Sonar and Underwater Acoustics

Preface

Implementation of the structure of sonar chains evidently entails the analysis of the functions to be carried out and therefore the analysis of operational needs. A constant of these analyses, which is without doubt over-simplistic in relation to all the possible developments, stems from the fact that all processing will be the determination of the kinematics of mobiles present and the help of objective designation.

The processes used from the available sensors should therefore ensure as far as possible these two main functions with which a certain number of secondary functions useful for the operator (tests, plotting sound fields, propagation losses, etc.) are often associated.

Historically, the processes have been associated with particular antennas: active antenna, passive antenna, interceptor antenna, etc. This distinction, justified at the time by the low calculating capacity available, is fading little by little today where a more global view of a sonar system can be considered because of the perceptible increase in the calculating capacities and the introduction of everything digital in signal processing.

This book is mainly aimed at technicians and engineers who work in the field of underwater acoustics and in particular in the field of sonar. Its purpose is to give the maximum amount of information on the diversity of techniques related to the study and development of systems in underwater acoustics. This work also forms an introduction for engineers beginning their careers and a brief outline of the problems which they will encounter and allows the more general public to attain the necessary notions to understand sonar and underwater acoustics.

The inspiration for this book is three volumes published in French which make up part of a summary of underwater acoustics written by the underwater branch of the Thales Group¹.

These three volumes are:

Volume 1: The marine environment (Le milieu marin)

Volume 2: The acoustic electric interface (L'interface acoustique électrique)

Volume 3: The chain of active sonar processing (La chaîne de traitement du sonar actif)

The author, Jean-Paul Marage, a former engineer with Thales, contributed over 40 years to the definition, study, development and perfection of high technology sonar processing systems, mainly for the military sector as well as for the public sector.

The main interest of the company at the origin of these books was the capitalization, transmission and distribution of the knowledge and experience which made up the essential basis of its profession centered on underwater acoustics, through a practical approach, with concrete examples of industrial applications.

This book is a reviewed, corrected and updated republishing of the aforementioned volumes. This new edition gives all the necessary information for the training of specialists in the processing of acoustic signals, which is the reason for its republishing and greater distribution.

All the variety of problems that designers and users of sonar systems encounter are grouped into this book. We therefore find a large, relatively detailed range of all the problems encountered in underwater acoustics with sonar application in mind. From the physical phenomena governing the environment and the corresponding sonar restrictions, to the techniques of transducer and antenna definition, as well as the associated types of signal processing. It is this which sets it apart from the original works and is one of its main concerns.

Of course, the advances in the techniques of signal processing, the digital technology and especially the tactical naval strategies currently being designed by military staff, propose increasingly complex and efficient systems (multi-statism, multi-platform, detection and treatment of multiple data streams obtained by satellite or other means, etc.), with most being confidential and protected.

The correctional and editing work was carried out in collaboration with Y. Mori, who was also an engineer with Thales and head of a test laboratory and environmental assessments.

The developments carried out are mainly practical, the bibliography giving works of further grounding allowing the reader to reinforce their theoretical knowledge in this field. Several appendices are provided citing theoretical aspects which are judged to be useful. These appendices come from summaries of books, articles and documents to which access is not easy, and sometimes even impossible, for every reader.

We would like to thank in particular Thales who authorized the use of the French edition in order for us to bring you our new book.

Jean-Paul MARAGE

Yvon MORI

June 2010

1 THALES UNDERWATER SYSTEMS S.A.S.

PART 1

The Marine Environment

Part 1

Introduction

Part 1 poses the problems of underwater acoustics and gathers the notions necessary for its understanding, along with an understanding of sonar, into several chapters. This is achieved by detailing the variety of problems that designers and users of sonar systems encounter.

It forms an introduction and outline of the problems that technicians will come across and sets the essential information out in a simple manner allowing them to understand the following chapters.

For this reason, this part's mathematical complexities have been deliberately reduced. The few formulae being, for the most part, given without demonstration or justification, relying upon the large bibliography. The theoretical aspects judged useful are given in the appendices.

The links between chapters are mainly acoustic: propagation, environment noises, artificially produced noises and sonar equations.

Chapter 1

Problematics

This chapter provides a brief history of the concept of acoustics through the ages. It explains the characteristics of acoustics, particularly underwater acoustics. This introduction allows us to arrive at the modern definition of a sonar system from a military and civil application viewpoint. Several generalized underwater acoustics problems are then discussed, including detection, information processing and underwater conflicts in the marine environment.

