireless net-

work because rarely all nodes are within the trans-

mission range of one another. This is especially true

for battery-powered underwater network nodes with

a barrier-type topology meant to monitor coastlines,

track migrating marine mammals, or protect habitats.

The main challenge of routing is to identify the chain

of nodes used to transfer data from a source to a des-

tination in the most efﬁcient manner.

Underwater acoustic waves propagate further than

electromagnetic or optical waves (Heard et al., 2009).

Thus, the primary medium for wireless underwa-

ter communications is acoustic waves. Unfortu-

nately, underwater acoustic communications are un-

predictable due to frequency-dependent attenuation,

time-varying multipath effects, large Doppler/delay

spreads, and limited bandwidth (Stojanovic and

Preisig, 2009). Underwater acoustics are such that

signiﬁcant losses occur due to absorption and spread-

ing (Urick, 1983). In certain underwater environ-

ments, the acoustic channel suffers from rapid vari-

ations (Blouin and Inglis, 2013). Certain links may

be unidirectional. Besides, acoustic paths bend due

to varying sound speed in the water column. Those

conditions make the selection of the most appropriate

underwater route a non-trivial task. Mobility through

the underwatermedium whose properties change both

in time and space exacerbate the problems.

Most underwater routing schemes are derived

from terrestrial radio communications. They are

rarely optimized for the underwater communication-

opaque environment and almost never tailored to

the limitations of the medium (Otnes et al., 2012).

Among such routing protocols, one ﬁnds both

proactive (SEAWEB (Rice and Ong, 2010), DSDV

(Perkins and Bhagwat, 1994)) and reactive (DSR

(Johnson and Maltz, 1996)) schemes. Proactive pro-

tocols actively maintain a routing table by period-

ically exchanging topology information. Reactive

protocols search for a topological route only when

there exists data to transmit. For battery-powered

sensors, reactive protocols are usually preferred from

an energy-conservation standpoint, but they do result

in additional delays. Many approaches perform ge-

ographic routing(Karp and Kung, 2000) where data

packets are progressively brought closer to the desti-

nation at every step. However, with such an approach

there is no guarantee of ﬁnding a path, thus possibly

leading to a communication void.

Geographic routing further breaks down in two

categories, i.e., location-awareor location-free,based

on the knowledge of the location of the destination

node. Most geographic routing protocols in fact re-

quire that each node knows its own location, which

in turn necessitates a sensor position estimation tech-

nique (Li et al., 2010) and its associated hardware and

software. Even though multihop transmissions may

reduce power consumption (Porto and Stojanovic,

113

Blouin S., Barbeau M. and Kranakis E..

Refracted Acoustic Communications in Wireless Underwater Sensor Networks with Mobility.

DOI: 10.5220/0005199501130121

In Proceedings of the 4th International Conference on Sensor Networks (SENSORNETS-2015), pages 113-121

ISBN: 978-989-758-086-4

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

2007), there exists an optimal hop length between

routers for a given set of transmission parameters

(Zorzi et al., 2008).

We propose a routing scheme exploiting down-

ward and upward refracting phenomena for under-

water networks with mobility. The contributions re-

late to theoretical results deﬁning the existence of re-

fracted paths between two underwater nodes and con-

ditions leading to interference with the ocean surface

and bottom. These results constitute the initial steps

in addressing a challenging routing problem that min-

imizes power losses for improving underwater com-

munications.

Section 2 reviews background material. Section 3

presents an approach to exploit refracted paths to es-

tablish point-to-point links. Section 4 generalizes the

approach to sensor networks with mobile nodes. Sim-

ulation results and concluding remarks are provided

in Sections 5 and 6. Additional material and proofs

are found in appendices.

2 BACKGROUND

2.1 Problem Deﬁnition

A generic underwater sensor network conﬁguration,

made of both static and mobile nodes, is illustrated in

Figure 1. The left-most static node 1 intends to send

a message to the right-most mobile node 6. In addi-

tion to variable separation distances between sensors,

energy losses due to underwater sound propagation

should be considered to efﬁciently solve the routing

problem.

Figure 1: Underwater sensor network.

Shallow-water acoustics are such that routing is

extremely challenging due to the multi-path effects,

often accompanied by frequency and delay spread

(van Walree, 2013). The various acoustic paths link-

ing source 1 to sink 6 can approximately be grouped

as occurring through either refraction (curved paths

due to the varying speed of sound through the water

column) or reﬂection (acoustic waves bouncing off

the bottom or the surface). Here, we focus on re-

fracted acoustic paths that do not encounter the sur-

face or bottom of the underwater environment. Such

paths are called direct paths.

