1. Understand why sound energy is employed for underwater surveillance and detection.

2. Understand the following terms associated with the fundamentals of sound production: wave motion & propagation, acoustic pressure, acoustic intensity, characteristic impedance.

3. Know the SI units employed in acoustic measurements and be able to convert to other measurement standards.

4. Understand the decibel method of measuring sound levels, including decibel calculations.

5. Be able to calculate intensity levels and sound pressure levels.

6. Be able to obtain an overall intensity level from individual sources.

7. Know how the following factors affect transmission loss: spreading, absorption, scattering, and bottom loss.

8. Be able to calculate appropriate values of transmission loss.

9. Understand the sources and effects of self-noise and ambient noise.

10. Understand the sources and effects of cavitation.

11. Be able to use Wenz curves to estimate values of ambient noise.

12. Be able to apply the concept of signal-to-noise ratio to underwater sound.

13. Know the basic sonar equation and its passive and active derivations.

14. Understand the significance of the terms in the sonar equations and their relationship to the figure-of-merit.

15. Be able to calculate figure-of-merit.

16. Be able to calculate values of sound speed based upon an understanding of temperature, pressure, and salinity effects.

17. Know the basic thermal and sound-speed structure of the ocean and how field observations of sound speed are made.

18. Be acquainted with the Ray Theory solution to the Wave Equation.

19. Understand the use of Snell's Law in determining ray path structure by calculating simple straight-line and curved ray paths.

20. Know the three basic types of sound-speed gradients and how they affect sound propagation to produce the following types of paths: surface duct, shadow zone, sound channel, convergence zone, and bottom bounce.

21. Demonstrate understanding of the theoretical and operational principles, applications capabilities and limitations of underwater sound through class discussion and problem solving.


The effectiveness of the present-day submarine depends upon its ability to remain undetected for long periods of time while it searches, tracks, or attacks from beneath the sea surface. This medium of concealment, however, is advantageous to the submarine only so long as it is not detected or deprived of its ability to detect. Before a submarine can be attacked, it must be detected and its subsequent positions determined within the requirements of the available weapons system. Detection and position fixing can take place in two ways. There may either be some radiation or reflection of energy from the submarine to the searcher, or else the submarine may disturb one of the natural, static, spatial fields, such as the earth's magnetic field, thereby betraying its presence.

The choice of energy to be used for underwater detection is determined by three factors:

1. Range of penetration in the medium.

2. Ability to differentiate between various objects in the medium.

3. Speed of propagation.

Of all the known physical phenomena, light has excellent differentiation ability and high speed of transmission, but its range in water is very limited, on the order of tens of meters, thereby restricting its operational usefulness. This is not to say that light will never be used in ASW, for continuing experimentation and perfection of laser light detectors might yet add this method to the arsenal. Radio frequency waves also are propagated with extreme rapidity and to great distances through certain mediums, but sea water is essentially impervious to them for most frequencies. VLF signals will penetrate only about 10 meters, whereas higher frequency penetration depths can be measured in millimeters. Magnetic and gravitational field distortions are detectable only at very short ranges because the anomaly diminishes proportionally with the inverse of the range cubed. While their detection range is greater than either light or radio frequency, it is only of the magnitude of several hundred meters and therefore is insufficient for normal surveillance.

Acoustic energy, while lacking the propagation speed of electromagnetic waves, is capable of being transmitted through the sea to distances that are operationally significant. Because of this, sound is the physical phenomenon used for antisubmarine warfare, underwater communications, and underwater navigation. It must not be inferred, however, that sound is a panacea. It too has significant limitations to its effective employment, all of which must be thoroughly understood by the operators of underwater sound equipment. The optimum use of sound requires a thorough understanding of its limitations so that these effects can be minimized. For example, sea water is not uniform in pressure, temperature, or salinity, and all these characteristics have important effects on sound propagation through the sea. The requirement for predicting these effects on sonar performance has become a necessity, and a difficult one at that.


All sound, whether produced by a cowbell or a complicated electronic device, behaves in much the same manner. Sound originates as a wave motion by a vibrating source and requires for its transmission an elastic medium such as air or water. For example, consider a piston suspended in one of these mediums. As the piston is forced to move forward and backward, the medium is compressed on the forward stroke and decompressed or rarefied on the return stroke. Thus, a wave motion or series of compressions and rarefactions is caused to move from the source out through the medium. In the fluid medium the molecular motion is back and forth, parallel to the direction of the piston's movement. Because the fluid is compressible, this motion results in a series of detectable pressure changes. This series of compressions and rarefactions, such as is produced by the piston, constitutes a compressional wave train. Another way of explaining the phenomenon of acoustic wave propagation is to consider the medium of transmission as a loosely packed collection of mass elements connected by springy bumpers. A disturbance of the elements at some point (e.g., piston motion) moves along in the fluid by the successive extension and compression of the springs as the elements swing back and forth, each communicating its motion to its neighbor through the connecting bumpers. In this way, the agitation of a cluster of elements is propagated through the medium even though the individual elements do no more than move about their equilibrium positions without actually migrating. The sound wave propagates parallel to the source resulting in a longitudinal wave. Recall from radar principles that the electromagnetic wave propagated perpendicular to the source, resulting in a transverse wave.

FFigure 8-1. Pictorial representation of simple longitudinal wave.

8.2.1 The Sound Wave

The sine wave of figure 8-1 is graphical representation of the compressional wave train. As the wave passes a given point, the fluid elements are compressed and extended in the manner depicted by the sine wave's oscillations above and below the static pressure. The compressions and rarefactions are so labeled on the curve. There are two important things to note on this curve. The first is the maximum amplitude of the sine wave, labeled P, which represents the maximum pressure excursion above and below the static or hydrostatic pressure that exists in the fluid at the location of the wave train. The second thing to note is that the wave train would be passing, or propagating at the speed of sound c in the medium. The units for speed of sound are meters per second. The relationship between the three acoustic quantities that can be derived from figure 8-1 is:

Frequency (f) = Speed of sound (c) (8-1)

Wavelength ()

This is exactly the same as described for electromagnetic energy in chapter 1 and thus will not be elaborated upon.

One final point should be made about the sine wave representation of figure 8-1. Though somewhat difficult to imagine and more difficult to describe pictorially, the displaced parallel lines in figure 8-1a represent the motion of the elements within the field as the wave train passes. As the elements are compressed and extended, their motion can also be mathematically described by a sine wave; however, the elements would be oscillating to and fro about their static position. The maximum amplitude would then be the maximum displacement from the static position. To provide an example of the order of magnitude of these displacements, consider that the faintest 1,000 Hz tone that can just be heard in air has pressure variations of only 2/10,000,000,000 of one atmosphere of pressure. The corresponding particle displacement is about 10-9 cm. By the way of comparison, the diameter of an atom is about

10-8 cm.

As this pressure disturbance propagates through the medium, the pressure at any point in the medium can be expressed as a function of the distance, r, from the source and time, t, since a particular wave departed the source.

P(r,t) = P(r)sin | 2 (r - ct) | (8-2)

| |

Note how the maximum amplitude, P(r), is dependent on the distance from the source. Neglecting wave interference effects, the amplitude of a pressure disturbance will always diminish, and equation 8-2 therefore represents a decreasing amplitude sinusoid wave. If, however, the distance is fixed, then pressure is solely a function of time, and equation 8-2 simplifies to

P(t) = PAsin [2(ft)] (8-3)

where PA is now the maximum amplitude at the range r.

8.2.2 Intensity

A propagating sound wave carries mechanical energy with it in the form of kinetic energy of the particles in motion plus the potential energy of the stresses set up in the elastic medium. Because the wave is propagating, a certain amount of energy per second, or power, is crossing a unit area and this power per unit area, or power density is called the intensity, I, of the wave. The intensity is proportional to the square of the acoustic pressure. Before giving the defining relationship for intensity, however, two variables must be explained. The value of peak pressure, P, as shown in figure 8-1 is not the "effective" or root-mean-square pressure. An analogy exists between acoustic pressure and the voltages measured in AC circuits. Most voltmeters read the rms voltage. The rms value of a sinusoidal voltage is simply the peak voltage divided by the square root of two. For example, the common 115-volt line voltage has a peak value of about 162 volts. In order to obtain the rms pressure, P must likewise be divided by the square root of two.

Pe = Prms = P (8-4)


In this text, the effective or rms pressure as measured by a pressure-sensitive hydrophone will be labeled Pe. The explanation of units for pressure will be fully discussed at a later time.

8.2.3 Characteristic Impedance

The second variable that must be explained is the proportionality factor that equates intensity to effective pressure squared. It consists of two terms multiplied together - fluid density, and the propagation speed of the wave, c. The quantity, c, is called the characteristic impedance; it is that property of a sound medium that is analogous to resistance or impedance in electrical circuit theory, where power equals voltage squared divided by resistance. Additionally, it can be illustrated by a simple example: When two dissimilar mediums, such as air and water, are adjacent to each other, the boundary between the two is called a discontinuity. When sound energy is traveling through one medium and encounters a discontinuity, part of the energy will be transferred across the boundary and part will be reflected back into the original medium. The greater the difference between the characteristic impedances, the greater will be the percentage of energy reflected. (The difference between the c values for air and water in SI units is approximately 1.5 x 106.) Thus, when sound is traveling through water and it reaches the surface, only a small amount is transmitted into the air. Most of the energy is reflected by the air/ocean boundary back into the water. Obviously, it is important to maintain a consistent set of units when comparing characteristic impedances, and care must be exercised when dealing with different sources of acoustic information.

With the concepts of rms pressure and characteristic impedance understood, it is now possible to formulate an expression for acoustic intensity, the average power per unit area normal to the direction of wave propagation.

I = pe2 (pcsea water 1.5 x 105 dyne-sec/cm3) (8-5)


The units of acoustic intensity are normally watts/m2. The importance of equation 8-5 is that it clearly shows the dependence of the power transmitting capacity of a train of acoustic waves on the pressure. If the rms pressure in the water can be measured, then the sound intensity can be determined. One way to do this is by means of a hydrophone, an electrical-acoustic device, much like a microphone, that transforms variations in water pressure into a variable electric voltage. Thus, after appropriate calibration, Pe can be read directly from a voltmeter attached to the output of a hydrophone.


8.3.1 A convenient system is needed in order to measure and discuss acoustic parameters. Pressure is defined as a force per unit area. Although many people are familiar with the British units of pounds per square inch (psi), it has long been the convention in acoustics to use metric units, namely newtons per square meter (N/m2), or dynes per square centimeter (dynes/cm2). Of the two metric units, the dynes/cm2 has been the most commonly used. It has an alternate name, microbar (bar), and is equivalent to approximately 1/1,000,000 of a standard atmosphere. For underwater sounds, a reference pressure of 1 bar was established from which all others were measured. The corresponding reference pressure for airborne sounds was 0.0002 bar, because this was the approximate intensity of a 1,000-Hz tone that was barely audible to human ears. The previously less commonly used N/m2 also has an alternate name, a Pascal (Pa), and the reference standard derived from this was the micropascal (Pa), which is equivalent to 10-6N/m2.