1.1. History

Before the Greeks, man had never gone beyond the practical observation of the effects of sound. The Chinese philosopher Fohi himself struggled, around 3,000 years BC, to liken the five notes of the range to the five components of nature: earth, water, fire, air and wind. In 500 BC, Zeno of Elea noted our inability to understand and explain sound, saying:

Since a bushel of millet grains make a sound when poured into a heap, each grain and each part of the grain, be it one ten-thousandth, should make its own sound.

It was Pythagoras, in 600 BC, who was one of the first people to study the science of acoustics. The phenomenon of the echo, reported by Roman writers, is used in ancient theater.

Aristotle carried out studies on sound at around 350 BC and wrote his treatise on physics.

The word acoustics means science relative to sound and comes from the Greek akoustikos. It is one of the most ancient sciences, however it was not until the Renaissance that we see the appearance of a large number of researchers interested in the phenomena involved. At the end of the 15th century, Leonardo da Vinci wrote:

If you stop your ship, then put one end of a blowpipe in the water and the other in your ear, you will hears ships far from yourself.

This was one of the first statements in history regarding passive sonars and the Renaissance was period during which the development of acoustics started gathering pace.

Towards 1600, Mersenne wrote Universal Harmony. He was one of the first people to measure the speed of sound in air. Towards 1700, Huygens' theory was extended to acoustics.

In 1827, Chladni determined the speed of sound propagation using the vibrations of rods and sound pipes. The first measurement of the velocity of sound in water was carried out in 1827 by Swiss physicist Daniel Colladon and French mathematician Charles François Sturm in Lake Geneva. They obtained a value of 1,435 m/s.

Figure 1.1. Measurement of the speed of sound in water in 1827

Lord Rayleigh published his Theory of Sound in 1887. This theory was used as a basis for the science of acoustics and still is today.

Towards 1912, Kennely introduced the notion of motional impedance and threw out the theory of quadripole electromechanics.

We must recognize that after several centuries of slow progress, however, the true beginning of underwater acoustics was triggered by a catastrophe rather than armed conflict. On the night of April 14–15th 1912, the transatlantic liner the Titanic collided with an iceberg and sunk, taking hundreds of lives. Following this event, studies were carried out to devise an iceberg detection system. In the United States in 1914 Fessenden was capable of detecting an iceberg from two nautical miles (1 mile = 1,852 m) away with the help of a dynamic loudspeaker. From this it was quickly perceived that, taking account of the political situations of the time, such information could also be used to detect submarines.

Over several decades, the progress of underwater acoustics was linked with the development of detectors; whether it was hydrophones that listened (equivalent to microphones) or transducers that emit (like a loudspeaker) but can also listen (the principle of reciprocity). The main physical phenomenon involved is piezoelectricity, i.e. the transformation of a mechanical stress (acoustic wave) into electricity and vice versa. It was Paul Langevin who first underlined this phenomenon in 1917 using a quartz crystal.

Currently, the piezoelectric materials used are amorphous ceramics, molded when hot into plates, spheres or cylinders, then polarized through the application of a strong electric field, barium titanate, lead titano-zirconate or lead niobate.

The first echoes of submarines were obtained in 1918 at a distance of 1,500 meters. The first active systems were baptized ASDIC (the abbreviation for Anti-Submarine Detection Investigation Committee) by the British and simply SONAR (SOund Navigation And Ranging) by the Americans, a word universally adopted today. They worked at around 30 kHz.

We soon realized that extenuation through propagation strongly decreased with frequency: from 2 dB/km at 20 kHz it becomes 0.5 dB/kilometer at 5 kHz. This therefore led to the lowering of working frequencies in order to increase range; a practice that persists today.

Taking into account the progress of electronics, we can see that after a period of development of sensors and dedicated small electronics there was a period of progress in signal processing that we carry out today on calculators.

Today a period of information processing (mulit-antennas, multi-targets, multi-platforms, etc.) with an ever-growing emphasis on post-treatment algorithms is added to electronic beamforming, matched filtering, spectral analysis, etc.

An example of a submarine detection system is shown in Figure 1.2 and corresponds to what is called major ship in anti-submarine warfare.