Overspread occurs when both frequency and de-

lay spreads disable the demodulation of the received

signal (Otnes et al., 2012). Reﬂected paths tend to

lead to larger overspread than refracted paths, espe-

cially when multiple reﬂections occur with a moving

sea surface. Moreover, every reﬂection transfers en-

ergy to the boundary it encounters, resulting in addi-

tional attenuation. Consequently, usage of refracted

paths should be preferred as they may provide sav-

ings in terms of power consumption, which is critical

for underwater battery-powered wireless sensors.

The prior arguments highlight the importance of

exploiting refracted paths as much as possible in the

underwater environment. Therefore, the main ques-

tion of interest is:

How does one exploit refracted paths to route

a message from node 1 to node 6 through

an underwater sensor network made of both

static and mobile nodes?

A unique feature of the aforementioned problem

pertains to the fact that the routing strategy needs

to adapt to the ever-changing network topology in-

duced by the presence of mobile nodes and also to the

varying sound speed proﬁle resulting from a dynamic

ocean.

The sound speed proﬁle describes the variation

of the speed of sound, c(z), as a function of depth,

z. Such proﬁle originates from variations in water

temperature, pressure and conductivity (thus salin-

ity) with depth. The sound speed value c(z) usually

ranges between 1400m/s and 1600m/s (Erbe, 2011),

but changes over both time and space. We deﬁne the

slope of the sound speed proﬁle as

g =

d c(z)

, (1)

with c(z) in (m/s), z in (m), and g in (1/s or second

−1

2.2 Assumptions

We assume the presence of the following features:

• First-order, i.e., linear, sound speed proﬁles.

• Flat bottom and surface oceanic boundaries.

• Sensory node with an acoustic transducer having a

relatively narrow beamwidth β and a controllable

directionality −π/2 < φ < π/2 with respect to the

horizon.

• Nodes equipped with pressure sensors providing

a depth measurement.

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• Global knowledge of relative distances separating

nodes.

Even though not always present in practice, the

ﬁrst-order sound speed proﬁle assumption is a com-

mon initial hypothesis in the shallow-water acoustic

literature (Katsnelson et al., 2012). Its main impact

is that the resulting sound speed proﬁles reduce to the

three categories shown in Figure 2: isovelocity (null

slope or g = 0), downward refracting (negative slope

or g < 0), and upward refracting (positive slope or

g > 0), where c

(m/s) denotes the sound speed at the

sea surface and g the sound speed proﬁle slope de-

ﬁned in Equation 1. The narrow beamwidth assump-

tion is similar to the focus-beam concept presented

in (Nicolaou et al., 2007). Depth measurement ca-

pability is assumed in depth-based routing schemes

(Hasannezhad et al., 2014). Awareness of relative dis-

tances is often assumed in location-aware approaches.

Figure 2: First-order sound speed proﬁles.

Given the ﬁrst-order sound speed proﬁle assump-

tion, the sound speed has a slope belonging to one of

these categories:

d c(z)







= 0, isovelocity,

< 0, downward refracting, or

> 0, upward refracting.

(2)

Given a sea surface sound speed c

(at depth z =

0), the linear evolution of sound speed with respect to

depth is expressed as

c(z) = c

+ gz, (3)

with the slope g as deﬁned in Equation 1. We denote

node i’s depth as z

and associate a sound speed c

deﬁned as

= c(z

) = c

+ gz

, (4)

such that c

> 0 for all i’s.

3 POINT-TO-POINT LINK

We derive conditions indicating the existence of an

acoustic refracted path between any two nodes. In

Section 4, this point-to-point link solution is extended

by searching all potential paths through the network

and selecting the path leading to the lowest transmis-

sion loss.

3.1 Straight and Curved Acoustic Rays

We leverage notions from the Acoustic Ray theory

and thus assumes that appropriate communication fre-

quencies apply (Urick, 1983). An isovelocity sound

speed proﬁle leads to straight acoustic rays whereas

non-isovelocity cases (upward and downward refract-

ing) generate curved acoustic rays.

We focus on curved acoustic rays. More speciﬁ-

cally, the up-coming solution applies to the downward

refracting situation, as it is often encountered in shal-

low waters (Katsnelson et al., 2012).