With such a profusion of reference standards and measurement systems, there were ample opportunities for misunderstandings as an operator or planner consulted different sources of acoustic information. In 1971 the Naval Sea Systems Command directed that thereafter all sound pressure levels should be expressed in the Systeme Internationale (SI) units of micropascals. Although all new publications contain the updated standards, older references will not until they are revised. To assist in making conversions until all publications are revised, table (8-1) summarizes some conversion values.

Throughout this text, the acoustic pressure reference standard, Po, is 1 Pa unless otherwise noted.

Table 8-1. Acoustic Reference Conversion Factors


1 bar = 1 dyne/cm2 = 0.1 N/m2

= 105 Pa = 10-6 atmospheres

1 Pa = 10-6 N/m2 = 10-5 bar

= 10-5 dyne/cm2 = 10-11 atmospheres

8.3.2 Sound Pressure Level

In theoretical investigations of acoustic phenomena, it is often convenient to express sound pressures in newtons/m2 and sound intensities in watts/m2. However, in practical engineering work it is customary to describe these same quantities through the use of logarithmic scales known as sound pressure levels. The reason is related, in part, to the subjective response of the ear. The human ear can hear sounds having pressure disturbances as great as

100,000,000 micropascals and as small as 10 micropascals. A problem is encountered when discussing pressures that vary over so great a range, in that the minimum audible disturbance is one ten-millionth that of the maximum. In underwater acoustics, useful pressures having even greater variations in magnitude are commonly encountered. In order to make the numbers more manageable, both in magnitude and for actual manipulation, logarithms are used rather than the numbers themselves. Suppose two acoustic signals are to be compared, one having a Pe, of 100,000,000 Pa and the other a Pe of 10 Pa. Their ratio would be

P1 = 100,000,000 Pa = 10,000,000 = 107

P2 10 Pa

In underwater acoustics, however, the attribute of primary interest is sound intensity, or power, rather than pressure. As with pressure, acoustic intensities are referenced to some standard intensity, designated Io, and the logarithm of the ratio taken. Intensity level is therefore defined as

IL = 10 log(I/Io) (8-6)

where IL is measured in dB. However, as there is only one intensity reference (10-12 watt/m2 in air) and many pressure references, IL must be able to be expressed in terms of pressure. By inserting equation (8-5) into equation (8-6), a new expression of IL can be obtained, which is based on pressure rather than intensity per sec.


IL = 10 log pc (8-7)




IL = 10 log Pe2 = 20 log Pe

Po2 Po

Under the assumption that the reference intensity and the reference pressure are measured in the same acoustic wave, then a new sound level can be defined called sound pressure level.

SPL = 20log Pe (8-8)


Since the voltage outputs of the microphones and hydrophones commonly used in acoustic measurements are proportional to pressure, acoustic pressure is the most readily measured variable in a sound field. For this reason, sound pressure level is more widely used in specifying sound levels, and this is also why only pressure references are used in underwater acoustics. Note that IL and SPL are numerically equivalent.

IL = 10logI = 20logPe = SPL (8-9)

Io Po


Io = P2



8.3.3 Decibels

SPL has the dimensionless units of decibels (dB). The decibel system was selected by acousticians for a number of logical reasons. First, it is a logarithmic system, which is convenient for dealing with large changes in quantities. It also simplifies computations since multiplication and division are reduced to addition or subtraction, respectively. Second, human senses have an approximate logarithmic response to stimuli such as light, sound, and heat. For example, the human ear perceives about the same change in loudness between 1 and 10 units of pressure as it perceives between 10 and 100 units of pressure. And finally, in the area of underwater acoustics, the primary interest is in ratios of power levels and signal levels rather than absolute numerical values. In the decibel system, the bel is the fundamental division of a logarithmic scale for expressing the ratio of two amounts of power. The number of bels to express such a ratio is the logarithm to the base 10 of the ratio. Acousticians decided the bel was a unit too large for application in their field, and subsequently adopted the decibel (1/10 bel) as their basic logarithmic unit. The conversion factors in table 8-1 can in themselves be cumbersome to use, but when expressed in dB, only addition or subtraction is required. When converting from a pressure referenced to 1 bar to one referenced to 1 Pa, simply add 100dB. When converting from 0.0002 bar to 1 Pa, simply add 26dB. If converting from 1 Pa to the others, merely subtract the appropriate values. Table 8-2 shows some representative values of conversions between different reference levels. Note that the new micropascal reference standard is small enough that negative values of decibels are rarely encountered.

As an aid in interpreting and understanding the decibel scale and its relation to intensity and pressure, it is useful to remember that

a factor 2 in intensity is + 3 dB

a factor of 0.5 in intensity is -3 dB

a factor of 10 in intensity is + 10 dB

a factor of 0.1 in intensity is -10 dB

a factor of 2 in pressure is + 6 dB

a factor of 0.5 in pressure is -6 dB

a factor of 10 in pressure is + 20 dB

a factor of 0.1 in pressure is -20dB

In understanding intensity levels and sound pressure levels, it is important to note that the decibel scale is a ratio of power or energy, no matter what quantities are being ratioed. A problem commonly arising in acoustic calculations is that of obtaining the overall intensity level after the individual intensities within the applicable bandwidths have been calculated. Such a situation is encountered in calculating a term in the sonar equations (to be discussed later) called noise level, which is actually a combination of ambient noise and self-noise. Because we are dealing with decibels, it is not possible to merely add similar intensity levels together and work with their sum. For example, two 30 dB signals combine to give a total intensity level of 33 dB, not 60 dB as might be expected. The reason for this is that, as shown above, doubling the intensity is represented in decibels by a + 3 dB change. The process is more complicated when dealing with levels of unequal value. Figure 8-2 can be used to determine the dB increase above a level of IL1, in terms of the difference IL1 - IL2, to be expected when IL1 and IL2 are combined. This process can be expanded to include any number of intensity levels. However, when dealing with more than two intensities, it is often easier to use anti-logs, to convert each IL back to its intensity units, add all the intensities together, then reconvert to dB levels. Either method may be used and should result in the same numerical value of noise level.

Although standardization has been reached for measuring intensities, such is not the case for other quantities. Ranges are expressed in yards, kilometers, and nautical miles. Depths are given in feet, meters, or fathoms. Sound speed is stated in feet per second or meters per second and ship speed in knots.

Figure 8-2. Nomogram for combining dB levels.

Temperatures are commonly specified in degrees Fahrenheit or degrees Celsius. These diverse units should warn the user to exercise due caution when discussing the various facets of underwater sound to ensure that misunderstandings do not occur. In this text, the SI units of measure will be used wherever possible.


From physics it will be remembered that when gas is the transmitting medium, the denser the gas, the slower the speed of sound, and yet the speed of sound in water is about four times greater than that in air. Although this seems contradictory, it is not, because there is another more important factor that influences the speed of sound. In truth, the speed of sound is determined primarily by the elasticity of the medium and only secondarily by the density.

8.4.1 Bulk Modulus

Elasticity is defined as that property of a body that causes it to resist deformation and to recover its original shape and size when the deforming forces are removed. Of specific concern is volume elasticity or bulk modulus - that is, the ratio of force per unit area (stress) to the change in volume per unit volume (strain).


Bulk Modulus = Stress


In order to bring about a change in the volume of a liquid, it is necessary to exert a force of much greater magnitude than is required to bring about an equivalent change in the same volume of air. Therefore, the value of bulk modulus is much greater for a liquid than for a gas. This bit of information, however, is meaningless until it is applied in the formula for the speed of sound.

The speed of sound, c, in a fluid is equal to the square root of the ratio of bulk modulus to density. Thus,

c = Bulk Modulus (8-10)


Although seawater is almost a thousand times denser than air, the enormous bulk modulus of water is the more important factor determining sound speed. Of concern, however, are not the differences of the two mediums but the conditions in water that cause changes in sound speed. Contrary to the assumptions made up to this point, the ocean is not a homogeneous medium, and the speed of sound varies from point to point in the ocean. This variation in sound speed is one of the most important characteristics affecting the transmission of sound. The three main environmental factors affecting the speed of sound in the ocean are salinity, pressure, and temperature.

8.4.2 Salinity

Salinity, which on the average ranges from 32 to 38 parts per thousand (ppt), is fairly constant in the open ocean. A change of salinity will cause a small corresponding change in density with a resulting change in bulk modulus, causing variation of sound speed. The greatest variation in salinity in the open ocean exists in the vicinity of "oceanic fronts," which are narrow zones separating water masses of different physical characteristics, usually exhibiting very large horizontal gradients of temperature and salinity (figure 8-3). Even greater variation in salinity can be expected around the mouths of rivers, heavy ice, and in areas of extraordinary rainfall (e.g., the monsoon) where a layer of fresh water overrides a layer of salt water. A change in salinity of one part per thousand will result in a change in sound speed of approximately 1.3 meters per second.

8.4.3 Pressure

Pressure in most circumstances is more important than salinity, but in the sea its change is constant and thus predictable. It also causes a change in bulk modulus and density, and the result is an increase in sound speed of 0.017 m/sec for every meter of depth increase. This slight change, which is important when temperature remains constant, causes a sound beam to bend upward at great depths as will be discussed later.

8.4.4 Temperature

Temperature, the foremost factor affecting sound speed, usually decreases with depth, and this leads to an accompanying decrease in sound speed at the rate of approximately 3 m/sec per degree Celsius. Below a depth of about 1,000 m, however, temperature is

fairly constant, and the predominant factor affecting sound speed becomes pressure. At first glance it would seem that a temperature decrease would increase sound speed due to the increased water density, but not so. As the temperature of a medium decreases, bulk modulus decreases while density increases. Considering these effects in terms of the sound speed formula in equation (8-10), it is clear that a decrease in temperature brings an attendant decrease in sound speed. It also should be noted that temperature differs bulk modulus and density at a variable rate. A change in temperature at one point on the scale, therefore, affects sound speed differently than an equal change at another point on the scale. It should be noted that the effect of temperature is relatively large compared to the other factors. It takes a depth change of about 165 meters to cause the same change in sound speed as a one-degree temperature change. As will be discussed, temperature is therefore the only factor normally measured and evaluated under operational conditions.

8.4.5 Sound Speed Equation

Dealing with these three factors to arrive at values for bulk modulus and density, and thence sound speed, is very cumbersome. To overcome this, numerous empirical relationships have been developed for converting the three factors directly to sound speed. A simplified version of such sound speed equations developed by Wilson in 1960 is present below.

c = 1449 + 4.6T + 0.055T2 + 0.003T3

+ (1.39 - 0.012T)(S - 35) + 0.017d (8-11)


T = temperature in degrees Celsius

S = salinity in parts per thousand

d = depth in meters

Given accurate temperature, salinity, and depth data, this equation is accurate within 0.6 meters/sec, 96 percent of the time. By way of contrast, the equation for the speed of sound in air is approximately

c = 331.6 + 0.6T

In making calculations involving the transmission of sound through the sea, it frequently is adequate to use a standard speed rather than the more accurate value given by equation 8-11. Although in seawater c can vary from a low of about 1,420 m/s to over 1,560 m/s depending on environmental conditions, a standard speed of 1,500 m/s may be assumed for computation purposes unless otherwise noted.