Figure 1.2. Submarine detection systems

1.2. Underwater acoustics

As it is meant here, underwater acoustics covers sonar systems, i.e. the techniques that use the waves of mechanical vibrations in order to transmit and receive information in the marine environment.

Figure 1.3. Use of the marine channel

Of all kinds of energy, it is mechanical vibrations that propagate best in water. Electromagnetic waves abate so quickly that the ranges obtained by using them are ridiculous for most of the intended applications. Sometimes classed as acoustics, these waves are therefore the main means of investigation of the underwater environment.

In a general sense, underwater acoustics look to exploit the marine channel, in a broad sense of the term, as shown in Figure 1.3. By starting use the marine channel, we must consider it as a propagation channel and attempt to understand it.

1.2.1. Communications channel

The marine channel can be a communication channel in its classic meaning; we shall come back to this point later. Information must be transmitted between two points, both of them situated in an underwater environment: it is a problem of underwater communication.

A communications channel can also be an obligate transmission channel. This is the case in oil prospecting, where the section a company is interested in is beneath the seabed and the liquid part has to be traversed (because of the depth of the solid layer) more than desired.

These two types of channel are illustrated in Figure 1.4.

Figure 1.4. Transmission channel

1.2.2. Knowledge of the channel

The sonar system can be used to find the marine channel or knowledge about it. The word sonar relates to the use of acoustic waves in water to aid navigation and obtain information. It is possible to deduce two types of characteristics of the channel (or rather incidents in the channel) sought: the seabed and the particularities:

– Navigation: seabed. The knowledge of the seabed evidently takes on a crucial character for navigation. The corresponding pieces of equipment are called sounders.

– Ranging: particularities. By considering the notion of the channel in a general sense, the particularities of the channel can be: fish, which leads to fishing sonars; mine hunting sonars for mines; and surface ships and submarines, which are the targets of large sonar systems.

1.3. Applications

The applications of underwater acoustics are numerous, even if the economic weight of the field is not very significant. The applications can be classed by the outcomes sought. Applications are divided into two sectors: civil and military.

1.3.1. Civil applications

Four civil applications have already been cited:

– the measurement of the seabed with sounders;

– the detection and localization of shoals of fish with fishing sonars;

– oil prospecting with the aid of large linear antennas called flutes;

– the transmission of information with the help of underwater communication systems between, for example, a surface ship and an underwater robot.

Other applications can be cited:

– marine mapping for navigation;

– aiding navigation with a sounder or owing to a positioning in relation to fixed beacons;

– oceanography, which is not a true application, however;

– hydrography.

1.3.2. Military applications

The different military applications of underwater acoustics are perhaps more interesting because of the complexity of the systems involved. Other than a few tasks already mentioned (communication, aiding navigation, etc.), the different missions allocated to a sonar system can be:

– the detection, localization and recognition of small objects, in general laid on the seabed;

– guiding an underwater weapon (torpedo);

– the interception of sonar emissions;

– tracking, an operation that requires a scanning function, and is therefore a detection function, accompanied in general by monitoring and classification;

– attacking, sometimes classed as an estimation of elements-objective, where the preceding functions are put into action and where localization takes on a particular importance.

Two sorts of sonar systems exist: active and passive.

Passive systems, see Figure 1.5, seek to detect noises radiated by the target.

Figure 1.5. Passive listening

It is passive in the sense that it does not emit any signal, it is content with listening. This is the technique of submarine detection, which has the principal advantage of discretion.

As for an active system, it emits a signal and bases its detection on the signal reflected from a possible target. This is illustrated in Figure 1.6. It is the same type of approach as most radar systems.

Figure 1.6. Active listening

1.4. Comparison with radar

A quick comparison with radar is informative and will allow us to introduce several important notions.

Waves

A sonar uses mechanical waves and a medium is necessary. It is therefore logical to consider that this medium is of great importance (for noise, propagation, etc.).

A radar uses electromagnetic waves that do not need a medium for support: they can propagate in a vacuum.

Velocity

Mechanical waves propagate in water with a velocity of around 1,500 m/s (330 m/s in air). Electromagnetic waves propagate at the speed of light, so 300,000,000 m/s. This ratio of around 2.10⁵ between the two velocities has three significant consequences.

Algorithms

The sonar signal processor has more time than its radar equivalent to process the signals and apply the appropriate algorithms to its observation.