An important result originating from Snell’s Law

stipulates that in the presence of a linearly varying

sound speed proﬁle of constant slope g, any acoustic

ray departing from a source node describes an arc of

a circle (Urick, 1983). Given an underwater source

node i, the center of all circles resulting from dif-

ferent departure angles resides on a line of centers

whose vertical distance from the source node location

is computed as

= −

. (5)

According to Equation (5), an isovelocitysound speed

proﬁle (g = 0) generates arcs of an inﬁnite radius,

thus straight rays, whereas downward (respectively,

upward) refraction, i.e., g < 0 (resp., g > 0) locates

the line of centers under (resp., above) the sea surface.

3.2 Potential Refracting Scenarios

Figure 3 shows the concepts of the line of centers as

well as a generic case with two underwater nodes, i

and j. Node i is the source and node j the sink. Note

that the water depth is represented by D. The sep-

aration distance between nodes i and j is d

. Such

a conﬁguration leads to four potential and mutually-

exclusive cases when an acoustic ray departs from

node i:

• Case 1: The acoustic ray hits the sea surface be-

fore reaching node j.

• Case 2: The acoutic ray overshoots node j.

• Case 3: The acoutic ray directly hits node j.

• Case 4: The acoustic ray undershoots node j, or

hits the sea bottom before reaching node j.

RefractedAcousticCommunicationsinWirelessUnderwaterSensorNetworkswithMobility

115

Figure 3: Potential downward-refracted acoustic paths.

3.3 Geometric Solution for Cases 1-to-4

Figure 4 shows (in blue) the two rays acting as the

outer envelope for all acoustic rays departing from

node i. Given a beamwidth β and a direction φ (with

respect to the horizon), the envelope determines the

angles α

and α

where

= φ, (6)

= φ+ β, (7)

such that −π/2 < α

,α

< π/2 which means that

acoustic rays are prevented from traveling along the

vertical axis and solely interacting with boundaries.

α =

= +β

Figure 4: Acoustic rays departing from node i.

The rationale here is that if both value α

,α

fall

in the same case, 1, 2, or 4, of Figure 3, then no in-

termediate directionality angle α

contained between

and α

results in an acoustic ray departing from

node i and reaching node j by a refracted path. Simi-

larly, if α

coincides with case 4 and α

matches case

2, then there exists an intermediate directionality an-

gle α

≤ α

for which an acoustic ray departing

from node i reaches node j by a refracted path.

Assuming an acoustic ray departing from node i

at an angle α

where α

≤ α

, it is possible to

compute the exact location of the corresponding cir-

cle center with simple geometry as illustrated in Fig-

ure 5. Given that the acoustic ray departing from node

i forms a right angle with the line joining node i and

its circle center, we can derive R

, the radius of curva-

ture, and d

, the horizontal distance separating node i

from the center of the circle describing the curved

acoustic ray as follows

cosα

, (8)

= R

tanα

, (9)

where R

originates from Equation (5). Note that the

absolute value of R

in Equation (8) is taken so that

the radius of curvature remains positive and to recon-

cile equations in references (Urick, 1983) and (Lur-

ton, 2010) for downward and upward refracting sce-

narios.

line of centers

acoustic

ray

Figure 5: Center of curvature of an acoustic ray.

We next solve the geometry problem and identify

conditions leading to Cases 1 through 4.

3.3.1 Case 1

Using node i’s location as the center of an (x,z) co-

ordinate system, it can be seen from Figure 5 that if

the top of the circle centered at (d

) and of radius

is above or at sea level then the acoustic ray hits

the sea surface, otherwise it does not. Consequently,

if R

+ z

, the depth of the circle center, exceeds the

length of the curvature radius R

, i.e.,

+ R

> R

, (10)

then the refracted acoustic ray avoids the sea surface.

Using the previous argument and the development

found in Appendix, we obtain the following result.

Proposition 1

Assume an underwater node

at depth

in the pres-

ence of a linear sound speed variation as deﬁned in

Equation (4). A refracted acoustic ray departing from

node

at an angle

|α

| < π/2

avoids the sea surface if

and only if

< α

max

, (11)

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where

max

= arccos



−

sign

(g)





. (12)

The angle α

max

of Proposition 1 represents the

largest departing angle that an acoustic ray can have

to avoid the sea surface.

