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CHAPTER 1

BASIC PRINCIPLES OF UNDERWATER SOUND

Introduction

 

One of the most critical problems encountered by the Allies early in World War II was the submarine menace. Almost five thousand merchant ships were sunk and more than twenty million tons of war supplies were lost by enemy action. The struggle against enemy submarines was successful because we were able to detect and locate them whether they were surfaced, submerged, underway, or lying in wait.

The majority of enemy submarines attacked were detected and located by sonar. To date, sonar has been the most effective method of detecting completely submerged submarines. Other methods such as radio, radar, and infrared, have proved ineffective because their range of transmission in sea water is practically nil.

The development of SONAR during the period between the wars was an unspectacular, slow but steady conquest over the physical elements of the

  sea, culminating during and since World War II in one of the Navy's largest research and development programs. The word "SONAR" abbreviates SOund, Navigation, And Ranging, and includes all types of underwater sound devices used for listening, depth indication, echo ranging, ship-to-ship underwater communication, and other uses. The importance of sonar in naval warfare cannot be overemphasized.

This text is divided into two general parts-(1) a brief discussion of the physics of sound propagation in an ideal medium, followed by a presentation of the peculiarities and limitations of sea water as a medium for the transmission of sound, and (2) a general study of the design and function of representative sonar equipments.

In planning this text it has beep assumed that the reader has a knowledge of elementary physics, mathematics, and electronics.

 
Characteristics of Sound in an Ideal Medium
 
IDEAL MEDIUM

The peculiarities and limitations of sound transmission in sea water are understood more easily if sound is first thought of as being transmitted in an ideal medium. Such a medium is assumed to be homogeneous, infinite in extent, and perfectly elastic. A homogeneous medium has the same properties throughout, such as temperature, pressure, salinity, and density. The infinite extent of the medium permits omission of boundary reflections. Perfect elasticity means that the medium, when distorted or displaced, returns to equilibrium with no loss of energy by internal friction. It is obvious that the properties of sea water differ greatly from those of an ideal medium. Nevertheless, a discussion of sound transmission in the two media is advantageous.

 
SOUND AND PRESSURE WAVES

Most wave motion can be classified as either longitudinal or transverse. Sound waves are longitudinal and are characterized by the vibrating particles of the medium moving forward and backward parallel to the direction in which the waves are propagated.

The waves are composed of alternate compressions and rarefactions in the medium. The term "sound" is used in two senses-subjectively, it denotes the auditory sensation experienced by the ear, and objectively, it denotes the vibratory motion which gives rise to that sensation. This motion is often called the stimulus. All stimuli do not produce sensations of hearing, because the average ear responds to sounds in the frequency range from approximately 16 cycles per second to

 
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15,000 cycles per second. This range is known as the sonic frequency range. Frequencies above 15,000 cycles per second, although not within the range of response of the average ear, are useful and can be detected with proper instruments. These frequencies are known as ultrasonic frequencies. The dividing line is not sharply defined; many people-particularly young persons-can hear above 15,000 cycles per second; but some standard must be adopted, and 15,000 cycles per second is used as the arbitrary dividing line. Before rockets and aircraft attained speeds greater than the speed of sound, frequencies above 15,000 cycles per second were designated as supersonic. However, it is now agreed that the term "supersonic" designates velocities greater than the velocity of sound and the term "ultrasonic" means frequencies above 15,000 cycles per second.

EQUATION OF WAVE MOTION

The period, T, of a vibrating particle in a medium is the time in which it completes one vibration, and the frequency, F, is the number of vibrations completed per second. Frequency is expressed as "cycles per second (cps)," "kilocycles (kc)," and "megacycles (mc)." In such units "cycles" is understood to mean "vibrations per second." The maximum displacement from the undisturbed position is called the amplitude of vibration.

Two wave motions vibrating with the same frequency have definite phase relations. They are in phase when they continue to pass through corresponding points of their paths at the same time. For any other condition they are out of phase. They are in phase opposition when they reach their maximum displacement in opposite directions at the same instant.

The wavelength is the distance, measured along the direction of propagation, between two corresponding points of the wave train.