8.4.6 Field Observations of Sound Speed

Knowledge of sound velocity is important to the ASW tactician and physical oceanographer because of the effect that variations in sound velocity have upon acoustic absorption and refraction. Two different devices are in use today for finding the speed of sound in the sea. Bathythermograph. The first device is called a bathy-thermograph. As previously stated, temperature is the predominant ocean variable affecting sound speed. Not only is it relatively easy to measure, but when applied to empirical relationships such as equation (8-11), sound speed can be computed. Older BT systems employed a mechanical device that was lowered on a cable and the temperature was scribed on a smoked piece of glass. This had a number of inherent disadvantages that have been overcome through the development of the expendable bathythermograph (XBT), which does not require retrieval of the sensing unit. A diagram view of an XBT is shown in figure 8-4. It consists of a thermistor probe that is ejected from the launching platform and sinks at a known non-linear rate. The XBT is connected to a special recorder on board the launching platform by a fine wire. As it sinks, ther thermistor changes its electrical resistance with changing temperature, and as a result a temperature vs. depth trace is obtained. Because the wire uncoils from both the probe and its launcher, there is no tension on the wire until the probe has sunk to its full depth. At this point, the wire snaps and the recording stops. Variants of the basic XBT have been developed for use aboard submarines and from aircraft through incorporation into sonobuoys.

When the XBT temperature vs. depth trace is converted to sound speed vs. depth, it produces a sound speed profile very similar to that obtainable from a sound velocimeter, and is of sufficient accuracy for operational requirements. Sound Velocimeter

The second and most accurate method is the sound velocimeter. Its principle advantage is that it can measure sound speed directly, without need for conversions, by transmitting a pulse of sound over a very short path on the order of 1/2 meter or less. When the pulse arrives at the receiver, another pulse is then triggered from the transmitter; this is known as the "sing-around" or "howler" principle. The faster the speed of sound in the water in which the velocimeter is submerged, the faster the pulse will travel and the sooner it will arrive at the receiver to trigger the succeeding pulse. Since nearly all the time delay between pulses occurs as acoustic delay in the water, the PRF of the pulses is determined by the local sound speed and is directly proportional to it. Thus, knowing the path length and observing the PRF can lead directly to computation of sound speed. Until recently sound velocimeters were expensive and awkward to use, thus eliminating their use tactical-ly. The recent development of the expendable sound velocimeter (XSV) has made it possible to reduce sound velocity measurement er-rors to less than .25 meters per second at reasonable expense with-out reduction of the mobility of combatant units. Today's sophis-ticated sonars and acoustic navigation systems can provide improved information in many oceanic regions when actual sound-velocity pro-files are used rather than extrapolated sound velocity values based on temperature profiles and assumed salinity data. Based on the variability in the sea with time, a policy of regular periodic mea-surement of the velocity profile is required during an ASW opera-tion. Normally, one or two ships in the force are assigned the bathythermograph guard duty. These ships periodically measure the temperature or velocity profile, and promulgate it to all ASW units in company. These sound velocity profiles are essential in deter-mining the sound propagation paths available.

8.4.7 Typical Sound Speed Profiles

It is important to remember that while temperature is the dominant factor, the sound-speed profile is really a composite of the pressure, salinity, and temperature profiles as shown in

figure 8-6. In the area of ocean fronts, where salinity may vary up to 3 ppt. from assumed values, the use of temperature data alone may result in an error of up to 4.2 meters per second in the calculation of sound speed.

Figure 8-6. Graphical relationship of sound speed

to pressure, salinity, and temperature

A typical composite deep-sea sound-speed profile is shown in greater detail in figure 8-7. The profile may be divided into four major layers each having different thermal characteristics. Just below the sea surface is the surface layer, in which the speed of sound is susceptible to daily and local changes of heating, cooling, and wind action. The surface layer may contain isothermal water that is formed due to mixing by the action of wind as it blows across the water. Below the surface layer lies the seasonal thermocline - the word "thermocline" denoting a layer in which the temperature changes rapidly with depth. The seasonal thermocline is characterized by a negative sound-speed gradient that varies with the seasons. During the summer and fall, when the near-surface waters of the sea are warm, the seasonal thermocline is strong and well defined; during the winter and spring, and in the Arctic, it tends to merge with, and be indistinguishable from, the surface layer. Underlying the seasonal thermocline is the permanent thermocline, which is affected only slightly by seasonal changes. Below the permanent thermocline and extending to the sea bottom is the deep isothermal layer, having a nearly constant temperature of about 4oC, in which the speed of sound has a

positive gradient because of the effect of pressure on sound speed.

Between the negative speed gradient of the permanent thermocline and the positive gradient of the deep isothermal layer, there is a speed minimum toward which sound traveling at great depths tends to be bent or focused by refraction. This is the deep sound channel and will be discussed later. The refraction of sound, however, is much more complex than this simple four-layer ocean model would indicate. There are ocean eddies, fronts, interfaces between currents, underwater mountains and ridges. For instance, a submarine detected in the Labrador Current but crossing into the Gulf Stream has been compared to a person going out of an open field and disappearing into the nearby woods.


The propagation of sound in an elastic medium can be described mathematically by solutions of the wave equation using the ap-propriate boundary and medium conditions for a particular problem. The wave equation is a partial differential equation relating the acoustic pressure P to the coordinate x, y, z, and the time t, and may be written as

2P = c2 (2P + 2P + 2P) (8-12)

t2 (x2 y2 z2)

8.5.1 Normal-Mode Theory

There are two theoretical approaches to a solution of the wave equation. One is called normal-mode theory, in which the prop-agation is described in terms of characteristic functions called normal modes, each of which is a solution of the equation. The normal modes are combined additively to satisfy the boundary and source conditions of interest. The result is a complicated math-ematical function which, though adequate for computations on a digital computer, gives little insight, compared to ray theory, on the distribution of the energy of the source in space and time. Normal-mode theory is well suited for a description of sound prop-agation in shallow water, but will not be discussed in this text.

8.5.2 Ray Acoustics

The other form of solution of the wave equation is ray theory, and the body of results and conclusions therefrom is called ray acous-tics. The essence of ray theory is (1) the postulate of wave

fronts, along which the phase or time function of the solution is constant, and (2) the existence of rays that describe where in space the sound emanating from the source is being sent. Like its analog in optics, ray acoustics has considerable intuitive appeal and presents a picture of the propagation in the form of the ray diagram.

For almost all operational problems, the sound-speed gradient, with respect to horizontal changes of location, can be assumed to be zero. The major gradient of interest is the vertical gradient, dc/dz, where dz is the amount of depth change. If a source of sound at the surface of the sea radiates omnidirectionally, a wave front expanding from this source in all directions transfers energy from one particle in the water to another, and by this means the wave is propagated. If some point on this wave front is selected, and from it a line is drawn in the direction of energy propagation, then connecting these points as the wave expands in space will result in a line called a ray, as illustrated in figure 8-8.

A sound wave, or ray, which enters another medium or layer of the same medium having a different characteristic impedance, will undergo an abrupt change in direction and speed. Depending upon the angle of incidence and the abruptness of change in c, a portion of the impinging acoustic energy will be reflected off the medium boundary, and a portion will be refracted or bent passing

through the boundary. A sound ray will always bend toward the region of slower sound speed.

One of the most important practical results of ray theory is Snell's Law, which describes the refraction of waves in mediums of variable speeds. Snell's Law states that the angle of incidence, 1, at a boundary is related to the angle of refraction 2, by the following expression:

sin 1 = c1 (8-13)

sin 2 c2


c1 = sound speed in medium 1

c2 = sound speed in medium 2

If the wave is considered to be passing through three horizontal layers or strata, in each of which the sound speed is considered to be constant, then Snell's Law can be rewritten as

c1 = c2 = c3 = cn (8-14)

cos 1 cos 2 cos 3 cos n


cn = speed of sound at any point in the medium

n = angle made with horizontal at that point

Note that the angle in equation 8-14 is the complement of the angle usually expressed in Snell's basic law. It is commonly referred to as the grazing angle or angle of inclination. This expression is the basis of ray computation used by most computers, since it enables a particular ray to be "traced out" by following it through the successive layers into which the speed profile may have been divided. In a layered medium having layers of constant speed, the rays consist of a series of straight-line segments joined together, in effect, by Snell's Law.

In practice, however, temperature does not change abruptly, but rather the gradient will normally decrease or increase at a measurable rate. For such a situation, the sound speed at any depth z would be given by

c(z) = co + gz (8-15)


co = speed at the surface or transducer depth

g = speed gradient dc/dz between the surface and depth z

The net result is that, in reality, ray traces appear as curves rather than straight lines. By combining equations (8-14) and (8-15), an expression can be developed for the radius of curvature R of any ray at any point along the ray path, as shown by equation (8-16) and figure 8-11.

R = -co = c (8-16)

g gcos

Under operational conditions, values of R are very large, approaching several tens of kilometers.


8.6.1 Thermal Structure

The thermal structure of the ocean governs the refractive con-ditions for a given water mass. Despite infinite vertical temperature variations in the ocean, the temperature structure normally can be related to three basic types: (1) isothermal,

(2) negative temperature gradient, and (3) positive temperature gradient. In discussing sound propagation, it is customary to use the temperature profile as an indicator of sound speed conditions at various depths, because it has the greatest effect. It must be remembered, however, that changes in sound-beam direction result from changes in the sound-speed profile, which is influenced not only by temperature but pressure and salinity as well.

8.6.2 Direct Path

In an isothermal condition, the water's temperature is almost constant. If there is a slight decrease in temperature, and it is just balanced out by the pressure increase, the result is an iso-sound-speed condition. This causes a straight-line ray, leaving the source in lines that continue with little or no change in angle. Long ranges are possible when this type of structure is present.

When there is a negative temperature gradient, sound speed decreases with depth, and sound rays bend sharply downward. This condition is common near the surface of the sea. At some horizon-tal distance from the sound source, beyond where the rays bend downward, is a region in which sound intensity is negligible (fig-ure 8-22); it is called a shadow zone. The magnitude of the tem-perature gradient determines the amount of bending of the sound beam and thus the range of the shadow zone. For example, if the decrease in temperature to a depth of 10 meters totals 2oC or more, the shadow zone would begin beyond a horizontal range of 1,000 met-ers due to the sharp curvature of the sound beam.

When the temperature of the water has a positive gradient, sound speed increases with depth, and sound rays are refracted upward.