If an active sonar system has a range of 30 km, we would be able to emit, without superposition, a signal every 2.30,000/1,500 = 40 seconds. If a radar system has the same range of 30 km, we would be able to emit a signal every 2.30,000/300,000,000 = 0.0002 seconds. The ratio between the available processing time is 200,000.

Beamforming

The low value of the velocity of acoustic waves in the underwater environment requires a certain waiting period in order to complete a general survey.

Let us assume that with the help of an antenna, we would like to monitor the whole horizon (360°) by steps of 10° and that the expected range of the system is 12 km. We would have to wait 2.12,000/1,500 = 16 seconds in each direction and a general survey would therefore take 16.360/10 = 576 seconds, or a little less than 10 minutes.

Even though the first active sonar systems possessed such mechanical beamforming (the antenna was turned by hand), this significant time interval between two successive recurrences quickly led sonar operators to develop electrical beamforming (electronic scanning in radar) which, with the help of multiplexing, forms all the lines of the horizon at the same time. The complexity of this operation is obviously multiplied by a factor in the order of the number of lines formed, 36 in the previous example.

Target speed

The difference between the two velocities – a ratio in the order of 10⁵ – is much larger than the difference between the two speeds of the two types of targets for sonar and radar systems. Taking the Doppler effect into account, which includes the relative speeds between the carrier of the emission reception antenna and the target, allows us to say that boats go faster than planes. Typically the maximum values of the relative speeds are in the order of 5 Mach for radar and 60 knots for sonar (1 knot = 0.5 m/s and 1 Mach = 330 m/s, or at the speed of sound in air).

In the first case, the Doppler effect is equal to:

whereas for sonar, we get:

The Doppler effect is almost 4,000 times greater in sonar than in radar.

1.5. Submarine detection and warfare

It is common to distinguish submarine detection, which covers detection in a broad sense of the term (detection, localization, monitoring, tracking) from submarine warfare, which includes, but is not limited to, underwater weapons (mines, torpedoes). In France, the (governmental) organizations studying submarine warfare are:

– the study group concerned with research in submarine detection (GERDSM) and the management of constructions and naval weapons (DCAN) in Toulon, for everything concerning submarine detection;

– the Atlantic submarine study group (GESMA) of the DCAN in Brest, for everything involving mine-hunting sonars;

– the organization of constructions and naval weapons (ECAN) in Saint-Tropez, which studies and develops torpedoes, and therefore the acoustic part, homing device or acoustic head and the relative electronics.

1.6. Submarine detection

Among the targets that submarine detection systems are interested in, one thing is of particular importance: submarines.

Two main types of submarine exist: sub-surface ballistic nuclear (SSBN) submarines and attack submarines, whether nuclear (sub-surface nuclear, SSN) or conventional (diesel). SSBN submarines are the main deterrent of the several countries that possess them (USA, Russia, UK and France).

Due to the inefficiency of submarine detection, these are in fact the only discrete carriers of strategic nuclear weapons. Attack submarines are also an inconvenience since, among other things, they paralyze forces (e.g. the blockade of Argentinean warships by a UK SSN submarine during the conflict over the Falklands after the sinking of the General Belgrano) or convoys (e.g. German U-Boats during the Second World War).

What's more, surface ships are detectable by means other than acoustics: infrared, electromagnetic (radar) or visible means situated on different potential carriers, including satellites.

1.7. Submarine detection: a veritable challenge

Submarine detection is a real challenge. In simple terms, submarine detection is about finding an information carrier signal in noise. Yet the level of signal decreases, as much in passive as in active, whereas the level of noise increases.

Decrease of the active signal

Using the case of radar, the general public is aware that there are chiefly two ways of decreasing the radar cross-section, meaning the surface of the plane seen by the radar:

– the shape: the conception of a plane with angled forms and no surface area perpendicular to the axis of the incoming radar signal;

– the materials: the use of absorbing materials or paintwork.

For obvious reasons relating to restrictions in hydrodynamics and spaciousness, the cross-section of a submarine tends to remain circular and the use of bizarre shapes in the design of a submarine is not easy. On the other hand, so-called anechoic materials exist that decrease what we call the index of the target at certain frequencies, meaning the proportion of acoustic energy reflected is reduced compared with the incident acoustic energy.