Example 1. The network conﬁguration of Figure 1 is

speciﬁed through a node depth array Z = {z

,... ,z

}

and a node-to-node horizontal separation distance ar-

ray d = {d

,... ,d

} with values

Z = {150, 135,147,125,150,143} − λ,

d = {400, 1000,1600,1900,2600+ δ},

where λ is a depth offset for all sensors, while δ is

a range offset for the mobile node only. Environ-

mental conditions are D = 200 m, c

= 1500 m/s,

and g = −0.25 1/s. All acoustic transducers have a

beamwidth of β = 10

and their directionality angle

φ takes different values through the following three

scenarios.

Scenario 1:

With λ = 0, δ = 0, φ = −β/2, Equations

(6) and (7) give α

= −α

= −5

and α

max

of Equa-

tion (12) is (in degrees)

Node i

1 2 3 4 5 6

max

12.8 12.2 12.7 11.7 12.8 12.5

As α

max

> max(α

,α

) = 5

for all i’s, Proposition 1

implies that none of the acoustic rays departing from

any node will reach the sea surface.

Scenario 2:

By making the environment shallower

with λ = 121, Equation (12) now gives

Node i

1 2 3 4 5 6

max

(degrees) 5.6 3.9 5.3 2.1 5.6 4.9

so that some acoustic rays departing from nodes 2, 4

and 6 violate the condition of Proposition 1.

Scenario 3: For this scenario, the λ = 121 value still

applies. Slightly changing the directionality angle to

φ = −β/2− 3

on all nodes alters α

and α

such that

= −8

and α

= 2

. Given that α

max

values of

Equation (12) remain unchanged from the previous

scenario, max(α

,α

) < α

max

and no acoustic rays

departing from any node hits the sea surface as per

Proposition 1.

Example 1 is instructive in that (a) controlling the

directionality angle φ can help ensure that no acoustic

ray disperses its energy by interacting with the sea

surface, and (b) there exists a technique for altering φ

as captured in the next result.

Corollary 2

Assume an underwater node

at depth

in the pres-

ence of a linear sound speed variation as deﬁned in

Equation (4). Given a ﬁxed beamwidth

, Equation

(11) of Proposition 1 holds if the directionality angle

satisﬁes

φ < α

max

− β (13)

The proof of Corollary 2 follows from the fact

that Equation (13) is equivalent to φ+ β = α

< α

max

based on Equation (7).

3.3.2 Cases 2, 3, and 4

Assuming that Equation (11) holds, Cases 2, 3, and 4

can be solved using a similar development. Denoting

node j’s depth as z

, Figure 5 displays the geome-

try involved in identifying the distance L separating

node j’s location from the center of the circle associ-

ated with an acoustic ray leaving node i at an angle

. Note that Figure 5 represents the particular case

where d

> d

Expressing the distance L mathematically gives

L =

− d

)

+ (z

+ R

− z

)

. (14)

From Figure 3 and Equation (14), the remaining cases

correspond to

L < R

, ray overshoots node j (Case 2),

L = R

, ray reaches node j (Case 3),

L > R

, ray undershoots node j (Case 4).

(15)

For Case 3, the following result holds.

Proposition 3

Assume two underwater nodes

i, j

at depths

the presence of a linear sound speed variation as de-

ﬁned in Equation (4). A refracted acoustic ray depart-

ing from node

at an angle

|α

| < π/2

and reaching

node

exists if and only if

L = R

, (16)

which is equivalent to an acoustic ray departing angle

= arctan





+ (∆z

)

+ 2∆z





. (17)

The proof for Proposition 3 can be found in Ap-

pendix. Equation (17) is a closed-form solution indi-

cating which angle α

results in a refracted path be-

tween node i and node j. Using the geometry shown

in Figure 5, it is possible to compute the arc length S

of the refracted acoustic path linking node i to node j

through

= R

θ, (18)

RefractedAcousticCommunicationsinWirelessUnderwaterSensorNetworkswithMobility

117

where R

is derived as per Equation (8) and the arc

angle span θ (rad) shown in Figure 5 is given by

if d

> d







θ = α

+ 0.5π− η

η = arctan



+ R

− z

− d



if d

< d







θ = α

− 0.5π+ η

η = arctan



+ R

− z

− d



(19)

with α

from Equation (17) and η shown in Figure 5.

4 NETWORK ROUTING

A refracted acoustic route between a source node 1

and a destination node 6 exists if there is a sequence

of refracted paths linking node 1 to node 6.