Longer ranges are attained with this temperature structure than with a negative gradient because the rays are refracted upward and then reflect off the surface. Unless the surface of the sea is very rough, most of the rays are repeatedly reflected at the surface to longer ranges.

Circumstances usually produce conditions where combinations of temperatures occur. One of these combinations includes a layer of isothermal water over water with a negative gradient. Approximately 90 per cent of all the bathythermograph records from all over the world show this type of thermal structure. One ray, labeled "the critical ray," becomes horizontal at the boundary or division between the isothermal layer and the negative gradient. The speed of sound is a maximum at this boundary point.

Consequently, we define the layer depth (z) as that depth of greatest sound speed (c) above the seasonal thermocline (see figure 8-7). One half of the critical beam bends toward the upper region at a reduced speed, and the other half bends toward the lower region at a reduced speed. The angle that the critical ray makes with the horizontal at the point of projection is called the critical angle.

All rays in the sound beam directed at an angle less than the critical angle will follow paths entirely within the isothermal layer and will be bent upward to the surface. All rays directed at an angle greater than the critical angle follow paths that penetrate the boundary and are subsequently refracted downward. No rays enter the region bounded by the two branches of the split critical ray, and for this reason it is also called a shadow zone. Sharp shadow zones are not fully developed because of diffraction and other effects, though the sound intensity in this area is quite low. Submarine commanders deliberately use this phenomenon, when it exists, to attempt to escape detection when approaching a target. The optimum depth for close approach to a target with minimum probability of counter-detection is approximately

Best depth = 17 Z (8-33)

where z is the layer depth in meters.

This is accurate down to a layer depth of 60 meters. Below that, the best depth for approach is a constant 60 meters below layer depth.

8.6.3 Convergence Zone

In the deep ocean, temperature usually decreases with depth to approximately 1,000 meters. Deeper than this, temperature is a constant 4oC and sound speed increases as a result of pressure. A negative speed gradient overlays a positive speed gradient, allowing equal speeds at two different depths with slower speed conditions in between. Sound originating in the thermocline, traveling nearly parallel to the surface initially, thus bends toward greater depths. But as the detected sound penetrates into deep water, it passes the region of minimum sound speed and enters the deep isothermal layer. Now the gradient in sound speed operates in the other direction; the sound path bends upward rather than downward, and the sound returns to the surface. This produces a convergence zone, where the sound waves concentrate as if they had been focused. It typically lies at a distance of about fifty kilometers from the source. Beyond this convergence zone is a second zone of silence, where again the acoustic waves diffract downward; then another convergence zone, fifty kilometers out, and so forth. The mapping of these zones is a routine part of submarine operations; by measuring deep-water temperatures witha bathythermograph, one obtains data that readily allow a computer to calculate the appropriate sound-wave paths.

When water with a negative speed gradient overlays a positive speed gradient, a sound channel is produced. Under these circumstances, any sound signal traveling in this area is refracted back and forth so that it becomes horizontally channeled. Sound rays originating with an initial upward inclination are refracted upward. Rays from a sound source in this layer that make a small angle with the horizontal are roughly sinusoidal, crossing and

recrossing the layer of minimum speed. This reinforcement of rays within the sound channel can continue until the sound is absorbed, scattered, or intercepted by some obstacle. Sounds traveling in this manner sometimes are received at extremely great distances from the source. These long ranges occur primarily as a result of two factors: absorption is small for low-frequency sound and most of the sound energy from a sound source at the axis is confined to the channel.

Under certain circumstances, a sound channel can exist near the surface of the sea. In a surface layer with a strong positive

temperature gradient the upward bending of sound rays combined with reflections from the surface will form such a channel. Sonar ranges many times greater than normal have been observed where sound channels exist. However, the conditions that produce such sound channels near the surface are rare and not very stable.

The region of minimum sound velocity, at a depth exceeding a kilometer, is the deep sound channel. It acts as an acoustic waveguide; sound propagating either upward or downward encounters sound-velocity gradients and bends back into this channel. Within the deep sound channel, sound undergoes only a cylindrical spreading loss, in which intensity drops off only as the first power of the distance. If sound spread uniformly in all directions, known as spherical spreading, this falloff would follow the square of the distance. The concept of cylindrical and spherical spreading and their importance to sonar is presented in the next section. Figure (8-18) shows the ray paths existing for a deep sound channel. Rays A and B are bounded by the sound channel. Other rays may follow propagation paths that are reflected at the sea surface and bottom (ray C), or refracted and reflected from either the sea bottom or the sea surface (rays D and E). Ray D is commonly called the refracted surface reflected (RSR) path, while ray E is called the refracted bottom reflected (RBR) path.

Figure 8-18. Sound Velocity Profile and Typical Ray

Path Diagram for a Sound Source (or Receiver) on the Axis

of the Deep Sound Channel

8.6.4 Bottom Bounce

In addition to being refracted by varying conditions in the medium, sound can be reflected in the manner of a light beam striking a mirrorlike surface and lose little of its intensity. The two surfaces that can produce this type of reflection are the surface of the water and the bottom of the ocean. Rarely if ever are these surfaces smooth enough to give a mirror reflection, but in many instances the majority of the sound is reflected as a beam. Some sonars make use of this phenomenon which is called bottom bounce. A beam is directed against the bottom from which it is reflected to the surface of the water. From the surface it is reflected back to the bottom again. Thus, the ray bounces from one to the other until its energy is dissipated or until it strikes a target and returns to the sonar. As with reflected light, the angle of reflection is equal to the angle of incidence. Obviously, ranging over a fairly flat bottom is more effective than ranging over a rough or sloping bottom. This path is highly dependent upon depth and absorption of sound by the ocean bottom.

It is obvious, then, that many sound paths are available, as indicated in the composite drawing in figure 8-20. For simplicity in the figure, the bottom-bounce path is depicted as a single beam, whereas in the real world, a wider beam corresponding to a multidirectional sound source would be the normal case.

Figure 8-20 Three Typical Sound Paths Between

a Source and a Receiver in Deep Water in Mid-North Atlantic Ocean

and Nominal Values of One-Way Propagation Loss

The convergence zone path is shown as having depth based on the concept of a wide sound beam or "bundle" of sound rays emanating from a source. By definition, the convergence zone is formed when the upper ray of the bundle becomes horizontal. This is the depth at which the sound velocity is equal to the high of the velocities at either the surface or at the bottom of the surface layer. The difference in depth between the ocean bottom and the depth at which the upper ray becomes horizontal is called the depth excess. The depth excess available defines the depth of the bundle of rays or "amount of sound" that forms the convergence zone path. As the sound energy travels upward and approaches the surface, the path narrows, tending to focus the sound energy resulting in convergence gains. A maximum of 1200 ft depth excess is required to produce an operational useful convergence zone with reliable convergence gains. (As velocity increases with depth, the corresponding amount by which the velocity of the bottom ray exceeds that of the top ray in the convergence zone path bundle is called velocity excess.) These concepts are depicted in Figure (8-21)

Although many paths are available, only by observing the environment carefully and paying close attention to his equipment will the operator be able to use them to best advantage and not find them a liability.

Figure 8.21 Depth Excess and Velocity Excess Concepts

for the Convergence Zone Sound Path


The sea, together with its boundaries, forms a remarkably complex medium for the propagation of sound. It possesses an internal structure and a peculiar upper and lower surface that create many diverse effects upon the sound emitted from an underwater source. In traveling through the sea, an underwater sound signal becomes delayed, distorted, and weakened. The transmission loss term in the sonar equations expresses the magnitude of these effects.

8.7.1 Transmission Loss

Consider a source of sound located in the sea. The intensity of the sound can be measured at any point in the sea, near to or far from the source. For purposes of measuring intensity at the source, the intensity measurement is generally taken at one unit distance from the source and labeled Io. The intensity can then be measured at any distant point where a hydrophone is located and denoted I. It is operationally significant to compare the two values. One way to do this is to form the ration Io/I. Note that if the ratio, denoted n, is greater than unity, the intensity at the source is greater than at the receiver, as would be expected. If n = Io/I, then

10 log n = 10 log Io - 10 log I

= sound intensity level at the source minus sound

intensity level at the receiver.

The value 10 log n is called the transmission loss and, of course, is measured in decibels. Most of the factors that influence transmission loss have been accounted for by scientific research, and can be grouped into two major categories: spreading and attenuation.

8.7.2 Spreading Loss

To understand spreading loss, it is convenient to imagine a theoretical ocean that has no boundaries and in which every point has the same physical properties as any other point - i.e., an infinite, homogeneous medium. In such a medium, sound energy would propagate from a point source in all directions along straight paths and would have a spherical wave front.

Under these conditions the change in power density with distance from the point source would be due only to the spherical divergence of energy. Note that there is no true loss of energy as might be implied, but rather the energy is simply spread over a progressively larger surface area, thus reducing its density. For this model, the amount of power spread over the surface of a sphere of radius r, centered at a point source, is expressed by

Pt(watts) (8-17)

Power density (watts/m2) at r = 4r2

where P, is the acoustic power level immediately adjacent to the source. This concept of power density (watts/m2) was used to develop the radar equation in chapter 2. As stated before, the units of acoustic energy are watts/m2. Therefore, equation 8-17 can be written

It = Pt (watts/m2 = P2e

4r2 pc

where Pe, is measured at range r, and Pt is the acoustic power level immediately adjacent to the source.

The intensity of the sound immediately adjacent to the source is measured, by convention, at a unit distance (1 meter) from the source. It will be labeled I1 and is given by

I1 = Pt


The acoustic intensity (Ir) at some distance, r, from the source will be less than the acoustic intensity (I1) at 1 meter from the source. This is the result of spreading a fixed amount of power over a geometrically increasing surface area (a sphere, figure 8-22). The reduction of the acoustic intensity as a function of distance, r, is shown in the ratio


Ir = 4(r)2 = 1

I1 Pt r2


However, the ratio Ir to I1 at any significant range is typically so small that these values are best examined on a logarithmic scale using the decibel.

10 log Ir = 10 log 1 = 10 log(1) - 10 log r2 = -20 log r

I1 r2


10 log Ir = 10 log I1 - 20 log r

The reduction in acoustic intensity (I1) due to range (spreading)

is called the transmission loss (TL), and for spherical spreading

TL = 20 log r (r in meters) (8-18)


10 log Ir = 10 log I1 - TL

The ocean is not an unbounded medium, however, and all sources are not omnidirectional point sources. For sources that radiate energy only in a horizontal direction, sound energy diverges more like the surface of an expanding cylinder. Also, since the ocean is bounded at the surface and bottom, cylindrical divergence is usually assumed for ranges that are large compared to the depth of the water or for when sound energy is trapped within a thermal layer or sound channel. For this model, the acoustic intensity of energy at the surface of a cylinder of radius r is expressed by

Ir = Pt watts/m2 (8-19)


where h is the vertical distance between upper and lower boundaries

(see figure 8-22). The transmission loss is therefore

10 log Ir = 10 log 1 = 10 log (1) - 10 log r = - 10 log r

I1 r


l0 log Ir = 10 log I1 - 10 log r


TL = 10 log r (8-20)


10 log Ir = 10 log I1 - TL

Equation (8-20) represents cylindrical divergence of sound energy, sometimes referred to as inverse first power spreading. Except at short ranges, it is the most commonly encountered type of spreading loss. It should be noted that the loss of intensity of a sound wave due to spreading is a geometrical phenomenon and is independent of frequency. As range increases, the percentage of intensity lost for a given distance traveled becomes increasingly less.