Decrease of the passive signal

A passive sonar looks to detect noises radiated in an involuntary manner by a target, in our example a submarine. These radiated noises are mainly produced by:

– the engines on board (motors, back-ups, pumps, etc.);

– the phenomenon of cavitation (creation of bubbles that implode around the propellers);

– transitional and impulsive sources (door closing, dropped hammer, etc.);

– hydrodynamic phenomena (turbulent boundary limit).

It is evident that all developed nations have made an effort during recent years to decrease this type of noise, as illustrated in Figure 1.7 which shows the evolution of the level of noises radiated by typical examples of SSBN and SSN submarines (the levels shown here are only to give an idea of typical levels).

It is interesting to note that SSBN submarines – which are large machines of 10,000 tons for the Soviet Typhoon class SSBN, with an order of 10 MW of power installed – only radiates a few fractions of Watts into the environment.

Figure 1.7. Evolution of the noise radiated by different types of submarine

Increase of noise

Parallel to this decrease in the level of signal that we wish to detect, we must note that nuisance noise has a tendency to increase. We will take the noise of maritime traffic as an example.

One of the disruptive noises in submarine acoustics is ambient noise, meaning the noise that existed before the apparition of a sonar system (the other classes of noises are those emitted and radiated by yourself and, in active sonar, reverberation). Ambient noise is produced by natural sources (biological, precipitation, agitation of the sea, etc.) and artificial sources, such as those related to industrial activity (near oil rigs, near to ports, etc.) and those related to navigation by commercial or tourist ships (traffic). It is obvious that this last source of activity (traffic) is increasing and, therefore, so is the corresponding noise.

1.8. Overcoming the effects of the ocean

In order to develop an underwater detection system we need to overcome the (detrimental) effects of the marine environment. These effects we can aggregated into four groups: acoustics, propagation, noise and signal.

1.8.1. Acoustics

As we have already said, it is acoustic waves that propagate best (or least worst) in water. We must therefore develop sensors that transform:

– electrical energy at emission, that is readily available and able to be stocked on board, into mechanical energy: these are transducers (or projectors);

– an acoustic wave, at reception, into an electrical signal that we can easily process, thanks to electronics: these are hydrophones.

Additionally, as these sensors are not generally directive and we wish to obtain directional information, we cluster these sensors into an antenna in order to give priority to given observational directions.

1.8.2. Propagation

As opposed to electromagnetic waves that propagate in an almost straight line, acoustic waves propagate in a fashion that we qualify as being curious. In fact, their propagation depends on their velocity which, in itself, depends on the pressure, temperature and salinity of the seawater.

Figure 1.8, taken from the work of R.J. Urick [URI 03], illustrates different types of propagation by rays, following the immersion of the emissive source. We can note that important zones of the environment are not visible to the antenna and the position of these zones depends on the immersion of the latter (1 ft = 0.3048 m).

Figure 1.8. Propagation of sound rays in water as a function of immersion of the emmitive source

We see that the proportion of these zones not insonified by direct rays is reduced when we increase the immersion of the source. It is this which has lead to the notion of an immersed source, the antenna being located in a towed housing body (a fish) behind the boat. This is a variable depth sonar (VDS).

The attenuation of sound in water is added to loss by geometric divergence, as illustrated in an ideal way in Figure 1.9. This extenuation strongly increases with frequency. Sound will be carried further if we use low frequencies.

Figure 1.9. Attenuation of sound in water (in dB/km)

As the dimension of elementary sensors (mainly transducers) and antennas increases when the frequency decreases and as, moreover, a wave cannot distinguish objects whose size is smaller than its wavelength, the ranges of frequencies used will depend on the intended application. This is illustrated in Figure 1.10.

Figure 1.10. Frequency ranges in underwater acoustics

1.8.3. Unknown noise

The underwater acoustic channel introduces adverse noises that are illusory. We want to know what these noises are. This leads to the use of techniques of statistical signal processing, which we will come back to. The observation is modeled like a random variable that depends on the time and geographical location where the measurement is carried out, and eventually other parameters. This is what leads to the use of the theory of detection.

1.8.4. Unknown signal

In the use of this theory of detection, we cannot assume that only the noise is unknown. The signal is also unknown, because we do not know its moment of arrival (the position of the target is, in general, unknown). This is what leads us to use of the theory of estimation and to reason with statistical parameters such as the mean, variance and standard deviation. The techniques of signal processing therefore have a certain importance.