Underwater transmission loss, TL, is a measure

of the rate at which the sound energy is lost (Urick,

1983) and a common computation is

TL = N log

r− a · r, (20)

where N is the coefﬁcient of geometric spreading, r

the range or straight-line distance (in meters) from the

acoustic source to the destination, and a the absorp-

tion coefﬁcient. Given the relatively close proximity

between network nodes, it can be assumed that spher-

ical spreading dominates and is captured by inject-

ing N = 20 in Equation (20). Frequency-dependent

values for a, the absorption coefﬁcient, are shown in

Appendix for atmospheric pressure and a temperature

of 5

Celsius (Kinsler et al., 1982) (Kinsler et al.,

2000). For commercial acoustic modems operating

in the 10-30 kHz frequency band (Hasannezhad et al.,

2014), the absorption coefﬁcient a varies linearly in

the 0.0008 − 0.0076 db/m range.

We modify Equation (20) to compute transmission

loss along a refracted path from node i to j, with arc

length given by Equation (18), as follows

∗

i→ j

= N log

− a· S

. (21)

Based on Equation (18) for S

, Equation (8) for R

Equation (5) for R

, Equation (19) for θ, and Equation

(4) for c

, TL

∗

i→ j

is a nonlinear function h(·) with the

following input variables

∗

i→ j

= h(a,c

,g,N, z

,α

) (22)

with α

from Equation (17) and where all other vari-

ables are measured quantities or constants.

5 SIMULATION RESULTS

We apply the previous results of Section 3 to the un-

derwater network conﬁguration of Figure 1.

Example 2. Similar to Example 1, environmental

conditions are D = 200 (m), c

= 1500 (m/s), and

g = −0.25 (1/s). In Equation (21), N = 10 given

the shallow water context and a = 0.0043 as acous-

tic communications are assumed to occur around 25

kHz. The network conﬁguration is speciﬁed by a node

depth array Z = {z

,... ,z

} and a separation distance

array d = {d

,... ,d

} deﬁned as

Z = {150, 135,147,125,150,143} − λ,

d = {400, 1000,1600,1900,2600+ δ},

where λ = 0 and δ = 0. From the distance array given

above, any node-to-node distance d

can be derived

as shown in Appendix. All sensors have a beamwidth

of β = 10

and directionality angle φ = −0.5β. Table

1 lists all sixteen different conﬁgurations of forward

routes departing from node 1 and reaching node 6.

Table 1: Possible forward routes between node 1 and 6.

# Route

1 1 → 6

2 1 → 2 → 6

3 1 → 3 → 6

4 1 → 4 → 6

5 1 → 5 → 6

6 1 → 2 → 3 → 6

7 1 → 2 → 4 → 6

8 1 → 2 → 5 → 6

9 1 → 3 → 4 → 6

10 1 → 3 → 5 → 6

11 1 → 4 → 5 → 6

12 1 → 2 → 3 → 4 → 6

13 1 → 2 → 3 → 5 → 6

14 1 → 2 → 4 → 5 → 6

15 1 → 3 → 4 → 5 → 6

16 1 → 2 → 3 → 4 → 5 → 6

Figure 6 shows the α

values of Proposition 3

insuring a direct refracted path between two nodes

whose numbers are given on the horizontal axis.

1−2 1−3 1−4 1−5 1−6 2−3 2−4 2−5 2−6 3−4 3−5 3−6 4−5 4−6 5−6

−6

−4

−2

Single−hop link (Departing node # − Receiving node #)

Acoustic ray departing angle (degree)

Figure 6: Angles ensuring a refracted path between nodes.

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Given that acoustic rays are restricted to depart at

angles between +5

and −5

(indicated by red dashed

lines in Figure 6), only links 1 → 2, 2 → 3, 3 → 5,

4 → 5, 4 → 6, and 5 → 6 satisfy this constraint and are

indicated by hollow circular markers. Consequently,

only route # 13 of Table 1 is actually feasible given

the present constraints.

Figure 7 shows the summation of individual trans-

mission loss along the routes of Table 1 using Equa-

tion (21) for each individual single-hop link. Route #

13, the only feasible one, has an overall transmission

loss of 204 dB, which is signiﬁcant.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

100

150

200

250

Route number

Total transmission loss (dB)

Figure 7: Overall transmission loss over all routes.