8.7.3 Attenuation Loss

Attenuation of sound energy in seawater arises principally through the action of two independent factors, absorption and scattering, with an additional contribution from bottom loss. Absorption. The primary causes of absorption have been attributed to several processes, including viscosity, thermal con-ductivity, and chemical reactions involving ions in the seawater. (1) The viscosity of the medium causes sound energy to be convert-ed into heat by internal friction. (2) Some sound energy is con-verted into heat because sound waves alternately raise and lower the temperatures. (3) Suspended particles are set to oscillating by the sound waves and in this process some of the sound energy is dissipated in the form of heat. This is especially the case if the particles are air bubbles. While each of these factors offers its own unique contribution to the total absorption loss, all of them are caused by the repeated pressure fluctuations in the medium as the sound waves are propagated. They involve a process of conver-sion of acoustic energy into heat and thereby represent a true loss of acoustic energy to the environment.

Experimentation has produced a factor , called the absorption coefficient, which when multiplied by the range gives the total loss in dB due to absorption. Water temperature and the amount of magnesium sulphate (MgSO4) are important factors influencing the magnitude of , because the colder the average water temperature and the greater the amount of MgSO4 present, the greater will be the losses due to absorption. However, it is the frequency of the sound wave that causes the most significant variation in the absorption coefficient. While the formula will change slightly with temperature and geographical location, an equation for the value of in decibels per meter for seawater at 5oC is

= 0.036f2 + 3.2 x 10-7f2 (8-21)

f2 + 3600


f = frequency in kHz

While this formula is rather cumbersome, the important thing to observe is that increases roughly as the square of the frequency. This relationship is of major importance to the naval tactician. It tells him that if higher frequencies are chosen for sonar operation in order to achieve greater target definition, the price he must pay is greater attenuation. The higher the frequency, the greater the attenuation and the less the range of detection. For this reason, where long-range operation of sonar equipment is desired, the lower the frequency used the better. Figure 8-23 depicts typical values of the absorption coefficient of seawater at 5oC for varying frequencies.

To obtain the transmission loss due to absorption, is merely multiplied by the range in meters. Thus,

TL = r (8-22) Scattering

Another form of attenuation is scattering, which results when sound strikes foreign bodies in the water, and the sound energy is re-flected. Some reflectors are boundaries (surface, bottom, and shores), bubbles, suspended solid and organic particles, marine

life, and minor inhomogeneities in the thermal structure of the ocean. The amount of energy scattered is a function of the size, density, and concentration of foreign bodies present in the sound path, as well as the frequency of the sound wave. The larger the area of the reflector compared to the sound wavelength, the more effective it is as a scatterer. Part of the reflected sound is re-turned to the source as an echo, i.e, is backscattered, and the re-mainder is reflected off in another direction and is lost energy. Back-scattered energy is known as reverberation and is divided into three types: volume, surface and bottom.

Volume reverberation is caused by various reflectors, but fish and other marine organisms are the major contributors. Additional causes are suspended solids, bubbles, and water masses of markedly different temperatures. Volume reverberation is always present during active sonar operations, but is not normally a serious factor in masking target echoes. The one exception involves the deep scattering layer (DSL), which is a relatively dense layer of marine life present in most areas of the ocean. During daylight hours, the layer is generally located at depths of about 600 meters and does not pose a serious problem. At night, however, the layer migrates toward the surface and becomes a major source of reverberation. It is rarely opaque to sound when detected with a sonar looking down on it from directly above, as with a fathometer, but this is not the case with a search sonar transmitting in a more or less horizontal direction. By pinging horizontally, the sound waves encounter many more organisms, and the effect can vary from partial transmission of sound to total reflection and scattering, thereby hiding a submarine.

Surface reverberation is generated when transmitted sound rays strike the surface of the ocean, i.e., the underside of the waves. It is always a factor in active sonar operations, and is directly related to wind speed because it controls wave size and the angle of incidence.

Bottom reverberation occurs whenever a sound pulse strikes the ocean bottom. In deep water this condition normally does not cause serious problems, but in shallow water, bottom reverberation can dominate the background and completely mask a close target. The amount of energy lost through scattering will vary with the roughness of the bottom and the frequency of the incident sound.

Sound reflected from the ocean floor usually suffers a significant loss in intensity. Part of this loss is caused by the scattering effects just described, but most of it results from the fact that a portion of sound energy will enter the bottom and travel within it as a new wave, as illustrated in figure 8-24. The net result is that the strength of the reflected wave is greatly reduced. The amount of energy lost into the bottom varies with the bottom composition, sound frequency, and the striking angle of the sound wave. The total of these losses can vary from as low as 2 dB/bounce to greater than 30 dB/bounce. In general, bottom loss will tend to increase with frequency and with the angle of incidence. Soft bottoms such as mud are usually associated with high bottom losses (10 to 30 dB/bounce); hard bottoms such as smooth rock or sand produce lower losses.

While it is possible to derive equations that will compute precise values of TL, associated with each of these additional scattering and bottom loss factors, the ocean characteristics are so variable that there is little utility in doing so. It is customary, therefore, in operational situations, to make an educated guess as to their values and lump them together into one term "A," known as the transmission loss anomaly, which is included in the transmission loss equation.

8.7.4 Total Propagation Loss

It would be useful to have a simple mathematical relationship that would describe all the effects of the various factors influencing transmission loss as they occur in the ocean. But the state of the physical conditions encountered in the ocean are very complex and not at all easy to represent. A few mathematical models do exist that provide close approximations for some sets of conditions, but at present, no single model accounts for all the conditions encountered.

A simplified model used to obtain approximate values of transmission loss for the spherical spreading case is

TL = 20 log r + r + A (8-23)

and for the cylindrical spreading case

TL = 10 log r + r + A (8-24)

It is important to realize that sound transmission in the ocean is three-dimensional and that transmission loss versus horizontal range alone is not sufficient information for most operational situations. Areas of no sonar coverage occur at various intervals of range because of refraction, reflection, and interference between waves traveling via different paths. Therefore, while the TL equations are interesting and somewhat useful, they are not always totally accurate.


Background noise, like reverberation, interferes with the reception of desired echoes. Unlike reverberation, however, it does not result from unwanted echoes of the transmitted pulse but from active noise-makers located in the ship or in the water.

Noise produced by these sources is classified as self-noise and ambient noise. Self-noise is associated with the electronic and mechanical operation of the sonar and the ship. Ambient noise encompasses all of the noises in the sea.

8.8.1 Self-Noise

Self-noise is produced by noisy tubes and components in the sonar circuitry, water turbulence around the housing of the transducer, loose structural parts of the hull, machinery, cavitation, and hydrodynamic noises caused by the motion of the ship through the water. Machinery Noise. The dominant source of machinery noise in a ship is its power plant and the power distribution system that supplies power to the other machinery on the vehicle, such as compressors, generators, propellers, etc. Machinery noise is normally always present and is kept to a minimum by acoustically isolating the various moving mechanical components.

The gearing connecting the propellers is an important source of machinery noise. If the engine runs at a relatively low speed, as is the case with a reciprocating heat engine, gears may not be required between the engine and the propeller. High-speed power sources, however, such as steam or gas turbines, usually require reduction gears to the propeller. The frequency of the explosions in the cylinders of a reciprocating engine is not likely to be a source of ultrasonic (above 15 kHz) noise but might be an important source of low-frequency sonic noise. A more crucial source of noise in the reciprocating engine is the clatter of the valves opening and closing. In gas turbines, the noise generated is in the ultrasonic region at a radian frequency equal to the angular velocity of the turbine buckets.

Noise in the ultrasonic region is extremely important in sonar and acoustic torpedo performance. The noise produced by auxiliary units, such as pumps, generators, servos, and even relays, is often more significant than the power-plant noise. The large masses involved in the power plant usually keep noise frequencies relatively low. For this reason, a small relay may interfere more with the operation of a torpedo than an electric motor that generates several horsepower. Small, high-speed servomotors, however, may be serious sources of ultrasonic noise. Flow Noise. Flow noise results when there is relative motion between an object and the water around it. This flow is easiest to understand by assuming that the object is stationary and that the water is moving past it. Under ideal conditions such that the object is perfectly streamlined and smooth, water movement will be even and regular from the surface outward as shown by the flow lines in figure 8-25A. This idealized condition is called laminar flow and produces no self-noise. Irregular objects can achieve nearly laminar flow conditions only at very low speeds (i.e., 1 or 2 knots or below).

As flow speed increases, friction between the object and the water increases, resulting in turbulence (figure 8-25B) and progressively increasing noise due to fluctuating static pressure in the water. Thus we have, in effect, a noise field. If a hydrophone is placed in such a region, fluctuations of pressure will occur on its face, and it will measure the result in flow noise in the system.

As pressures fluctuate violently at any one point within the eddy, they also fluctuate violently from point to point inside the eddy. Moreover, at any given instant the average pressure of the eddy as a whole differs but slightly from the static pressure. Thus, very little noise is radiated outside the area of turbulence, and although a ship-mounted hydrophone may be in an intense flow-noise field, another hydrophone at some distance from the ship may be unable to detect the noise at all. Flow noise, then, is almost exclusiverly a self-noise problem.

Actually, not much information is known about flow noise, but these general statements may be made about its effect on a shipborne sonar:

1. It is a function of speed with a sharp threshold. At very low speeds there is no observable flow noise. A slight increase in speed changes the flow pattern from laminar to turbulent, and strong flow noise is observed immediately. Further increases in speed step up the intensity of the noise.

2. It is essentially a low-frequency noise.

3. It has very high levels within the area of turbulence, but low levels in the radiated field. In general, the noise field is strongest at the surface of the moving body, decreasing rapidly with distance from the surface.

4. The amount of flow noise can be related to the degree of marine fouling on the ship's bottom and sonar dome. Fouling is the attachment and growth of marine animals and plants upon submerged objects. More than 2,000 species have been recorded, but only about 50 to 100 species are the major troublemakers. These nuis-ances can be divided into two groups - those with shells and those without. Particularly important of the shell forms are tube-worms, barnacles, and mollusks because they attach firmly and resist being torn loose by water action. Nonshelled forms such as algae, hy-droids, and tunicates commonly attach to relatively stationary ob-jects and therefore do not cuase much trouble on ships that fre-quently get underway.