1.9. Sonar and information processing

A sonar system can be seen as a communication system which, in the way of Claude Shannon, is represented as follows (Figure 1.11):

Figure 1.11. Communication diagram

A signal is produced, modulated and then emitted into the environment. It propagates in this environment when a noise disturbs it. The reflected noise is then received by an antenna, is demodulated and a decision is made.

A sonar system can also be seen as a huge machine for compressing information, as illustrated in Figure 1.12.

Figure 1.12. Sonar model

The system in question is composed of 100 sensors, with a sample frequency of 10 kHz. We want the characteristics (distance, bearing and horizontal speed) for several targets, for example three. The input of the sonar receiver is made up of 10⁶ observations, whereas at its output it contains only nine values, be it a compression ratio of around 10⁵.

It is the presence of noise and an unknown modulation from the part of the channel that requires the use of probability methods, such as the statistical theories of detection and estimation.

Chapter 2

Sound Propagation in the Marine Environment

2.1. General points

Propagation in the marine environment will essentially be considered from a descriptive and practical angle here, in order to bring to light:

– the limitations and constraints that it imposes upon systems of detection and transmission;

– the theoretical models and methods that allow us to either study a given system and determine the conditions under which it should operate or analyze and evaluate sea trials.

We will only analyze the propagation of sound, leaving other phenomena that currently only have limited applications to one side, because of their very strong extenuation in seawater (radioelectric transmissions at very low frequencies, magnetic, electric or thermal detection of submarines over short distances, various firing systems, etc.).

2.2. Characteristics of the marine environment

The propagation of sound is fundamentally governed by the classic wave equation (see Appendix 2):

(2.1)

The velocity (or speed) of sound in water is therefore the most important parameter in the study of these phenomena. The sea is a far from homologous environment and the speed of sound is influenced by the temperature, pressure and salinity, as already mentioned.

All these factors vary according to the geographical locations, time and especially with immersion at a given point. The vertical variations of velocity are in fact much larger than the horizontal variations. This velocity can be known to 0.2 m/s, either through direct measurement or through the intermediary of empirical formulae linked with the temperature, pressure and salinity. Currently the most precise formulae are those of Wilson. The velocity grows almost linearly with temperature, pressure or in depth salinity. A practical formula is as follows:

(2.2)

with: T = temperature in °C; S = salinity in 1/1,000e; and D = depth in m.

It is important to know the general appearance of the velocity profiles in the different oceans and seas. We can distinguish two zones: one of them stable between 100 m and the seabed; the other very variable near to the surface. In the first, the velocity is linked to variations in temperature, which decrease steadily from the surface to the seabed in the Atlantic, Pacific and Indian Oceans. The influence of decreasing temperature is compensated by that of the increasing pressure and the velocity goes through a very distinct minimum at a depth of around 1,000 m in these three oceans. The value of this minimum is in the order of 1,490–1,500 m/s; the maxima at the surface and bed are in the region of 1,520–1,550 m/s.

The closed seas, such as the Mediterranean or the Black Sea, present certain particularities. In the Mediterranean, the temperature is practically constant at around 13°C below 100 m. The depth of minimum velocity is therefore around 100–150 m.

The surface layer of the oceans (0–100 m) is strongly affected by meteorological conditions (insolation and wind). Its temperature varies according to diurnal and seasonal cycles: low and uniform in winter (temperature gradient nil), it is normally high during summer (strong negative gradient). The prevailing wind mixes the surface layer and creates a superficial layer from 20–50 m thick where the temperature is consistent or even increasing (nil or positive gradient, followed by a strong negative gradient).

The existence of these two zones of different stability in the velocity profiles explains the importance of bathythermals taken at the same time as a trial or exercise, which we often complete with a mean curve for the deep zone of the profile. These curves are taken from statistical catalogs; there are correct statistics for the western Mediterranean, the Atlantic Ocean and the coasts off the south of Africa.

Although sound propagates over long distances, the sea is not an environment with losses. There is a non-negligible absorption of sound energy for the frequencies acousticians are interested in. This is due to a number of quite different phenomena: the thermal conductivity of water molecules (very low); the viscosity of water; the presence of dissolved salts in water (relaxation at around 100 kHz); the diffusion due to the heterogenities of water (bubbles, micro-organisms, convection currents, etc.); and other relaxation phenomena still unexplained, etc. This set of factors is grouped into one term of absorption, which essentially depends on temperature, salinity and frequency. Absorption is not very easy to measure and the values given in literature do not always correspond to the real situation.