From Figure 6, it can be seen that augmenting the

beamwidth β from 10

to 12

would incorporate links

1 → 3, 3 → 4 and 3 → 6 which in turn, enables route

# 3, i.e., 1 → 3 → 6, whose overall transmission loss

is 112 dB, a gain of 92 dB due to a 2

increase in

beamwidth.

6 CONCLUSION

This publication proposes a routing scheme exploit-

ing downward and upward refracting phenomena for

underwater networks with mobile nodes.

The proposed technique presents an approach us-

ing refracted paths to establish node-to-nodelinks and

extends those results to a routing problem in a sen-

sor network with mobile nodes. Both theoretical and

practical results are derived. In particular, necessary

and sufﬁcient conditions guaranteeing the existence

of a single-hop link through an acoustic refracted path

are derived.

Examples and simulations illustrate how impor-

tant are the network conﬁguration and the available

steering capability of the acoustic energy.

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APPENDIX

Proof of Proposition 1

+ R

> R

⇒ α

< α

max

: Using Equations (4) and

(5), the left-hand side of Equation (10) becomes

+ R

= z

−

= z

−

+ gz

= −

Given that c

> 0 for all i’s, that −π/2 < α = α

< π/2

or cosα > 0, and Equation (8), the right-hand side of

Equation (10) expands as

cosα

|g|

cosα

+ gz

|g|

cosα



|g|

+ z

sign(g)



cosα

|g|cosα

sign(g)

cosα

so that Equation (10) can be re-written as

−

|g|cosα

sign(g)

cosα

which after manipulations becomes

−

|g|cosα

sign(g)

cosα

−



sign(g)cosα



sign(g)

cosα

−

g sign(g)

[sign(g)cosα + 1] > z

sign(g),

sign(g)cosα + 1 <

sign

(g)

−c

cosα <

sign(g)



−c

− 1



= −

sign(g)





+ R

> R

⇐ α

< α

max

: Through contradiction,

we show that z

+ R

≤ R

⇒ α

≥ α

max

. Using the

previous development, it follows that

−

≤

|g|cosα

sign(g)

cosα

which through manipulations becomes

cosα ≥ −

sign(g)





thus

α ≥ arccos



−

sign(g)





= α

max

Proof of Proposition 3

(⇒) Assume that nodes i, j are linked through an

acoustic ray departing at angle |α

| < 0.5π from node

i. From Equation (8), the radius of curvature of the

acoustic ray departing from node i is R

. From Snell’s

Law and the constant slope g, the acoustic ray de-

scribes an arc of a circle meaning that the radius of

curvature for the entire arc remains equal to R

, thus

L = R

(⇐) If L = R

, then both nodes i, j are located on the

perimeter of a circle of radius R

. Consequently, both

nodes are linked by an acoustic ray given Snell’s Law

and the constant slope g. The departing angle α

fol-

lows from the relationship between R

and α

through

Equation (8) and the fact that |α

| < 0.5π.

Replacing L by R

in Equation (14) and putting it

to the square gives

)

= (d

− d

)

+ (z

+ R

− z

)

which after replacing R

and d

by their respective

expression in (8) and (9) leads to



cosα



= (d

− R

tanα

)

+ (z

+ R

− z

)

Re-labeling z

− z

in the previous Equation by ∆z

and expanding the right-hand side results in



cosα



= (d

− R

tanα

)

+ (∆z

+ R

)

cos

= d

− 2d

tanα

+ R

tan

+ ...

(∆z

)

+ 2∆z

+ R

Grouping all R

terms on the left-hand side gives



cos

− 1− tan



= d

− 2d

tanα

+ ...

(∆z

)

+ 2∆z

so that the new left-hand term equals zero thus leaving

the following expression

0 = d

− 2d

tanα

+ (∆z

)

+ 2∆z

which if re-arranged so as to extract α

leads to Equa-

tion (17).

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Node-to-node Distances

Node-to-node distances can be derived as follows

= d

− d

, d

= d

− d

= d

− d

, d

= d

− d

= d

− d

, d

= d

− d

= d

− d

, d

= d

− d

= d

− d

, d

= d

− d

Absorption Coefﬁcient

Figure 8 displays the absorption coefﬁcient for one

atmosphere and a temperature of 5 degrees Celsius.

−6

−5

−4

−3

−2

−1

Frequency (Hz)

Absorption Coefficient (dB/m)

Kinsler, et al., Fourth Edition [19]

Kinsler, et al., Third Edition [18]

Figure 8: Absorption coefﬁcient (1 atm., 5

C).

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