Most fouling organisms reproduce by means of the fertilization of eggs by sperm, and the resulting larvae drift with the currents. Depending on the species, the larval stage may vary from a few hours to several weeks. Unless the larvae attach themselves to a suitable surface during this period, they will not grow into adulthood.

Geography governs fouling by isolating species with natural barriers. Temperature is the most important factor governing distribution of individual species, limiting reproduction or killing adults.

On a local scale, salinity, pollution, light, and water movement affect the composition and development of fouling communities. Most fouling organisms are sensitive to variations in salinity, developing best at a normal seawater concentrations of 30 to 35 parts per thousand. Pollution, depending on type, may promote or inhibit fouling. Fouling algae and microscopic plants, the food supply of many fouling animals, are dependent upon sufficient light for proper growth.

To forestall fouling on ship bottoms and sonar domes, the Navy uses special paints with antifouling ingredients. The most common active ingredient of antifouling paint is cuprous oxide, and the effective life of the paint is from 2 to 2 1/2 years. Cavitation. As the speed of the ship or object is increased, the local pressure drops low enough at some points behind the object to allow the formation of steam. This decrease in pressure and the resulting bubbles of vapor represent the onset of cavitation. As the ship moves away from the bubbles, however, the pressure increases, causing the bubbles to collapse and produce a continuous, sharp, hissing noise signal that is very audible.

Because the onset of cavitation is related to the speed of the object, it is logical that cavitation first appears at the tips of the propeller blades, inasmuch as the speed of the blade tips is considerably greater than the propeller hub. This phenomenon, known as blade-tip cavitation, is illustrated in figure 8-26.

As the propeller speed increases, a greater portion of the propeller's surface is moving fast enough to cause cavitation, and the cavitating area begins to move down the trailing edge of the blade. As the speed increases further, the entire back face of the blade commences cavitating, producing what is known as sheet cavitation, as shown in figure 8-27.

The amplitude and frequency of cavitation noise are affected considerably by changing speeds, and changing depth in the case of a submarine. As speed increases, so does cavitation noise. As depth increases, the cavitation noise decreases and moves to the higher frequency end of the spectrum in much the same manner as though speed had been decreased. This decrease in noise is caused by the increased pressure with depth, which inhibits the formation of cavitation bubbles.

A given propeller at a given depth of submergence will produce no bubbles unless its speed exceeds a certain critical value. When the speed (RPM) exceeds this critical value the number of bubbles formed increases rapidly, but not according to any known law. The critical speed itself, however, depends in a simple manner on the depth of the propeller beneath the sea surface. This dependence is given by

No2 = constant No - critical speed

d d - depth

Thus, if a given propeller begins to cavitate at 50 rpm when at a depth of 15 ft, it will begin to cavitate at 100 rpm when at a depth of 60 ft.

Since all torpedo homing systems and many sonar systems operate in the ultrasonic region, cavitation noise is a serious problem. Torpedoes generally home on the cavitation noise produced by ships, and any cavitation noise produced by the torpedo interferes with the target noise it receives. Because the speed at which a vehicle can operate without cavitating increases as the ambient pressure is increased, some acoustic torpedoes are designed to search and attack from depths known to be below the cavitating depth. In the same way, a submarine commander, when under attack, will attempt to dive to depths at which he can attain a relatively high speed without producing cavitation. Quieting

It is important for an ASW vessel to operate with a very low noise level. Noise reduction methods can be applied to other military vessels, and to that matter to commercial vessels. For protection against acoustic mines and to reduce the risk of being detected early by sonar listening devices, noise minimizing is necessary. A low level of noise also permits own ship to receive a lower level of signal from the enemy. Strict levels on the limits of structureborne noise and of airborne noise levels improve the habitability for the ship's crew. Surface Quieting. Considerations must be given to noise emission planning in design of the ship. Attention must be paid to the layout of the propellers and the hull's appendages with regard to waterborne noise generation. Model tests on propellers are necessary to check cavitation and to shift the cavitation point to a relatively high speed. Above this speed a propeller air emission system can be used to reduce cavitation-generated waterborne noise. In the U.S. this bubble injection system is called Prairie Masker. Prairie is an acronym for propeller air-induced emission. A system on some U.S. vessels for injecting air through hull girths is called Masker. This reduces somewhat the flow noise. See figure 8-28.

Other measures can be taken to reduce noise:

- Reducation in structure borne noise transmission through the provision of low tuned flexible mounts for critical items combined with high impedance foundations.

- Encapsulation of all diesel prime movers, compressors and the gas turbines. Provision of high attenuation silencers for the exhaust ducts of all diesel prime movers and the gas turbines as well as for the combustion air inlets of the cruising diesels and the gas turbines.

Special attention must be paid to diesel units, the compressors, and converters positioned near the ship's sonar. As those are considered to be the most critical structureborne noise sources, they have been double elastic-mounted. In addition, the intermediate foundation frame structure of the diesel units is damped with reinforced concrete blocks at the foundation springs.

The design of the main reduction gear units is important to noise reduction. The rotating parts and meshes were machined to the highest accuracy to minimize gear contact noise. The reduction gear assemblies are hard elastic-mounted. In the design of elastic mounts attention must be paid to tuning with regard to the potential vibration exciting sources of the ship and its equipment. The behavior of the mounts under shock and rough sea conditions has to be examined. Submarine Quieting. Self noise is a limiting factor in sonar performance, both from a counter detection point of view, the point at which an ememy is able to detect you, and with respect to target detection and tracking. Thus, there has been a strong trend toward building quieter submarines.

Significant improvement was realized in using a "bed plate" or acoustically-isolated foundation for the gears, turbines, condens-ers and turbo generators of the propulsion plant. Its noise-iso-lating mountings, however, added a good deal of volume and weight when compared with the conventional alternative of mounting the machinery directly to the hull.

The reduction gearing, which converts the horsepower of the rapidly-rotating turbine shaft to that of the slowly-rotating propeller, is a particularly strong noise source. An attempt was made to replace this gearing with a heavy but quiet turbine-electric drive system, featuring a turbo generator that drives a low-noise electric motor of low rotational speed and high torque. This arrangement, however, proved to be less efficient than the standard geared-turbine system.

Significant headway has been made in reducing propeller cavit-ation. However, recent advances in submarine quieting, due to im-provements in propeller fabrication, have been neutralized as this technology has been more widely disseminated than hoped.

Further silencing gains have been found in the use of a natur-al-circulation nuclear reactor, pump-jet propulsors, and hull-mounted sound-absorbing acoustic tiles.

8.8.2 Ambient Noise

Ambient noise is background noise in the sea due to either natural or man-made causes, and may be divided into four general categories: hydrodynamic, seismic, ocean traffic, and biological. Hydrodynamic Noise. Hydrodynamic noise is caused by the movement of the water itself as a result of tides, winds, currents, and storms. The level of hydrodynamic noise present in the sea is directly related to the condition of the sea surface. As the surface becomes agitated by wind or storm, the noise level rises, reducing detection capability. Very high hydrodynamic noise levels caused by severe storms in the general area of the ship can result in the complete loss of all signal reception. Seismic Noise. Seismic noises are caused by land movements under or near the sea - as, for example, during an earthquake. They are rare and of short duration, and hence will not be elaborated upon. Ocean Traffic. Ocean traffic's effect on ambient noise level is determined by the area's sound propagation characteristics, the number of ships, and the distance of the shipping from the area. Noises caused by shipping are similar to those discussed under the heading of self-noise, with the frequencies depending on the ranges to the ships causing the noise. Noises from nearby shipping can be heard over a wide spectrum of frequencies, but as the distance becomes greater, the range of frequencies becomes smaller, with only the lower frequencies reaching the receiver because the high frequencies are attenuated. In deep water, the low frequencies may be heard for thousands of kilometers. Biological Noise. Biological noises produced by marine life are part of ambient background noise and at times are an important factor in ASW. Plants and animals that foul the ships are passive and contribute to self-noise by increasing water turbulence. Crustaceans, fish, and marine mammals are active producers of sounds, which are picked up readily by sonar equipment.

During and since World War II, a great deal of research on sound-producing marine animals has been carried out. The object was to learn all the species of animals that produce sound, their methods of production, and the physical characteristics of the sounds (frequencies, intensities, etc.). Sounds produced by many species have been analyzed electronically, and considerable physical data have been obtained.

All important sonic marine animals are members of one of three groups crustaceans, fish, and mammals.

Crustaceans, particularly snapping shrimp, are one of the most important groups of sonic marine animals. Snapping shrimp, about 2 centimeters long, bear a general resemblance to the commercial species, but are distinguishable from them by one long, large claw with a hard movable finger at the end. They produce sound by snapping the finger against the end of the claw. Distribution of snapping shrimp appears to be governed by temperature, and they are found near land in a worldwide belt lying between latitudes 35oN. and 40oS. In some places, such as along the coast of Europe, they range as far north and south as 52o. The largest colonies or beds of snapping shrimp occur at depths of less than 30 fathoms on bottoms of coral, rock, and shell. There are exceptions, however. They may, for example, occur as deep as 250 fathoms, and have been found on mud and sand bottoms covered with vegetation.

A shrimp bed is capable of producing an uninterrupted crackle resembling the sound of frying fat or burning underbrush. Frequencies range from less than 1 to 50 kHz. Noise is constant, but there is a diurnal cycle, with the maximum level at sunset. Over beds, a pressure level of 86 db re 1 Pa has been noted. Intensity drops off rapidly as the range from the bed increases. Lobsters, crabs, and other crustaceans may make minor contributions to background noise.

Fish produce a variety of sounds that may be placed in three categories, depending upon how the sounds are caused. The first category includes sound produced by the air bladder, a membranous sac of atmospheric gases lying in the abdomen. The sound is caused by the movement of muscles inside or outside the bladder or by the general movement of the body. The second division includes sounds produced by various parts of the body such as fins, teeth, and the like rubbing together. This noise is called stridulatory sound. The third class includes sounds that are incidental to normal activities, such as colliding with other fish or the bottom while swimming, biting and chewing while feeding, and so on.

The majority of the sonic fish inhabit coastal waters, mostly in temperate and tropical climates. Although fish are the most pre-valent, and therefore the most important, sound producers, their activity is not as continuous in intensity as that of snapping shrimp. The level of sound produced by them increases daily when they feed (usually at dawn and dusk), and annually when they breed. Fish sounds range in frequency from about 50 to 8000 Hz. Sounds of air bladder origin have most of their energy concentrated at the lower end of this spectrum, 75 to 150 Hz, whereas stridulatory sounds characteristically are concentrated at the higher end of the spectrum.

Marine mammals, in addition to returning echoes from sonar equipment, produce sound vocally and stridulously. Seals, sea lions, and similar animals posses vocal cords, and bark or whistle or expel air through their mouths and nostrils to produce hisses and snorts. Whales, porpoises, and dolphins force air through soft-walled nasal sacs, and the sounds produced in this way have been described as echo-ranging pings, squeals, and long, drawn-out moans similar to a foghorn blast. Other sounds probably produced by stridulation and attributed principally to whales, porpoises, and dolphins are described as clicking and snapping.