The depth of the oceans and their boundaries, the surface and the seabed are also important factors in sound propagation. We can divide the seas into two distinct groups:

– the continental shelf bordering the coasts; and

– the deep regions (more than 1,000 m) with an abrupt transition zone and often reduced area.

The depth of the continental shelf varies between 0 and 200 m and its length is often variable, typically between 0 and 100 km. The seabed is has relatively little slope, but is often uneven and stratified. The bathythermal conditions are often variable here depending on the season. The locations and stable part of the velocity profiles has often disappeared. Figure 2.1 illustrates a typical profile we can encounter – a contraction in the order of 60 has been applied to the horizontal scale in order to facilitate the readability of the changing profile.

In closed and shallow seas we find particular conditions, such as a much reduced salinity because of the flow of freshwater (Baltic Sea) or, on the contrary, high salinity because of evaporation (Red Sea).

The deep regions represent the oceans' core, as the values for the average depth of the main seas show: between 3,300 and 4,300 m for the Atlantic, Pacific, Indian and Antarctic Oceans. The seabed there consists of mountain chains and important regions of little-marked relief (abyssal almost completely flat plains, eroded hillsides, sedimentary basins and canyons), the seabed is stratified (numerous layers of sediments on a rocky base) except where this is not allowed by the slope of the ground.

Figure 2.1. Type of seabed profile¹

As we will see later, this possible separation into deep water and small beds enables us to consider two types of propagation that are quite different from one another.

The surface is characterized in a simple way by the state of the sea (force from 0 to 6) and in a more elaborate way by different statistical models outlining the properties of the wave height as a function of their location (x, y) and time (distribution of amplitudes and slopes; the Neumann-Pierson energy spectrum). There is a very obvious correlation between the level of noise from the sea and the wind. The correlation is less direct between the force of the sea and the wind.

Next to these macroscopic variations, the marine environment shows local fluctuations in temperature (or in density) due to the circulation current of the water masses. These irregularities lead to variations in the speed of sound (or in the refractive index), which are difficult to measure. As we will see later, they are mostly deduced from tests concerning the coherence of acoustic propagation. Only internal waves created by the meeting of different water masses (the straits of Gibraltar, for example) show a periodic variation of large amplitude and can therefore easily be studied through direct methods.

The term simatic comes from sima (magnesium-silicon). It is part of the upper mantle constituting the mainly basaltic and basic oceanic crust.

2.3. Models used

Solving the wave equation characterizing sound propagation in a non-absorbing medium has lead to two mathematical models: the theory of rays and that of normal modes.

The domain and validity of these models is limited by the following hypotheses: small movements, fluid at rest, adiabatic transformations, and slow variations in the elastic constants of the environment. The models are quite close to reality, except when we examine cases with signals of high amplitudes (explosions at short distances, for example).

In the theory of rays, we have separated the boundary effects (boundary limits) from the effects of the volume and we introduce them in the form of reflection coefficients of the surfaces involved. We prove that they are stable over time (Fermat's principle), follow the generalized Descartes' law and represent the trajectories followed by sound energy. A calculation of the sound field through this method therefore consists of determining the trajectory of the rays, energy losses and journey time along these rays.

The strong variations in velocity with depth on one hand and their relative stability in a given place for a certain period of time on the other, allow us to consider that velocity depends on depth alone for a given problem. We can therefore show that the rays are always contained in a vertical plane and that they follow the law:

(2.3)

where:

c(z) = velocity as a function of depth z; and

θ = angle between the ray and the horizontal.

It is suggested that we find a practical c(z) law in order to allow us to resolve an integral in the form:

which leads to the calculation of the coordinates of a trajectory. The formulae used do not allow us to generally represent the velocity profiles encountered in practice. We therefore consider the sea as a stratified environment, constituted of parallel layers where a c(z) law is valid.

The linear law (approximate profile with line segments) is by far the most frequently used because of its simplicity and ease with which we can obtain an approximate profile. It has the disadvantage of leaving discontinuities between layers (in the tangent to the profile), however, which creates the appearance of sometimes inconvenient secondary phenomena. The calculations are often weighed down with such phenomena and obtaining an approximate velocity profile with a good connection between layers is particularly tricky.

In the case of a linear approximation, the rays or sound channels are made of the arcs of a circle, of which we calculate the coordinates. In practice, to carry out a calculation of the sound field we define the depth z of the sound source S used and that of the sea to get a velocity profile, as well as a series angles starting from the source to which a series of rays will correspond to the power distribution (Figure 2.2).