8.8.3 Wenz Curve.

While difficult to determine accurately on an operational basis, ambient noise levels are nonetheless important factors to be considered in determining sonar performance. Figure 8-29is an example of a set of Wenz curves that can be used to estimate noise levels from a variety of sources within the frequency range of interest. Figure 8-30 is an expansion of the shipping noise/wind noise portion of the Wenz curves.


A SOSUS station is searching for an enemy submarine known to be producing sound at a frequency of 300 Hz. It is suspected that the sub is patrolling in shipping lanes that currently have 6 ft. seas. Use Wenz curves to determine an approximate value of ambient noise.

From figure 8-30 the ambient noise level due to shipping at 300 Hz is 65 dB and that due to 6 ft seas is 66 dB.

Using the nomogram, figure 8-2 combine the signals of 65 dB and 66 dB

IL1 - IL2 = 66-65 = 1

From the nomogram add 2.4 dB to 66 dB ... 68.4 dB.


The key to success in antisubmarine warfare is initial detection. For this work the major sensor in use today is sonar, both active and passive, and the present state of knowledge of the physical world suggests no change in this situation for many years. An understanding of sonar can only be achieved through a comprehension of the sonar equations and the concept called figure of merit. Many of the phenomena and effects associated with underwater sound may conveniently and logically be related in a quantitative manner by the sonar equations. For many problems in ASW, the sonar equa-tions are the working relationships that tie together the effects of the medium, the target, and the equipment, so that the operator can effectively use and understand the information received and provide prediction tools for additional information. Therefore, the purpose of this section is to spell out the sonar equations and figure of merit, to state the specifics of their usefulness, and to indicate how the various parameters in the sonar equations, in-cluding the figure of merit, can be measured.

8.9.1 Signal to Noise Ratio.

The sonar equations are based on a relationship or ratio that must exist between the desired and undesired portion of the received energy when some function of the sonar set, such as detection or classification, is performed. These functions all involve the re-ception of acoustic energy occurring in a natural acoustic back-ground. Of the total acoustic energy at the receiver, a portion is from the target and is called signal. The remainder is from the environment and is called noise. The oceans are filled with noise sources, such as breaking waves, marine organisms, surf, and dis-tant shipping, which combine to produce what is known as ambient noise. Self-noise, in contrast, is produced by machinery within the receiving platform and by motion of the receiving platform through the water. Further, in active systems, scatterers such as fish, bubbles, and the sea surface and bottom produce an unwanted return called reverberation, which contributes to the masking of the desired signal.

The function of the design engineer is to optimize the signal-to-noise (S/N) ratio for all conditions as detailed in the original design specifications of the sonar set. The operator, using his knowledge of the design specifications, his known ability in certain circumstances, the predicted conditions extrapolated from previously determined measurements, and actual on-board measurements, can then predict the detection probability.

In order to predict performance, the operator's interaction with the sonar set must be defined or quantified in a manner that provides a measure of predictability for varying signal and noise levels. This quantity, known as Detection Threshold (DT), attempts to describe in a single number everything that happens once the signal and its accompanying noise are received at the sonar. Detection threshold is defined as the signal minus noise level required inboard of the hydrophone array in order that an operator can detect a target. Actually, the business of detecting a sonar signal is a chance process for several reasons, one of which is that a human being is involved. The decision to call a target may be either right or wrong if a target is really present, then there is a detection; if a target is not present, then there is a false alarm. Hence, the definition of DT is normally qualified by adding the requirement that an operator "can detect a target on 50 percent of those occasions for which a target presents itself." Thus, if the average value of provided signal-to-noise equals the average of required signal-to-noise, a detection occurs in 50 percent of the times that a detection could occur. To summarize:

If average provided = average required, then detection probability is 50%

If average provided > average required, then detection probability is 50% to 100%

If average provided < average required, then detection probability is 50% to 0%

Note that the instantaneous value of the provided or required signal-to-noise can vary over a wide range due to the variability of operators, and an individual operator's moods, as well as time fluctuations in propagation loss, target radiated signal, and own ship noise. Hence, while the average value of provided signal-to-noise may be less than the average value of required signal-to-noise, at times the instantaneous value may be greater than the required value, and a detection may occur. Thus, a probability of detection greater than zero exists.

Putting this all together, it can be seen that if detection is to occur with a specified degree of probability, then the signal, expressed in decibels, minus the noise, expressed in decibels, must be equal to or greater than a number, the Detection Threshold, which also is expressed in decibels.

S - N > DT (8-25)

This equation is the foundation upon which all the versions of the sonar equations are based, and is simply a specialized statement of the law of conservation of energy.

The next step is to expand the basic sonar equation in terms of the sonar parameters determined by the equipment, the environment, and the target.

Two pairs of the parameters are given the same symbol (Own Sonar Source Level/Target Source Level, and Self-noise Level/Ambient-noise Level) because they are accounted for similarly in the sonar equations. This set of parameters is not unique, nor is the symbolism the same in all publications, but they are the ones conventionally used in technical literature. It should be noted as the discussion progresses how each of these parameters will fit into the mold of the basic sonar equation.

Parameter Active/Passive Symbol Determined by

Own sonar source level Active SL Equipment

Self noise level Active/Passive NL Equipment

Receiving Directivity Active (Noise Limited

Index Passive DI Equipment

Detection Threshold Active/Passive DT Equipment

Transmission Loss Active (2TL) TL Environment

Passive TL

Reverberation level Active RL Environment

Ambient Noise Level Active NL Environment


Target Strength Active TS Target

Target Source Level Passive SL Target


The transmission loss is frequency dependent as shown in Section 8.7. When calculating the acoustic signal resulting from a signal of interest superimposed on the ambient background noise of the ocean (signal to noise ratio), this calculation must be made using signals of the same frequency to be valid. Since sonar systems are designed to operate in specific frequency bands, calculations for these systems must relate to the design frequency band of the specific sonar to be valid.

8.9.2 Passive Sonar Equation

A passive sonar depends on receiving a signal that is radiated by a target. The target signal can be caused by operating machinery, propeller noise, hull flow noise, etc., but the same fundamental signal-to-noise ratio requirement must be satisfied. At the receiver, the passive equation begins as S - N > DT. If the target radiates an acoustic signal of SL (Target Source Level), the sound intensity is diminished while en route to the receiver because of any one or more of the following: spreading, ray path bending, absorption, reflection, and scattering. The decrease in intensity level due to this is called Transmission Loss (TL) and is also measured in decibels. Hence the intensity level of the signal arriving at the ship is

S = SL - TL (8-26)

Noise, N, acts to mask the signal and is not wanted. Therefore, the receiver is composed of many elements, sensitive primarily in the direction of the target so that it can discrimminate against noise coming from other directions. This discrimination against noise can be referred to as a spatial processing gain and is called the Receiving Directivity Index, DI. DI gives the reduction in noise level obtained by the directional properties of the transducer array. Therefore, in the basic equation, noise is now reduced and becomes

N = NL - DI (8-27)

There are two things to note in this simple equation: the first is that DI is always a positive quantity, so that NL - DI is always less than or equal to NL; the second is that the parameter NL represents both Self-noise Level and Ambient-noise Level, for by its definition it is the noise at the hydrophone location and can come from any, or all, sources.

The passive sonar equation can now be constructed in terms of signal and noise. When S and N are substituted from equations (8-26) and (8-27) into (8-25) the result is

SL - TL - NL + DI > DT (8-28)

which is the simplest form of the passive sonar equation. In words, equation 8-28 says that the source level of the target minus the loss due to propagation through the medium, minus the sum of all interfering noises plus improvement by the spatial processing gain of the receiver, must be equal to or greater than the detection threshold for a target to be detected with the specified probability of detection. However, the greater-than or equal-to condition is normally written as an equality. It is then understood to mean that if the left-hand side's algebraic sum is greater than DT, detection is possible with a greater probability than that specified by DT. If the sum is less than DT, detection probability decreases. Generally speaking, these two conditions imply that either detection is highly probable or seldom occurs. As a further aid to understanding the passive sonar equation, figure 8-31 illustrates the sonar parameters and indicates where

each term involved interacts to produce the desired results.

8.9.3 Active Sonar Equation

In an active sonar, acoustic energy is transmitted, and the received signal is the echo from the target. Two different, but related, equations are needed to describe the active sonar - one for an ambient-noise-limited situation and the other for the reverberation-limited situation. As developed previously, sonar performance is governed by the requirement that signal minus noise must be equal to or greater than detection threshold. The differ-ence in the two active sonar equations that satisfy this require-ment depends upon the characteristics of the noise that is actually present at the receiver when the signal is detected. The ambient noise may be described as either isotropic-i.e., as much noise power arrives from one direction as from any other - or as reverb-eration, in which noise returns primarily from the direction in which the sonar has transmitted.

Before developing the active sonar equations, the two types of noise should be briefly explained. Ambient noise consists of those noises present even when no sound is being radiated by a sonar. These include such noises as sea animals, machinery, propulsion noises generated by the echo-ranging platform, and the turbulence generated in the vicinity of the sonar. This type is the same as the noise level term discussed in the passive sonar equation. The second type, reverberation, consists of a multiplicity of echoes returned from small scatterers located in the sound beam and near the target when they reflect the transmitted energy. The combined effect of echoes from all of these scatterers produces a level of noise at the receiver that will tend to mask the returning echo from any wanted target. Noise-Limited. The development of the active sonar equation is similar to that for the passive equation. In other words, the formal sonar parameters will be fitted to the signal and noise terms of equation (8-25). If a sonar transmits an acoustic pulse with an initial source level of SL dB, the transmitted pulse will suffer a transmission loss in traveling to the target. The target will scatter acoustic energy, some of which will return to the sonar. The back-scattered intensity is called target strength and is related to the scattering cross section of the target. The returning echo will again undergo a propagation loss, and thus the signal at the sonar will be

S = SL - 2TL + TS (8-29)

As long as the source of the radiated energy and the receiver for the echo are located together, the transmission loss experienced is equal to twice the one-way transmission loss.

When the echo returns, under some conditions the reverberation background due to the initial transmission will have disappeared, and only ambient noise will be present. This noise will be identical to that described in the passive sonar equation, modified by the receiving directivity index. The fundamental relationship can then be expressed as

SL - 2TL + TS - NL + DI > DT (8-30)

which is the basic active sonar equation used when the sonar is operating in a noise-limited situation. Reverberation-Limited. If, on the other hand, the echo returns when the reverberation background has not decayed to a level below the ambient noise level, the background noise is given by RL. In this case, the parameter DI, defined in terms of an isotropic background, is inappropriate, inasmuch as reverberation is by no means isotropic. For a reverberation background the terms NL - DI are replaced by an equivalent reverberation level observed at the hydrophone terminals, and the sonar equation takes the form.