Figure 2.2. Example of sound propagation in the marine environment

The preceding method therefore allows us to determine the transmission losses in a non-absorbing fluid environment by taking into account at point R of reception, the rays issued from the source and passing through the reception point.

As for the propagation of electromagnetic waves, the notion of power at a point that would correspond to an infinite power density in the environment has no physical sense. We therefore substitute infinite power densitiy with power density at the point considered, which we can link to power thanks to the effective area of the emission or reception antennas that are never pinpointed.

The introduction of power density presents the additional advantage of introducing parameters linked to the propagative environment and the characteristics of the emission and reception antennas intervening in the power densities through the intermediary of their equivalent surfaces or their gain.

If we consider a force tube issued from emission point S and passing through reception point R containing a certain number of rays (see Figure 2.3) with:

– I0 = power density per unit of surface at a standard distance r0 from the source; and

– Ir = power density per unit of surface around the reception point;

where I0 and Ir are measured in W/m², the transmission losses of the non-absorbing fluid environment or losses through geometric divergence amount to the relationship:

(2.4)

In the acoustic environment the losses through divergence are accompanied by significant losses due to energy absorption by the environment, which is not purely elastic.

Figure 2.3. Force tube

We generally model the absorption by coefficient α dependant on frequency. A wave that propagates in the preceding force tube will incur losses when crossing an element of volume dv = S.dl. An energy proportional to the incident energy W and to the distance travelled dl (see Figure 2.4) is:

(2.5)

and since by definition of the power density I on cross-section S we get:

t being the period of observation, we arrive at:

Figure 2.4. Absorption model

If we integrate between 0 and r (the length of the ray considered between the origin of the rays and the point of reception) we get the following expression for the losses through absorption:

(2.6)

The total loss tl(r) endured during transmission is therefore given by the superposition of the divergence phenomenon and the absorption phenomenon, therefore:

(2.7)

where:

– d(r) represents the losses through geometric divergence determined by the propagative conditions or hypotheses made; and

– eα.r represents the losses through absorption associated with losses through divergence along the sound rays.

Figure 2.5 gives the typical absorption coefficients in seawater for the Atlantic Ocean and the Baltic Sea. For frequencies inferior to 100 kHz, the absorption coefficient can be represented by the relationship:

(2.8)

The expression of transmission losses is by construction a dimensionless quantity; it defines a ratio of power (or power density). It is current practice to introduce the decimal logarithms and decibels and to write (with lgx = decimal logarithm of x):

(2.9)

where:

(2.10)

with:

TL(r) = total transmission losses in dB;

D(r) = transmission loss through divergence referenced to r0 meters;

a0 = absorption coefficient in dB per r0 meters; and

r = distance measured in r0 meters.

We generally use the following for the reference distance:

(2.11)

Acousticians reserve the term sound intensity for power density; we therefore speak of sound intensity rescaled to 1 m during the measurement of power density produced by a sound source.

Figure 2.5. Absorption in seawater

Other terms accounting for reflective losses on the surface and at the bed must be added to this equation each time a ray hits one of these boundaries.

The reflection at the surface of the sea relates more to the study of diffusion than specular reflection.

Quite different theories (Rayleigh-Marsh and Eckart among others) have allowed us to analyze this problem; they both use the statistical model of the surface, mentioned in section 2.2. We can distinguish between three main regions: near to the normal incidence, where specular reflection of facets intervenes making up the water surface; near the grazing incidence, where the layer of air bubbles located near the surface is the main reflector; and finally intermediary angles, where reflection diffuses on the irregularities of the surface.

The Rayleigh-Marsh theory, which distinguishes specularly reflected energy from energy diffused in other directions, allows us to arrive at a representative formula, but with very limited application. It takes into account the frequency and height of the waves and the incident angle of sound energy.

Random fluctuations play a less important part in reflection off the seabed and the interpretation of reflection phenomena is mainly deterministic. The bed is considered to be a plane made of viscoelastic solid layers characterized by their density and the speed and absorption of longitudinal and transverse waves. It is possible to introduce gradients of speed and density into these layers.

This model, which corresponds to stratified sedimentary beds that cover a large area of the oceans, allows us to calculate the reflection coefficient of the bed in module and in phase as a