SL - 2TL + TS - RL > DT (8-31)

which is known as the reverberation-limited active sonar equation. Detailed quantification for the new term, RL, is difficult at best, for it is a time-varying function resulting from the inhomogeneties in the medium. One thing to note is that in the normal conversion from the basic equation to the active equations, the inequality again becomes an equality. As discussed under the passive sonar equation, it is understood that when the terms on the left-hand side exceed the detection threshold by a significant amount, detection is highly probable, and when it is significantly less than the detection threshold, detection seldom occurs. Figure 8-31 pictorially depicts the active sonar equations.

Of special interest in the active sonar equations is the term TS, and the fact that it usually is on the order of 15 to 25 dB. The variability of the value of TS is a function of the target aspect presented to the incoming signal. A beam target presents a greater reflective area then the bow, hence more energy is reflected from a beam target then from a bow target. Because TS is 10 times the logarithm of the reflected intensity divided by the inbound intensity, this statement apparently says that more energy is reflected than is incident, a condition clearly not possible.

The key lies in the definition of terms:

I reflected is the intensity, I, of the reflected signal measured one meter from the target, assuming the target is a point source.

I inbound is the intensity, I, of the signal inbound from transmitting ship to the target measured at a point on the target.

Intensity is actually power per unit area striking the target at some point, and thus the total sound power striking the target is I inbound times an effective area. If one assumes that the major portion of this power is reflected not from the original effective area (which is almost the same as the profile area of the target), but instead from a point source, it necessarily follows that the reflected energy computed in this way must be greater because of the reduced area from which the energy emanates. Thus I reflected is greater than I inbound if both are defined as indicated above.

In this case, there is no such wave as the one that is defined as originating from the point source. This construct is merely a convenient way of duplicating the actual measured value of I re-flected when the wave is 1,000 meters or more away from the point source enroute back to the transmitting ship. Thus, if one were to measure I reflected and I inbound both at 1,000 meters from the target, then I inbound would definitely be greater than I reflect-ed. Therefore, I reflected would have been computed to suffer a greater attenuation in traveling 1,000 meters from the constructed point source than I inbound will suffer in going 1,000 meters to the target. The explanation for this is the rapid attenuation due to spreading from the point source as compared to that undergone by the inbound wave, which is very near a plane wave when it is within 1,000 meters of the target.


The possible detection range of particular equipment should be known so that a tactician will then have a measure of the sonar's capability and a feel for what the sonar can do in a given tactical situation. Unfortunately, with no change in basic sonar configuration its detection capability measured in terms of range can increase or decrease severalfold simply because the ocean itself has changed. To state this another way, sonar equipment can only be designed to detect the arrival of a certain sound energy intensity. The range from which that sound intensity arrives is highly dependent on how much energy was lost en route, and therefore detection range alone is a poor measure of sonar capability. A better measure is the ability of the sonar to detect a certain level of sound energy intensity just outboard of its receiver. The key to this better measure of performance is to separate the sonar from the ocean in which it must operate. Only then can sonar capability be discussed in terms of the unchanging sonar hardware as distinguished from the ever-changing ocean.

The better measure for sonar capability is called figure of merit (FOM), and it equals the maximum allowable one-way transmission loss in passive sonars, or the maximum allowable two-way transmission loss in active sonars for a detection probability of 50 percent. Therefore, solving equations (8-28) and (8-30) for transmission loss, we get

Passive FOM = SL - NL + DI - DT (8-32)

Active FOM = SL + TS - NL + DI - DT (8-33)

This combination of terms is probably the most used performance parameter for sonars, and it is important to understand just what it means. The FOM of a sonar system is the maximum transmission loss that it can tolerate and still provide the necessary detection probability as specified by DT. FOM is improved by raising the source level (this can be accomplished by increasing the transmitted power in the active case or finding a noisier target in the passive case), decreasing the ambient noise level, increasing the absolute value of the receiving directivity index, and decreasing the detection threshold. The value of figure of merit is that, with no knowledge of the intervening propagation path between a ship and a target, a quantitative comparison of two different sonars can be made. The figure of merit may be utilized for comparing the relative performance of two passive sonars provided the calculations of the comparative figures of merit are made for the same frequency. The difference in FOM's represent additional allowable propagation loss that can be sustained by the sonar having the higher FOM and still make a detection; thus a longer detection range results.

8.10.1 Prop Loss Curves

Naturally, to the tactician, detection ranges are of prime importance, and although FOM can be interpreted in terms of range, it can be done only if the propagation losses involved are known. The measure of propagation loss (total transmission loss) in many parts of the ocean has proven that not only does sound propagation change with frequency and location but with season as well. Thus, the sound velocity profile will determine the propagation path(s) available, which along with frequency and location will determine transmission loss versus range. Therefore, in order to convert FOM to range, one must have a propagation loss curve for the frequency or the sonar (s) concerned and for the area and season of the year for which range prediction is desired.

Figure 8-33 is a typical propagation loss curve for the waters off Iceland in the summer for a frequency of 2 Khz. Note that the propagation losses for the three sound paths previously described are plotted. Multiple bottom bounce losses were measured and mul-tiple convergence zones were estimated.

Assume three different passive sonars with the following values of FOM at 2 Khz: 80, 95, 105. Based on Figure 8-32, the range in Kyds at which 50% probability of detection is predicted for each sonar for each sound transmission path in the Iceland area is as follows:

Sonar (FOM) Direct Bottom Bounce Convergence Zone

A 80 7.5 None None

B 95 21 50 81

C 105 36 90 160

Note that the sonar with the higher figure of merit permits the use of sound paths (bottom bounce and convergence zone) not available to the sonar with relatively low figure of merit. The higher the FOM the longer the detection range for a given path. Propagation loss curves can be made from a ray tracing program or actual measurements and smoothed data.

In summary, sonar performance is the key to ASW success, and figure of merit is the key to sonar performance. With knowledge of his sonar's FOM, a commanding officer can ensure that his equipment is peaked, and also predict detection ranges against possible enemy targets. The war planner can do likewise for either a real or hypothetical enemy. Because the changing ocean results in dramatic changes in propagation loss versus range, to state a sonar's capability in terms of range is only half the story, and may even be misleading. Using figure of merit, however, the sonar with the higher FOM will always be the better sonar when comparing sonars in the same mode.

8.10.2 Figure of Merit Sample Problem

Your sonar is capable of either passive or active operation. You are operating in the shipping lanes with a sea state of 2. Water depth is 200 fathoms. Using the following information, you must decide which mode to use. Intelligence information indicates that the threat will be a Zebra-class submarine (all dB are reference 1Pa).

Target parameters:

radiated noise source level 100 dB

radiated noise frequency 500 Hz

target strength 15 dB

target detection range l0,000 m

Sonar parameters: Active Passive

Source level 110 dB -

Frequency 1.5 kHz -

Self-noise at 15 kts 50 dB 50 dB

Directivity index 10 dB 8 dB

Detection threshold -2 dB 3 dB

In order to determine which sonar to use, it is necessary to calculate the FOM and total transmission loss for each mode.

First, calculate the total transmission loss for each mode. Since the desired target detection range of 10 km is much greater than the water depth of 200 fathoms, we will use equation (8-24) for cylindrical spreading:

TL = 10 log r + r + A

The quantity A is assumed to be zero since no information is available. The absorption coefficient () is calculated by substituting signal frequencies for each mode into equation (8-21):

= .036f2 + 3.2 x 10-7 f2 (where f is in kilohertz)

f2 + 3600

Active mode Passive mode

= .036(1.5)2 = .036(.5)2

(1.5)2 + 3600 (.5)2 + 3600 + (3.2 x 10-7)(1.5)2 +(3.2 x 10-7)(.5)2

= 2.32 x 10-5 = 2.6 x 10-6

TL = 10 log(10,000) + 2.32 TL = 10 log(10,000) + 2.6 x

x 10-5(10,000) 10-6(10,000)

TL = 40 + .232 TL = 40 + .026

TL = 40.232 (one way) TL = 40.026dB (Total TL)

2 x TL = 80.464dB (Total TL)

Note that TL depends only upon the detection range and the frequency of the signal.

Next calculate the FOM using equation (8-33) for the active mode and equation (8-32) for the passive case. Note that the only values that must be determined are the noise levels using the Wenz curves and nomograms for the different frequencies.

Active Mode Passive Mode

f = 1500 Hz f = 500 Hz

AN(shipping) = negligible AN(shipping) = 57dB

AN(sea state) = 58dB AN(sea state) = 61dB

Self-Noise = 50dB Self-Noise = 50dB

Using the nomogram, combine the signals for each mode:

58-50 = 8dB 61 - 57 = 4 dB

NL = 58 + .65 (from nomogram) AN = 61 + 1.5 (from nomogram)

NL = 58.65 dB AN = 62.5 dB

62.5 - 50 = 12.5 dB

NL = 62.5 + .25 = 62.75 dB

Note that when three noise signals are involved, a two-step signal-combining process is required. The resultant is always added to the higher signal level. Determining the FOM is now a simple matter of substituting the calculated and given values into the appropriate equations:

Active Mode Passive Mode

FOM = SL + TS - NL + DI FOM = SL - NL + DI - DT

- DT FOM = 100 - 62.75 + 8 - 3

FOM = 110 + 15 - 58.65 + FOM = 42.25 dB

10 -(-2)

FOM = 78.35 dB

Compare the FOM of each mode with the total TL for each mode to determine which mode is optimum for this target. The FOM for the active case is less than the total TL. Therefore, the active mode will give your ship less than 50% probability of detection. The FOM for the passive case is greater than the total TL. Therefore, the passive mode will give a greater than 50% probability of detection, which means that the passive mode should be used.


Of all the energy forms available, sound, even with its inherent disadvantages, is the most useful for underwater detection of submarines. It travels as a series of compressions and rarefactions at a speed equal to the product of its frequency and wavelength. The pressure of the wave can be expressed as a function of both time and its distance from the source. Acoustic power, called its intensity, is a measure of the amount of energy per unit area and has the units watts/m2. Acoustic pressure is expressed in micro-pascals. To make comparisons easier, both are normally converted to the logarithmic decibel system. The speed of sound in the sea is related to the bulk modulus and density of the water, which are affected by the temperature, pressure, and salinity. Temperature is the most important of these environmental factors, and therefore the thermal structure of the ocean is of significant tactical importance. The tracing out of sound paths in water is known as ray theory and is governed by Snell's Law. Various unique propagation paths can be identified according to the thermal structure of the water, but in practice such paths are a complex combination of simpler structures. Actual sound propagation through the sea is subject to geometric spreading and attenuation, both of which decrease the acoustic intensity at the receiver. The active and passive sonar equations are an expression of various factors determined by the equipment, the medium, and the target, which lead to an overall measure of sonar performance called figure of merit.


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