The general relation that exists among the frequency of vibration, the velocity of propagation, and the wavelength of wave motion in any medium is equally applicable to the propagation of sound waves in sea water. A body which is vibrating at a definite rate produces a disturbance that moves away as a wave in the surrounding medium. In the time, T, the vibrating body completes one vibration, and the wave advances a distance equal

  to its wavelength, λ, so the velocity of the wave is v=λ/T. Because the period, T, is the reciprocal of the frequency, F, it follows that the wave velocity is

v=Fλ.   (1-1)

In this equation the wave velocity, v, is determined completely by the properties of the transmitting medium and is independent of the frequency of the source and of the wavelength. When F changes there must be a corresponding change in λ so that the equation may be satisfied.

INTENSITY OF SOUND WAVES

The intensity of sound waves is proportional to the amount of energy passing per second through unit area at right angles to the direction of propagation. Both kinetic and potential energy are present in a sound wave. The average kinetic energy equals the average potential energy, and the total energy at any time equals twice the average of either kinetic or potential, or the total energy equals the maximum of either kinetic or potential. Therefore the total energy of the sound wave may be determined by computing the maximum kinetic energy of all the molecules which are moving back and forth out of their equilibrium positions as the wave passes. If the sound wave is simple harmonic motion, the maximum velocity, u, of a vibrating particle of the transmitting medium is 2πaF, where a is the amplitude, and F the frequency. The maximum kinetic energy of one particle, which also equals the total energy, E, of this particle, is

Emu2m(2πaF)2  ergs/particle.

Let the density of the medium be ρ gm/cm3. Then if the density, ρ, is substituted for the mass, m, the result is the energy density or the energy per unit of volume-

E=½ρu2=½ρ(2πaF)2
E=2π2ρa2F2  ergs/cm3.  (1-2)

The loudness of a sound wave, which determines the strength of sensation, and its ease of reception depend upon the intensity, I, which is the energy

 
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transmitted per second per unit area perpendicular to the direction of propagation. In 1 second the sound wave disturbs a volume of medium of length v, where v is the velocity of propagation. The intensity is therefore the energy in a column of medium of length v and unit cross section, the volume of which is v cm3. Equation (1-2) for energy per unit volume or the energy density must be multiplied by v to find the intensity-

I=2π2ρa2F2v ergs/cm2sec.  (1-3)

From this result it is seen that the intensity of sound is proportional to the (1) square of the amplitude, (2) density of the medium, (3) velocity of propagation, and (4) square of the frequency of vibration.

A more practical concept of sound intensity is in terms of pressure variations which occur at all points in the transmitting medium as the sound wave advances. The greater the pressure variations, the more intense is the sound wave. It can be shown that the intensity is proportional to the square of the pressure variation at all frequencies. For most practical purposes, sound intensity in terms of pressure units is preferred to energy density or energy flow because of the ease of measuring sound pressure.

  Decibel System

The values of pressure, p, encountered in practice, range from about 10-4 to 106 dynes/cm2. It is customary to express sound levels, L, in terms of the logarithm of sound-intensity ratios-

L=C log(I/Io) = C log(p2/po2),  (1-4)

where C is a constant that depends upon the units used, and Io is a specified value of sound intensity that is chosen as a standard of reference. In practice, C is taken as 10 and the corresponding unit for L is called the decibel (db). It follows that the pressure level of sound, or simply the sound level, L, is given by the equation,

L=10 log I = 20 log p db.  (1-5)

The logarithm is to the base 10.

Two reference units of pressure are in common use. These units are 1 dyne/cm2 and 0.0002 dyne/cm2. Both units have been used with underwater sound, but 1 dyne/cm2 is used here, unless otherwise stated. In keeping with international practice, the unit 0.0002 dyne/cm2 is used as a reference intensity for airborne sound.

If I exceeds Io, L is positive, or the sound level is said to be "up" L db with respect to reference level Io. If I is less than Io, L is negative, or the sound level is said to be "down" L db with respect to reference level Io.

 
Sound Propagation in an Ideal Medium
 
INVERSE SQUARE LAW

In a study of underwater sound it is important to understand how the sound intensity varies as the waves advance out from the source. Consider the most elementary condition-that of a very small radially pulsating sphere being placed in the medium. Its waves spread out spherically and affect the whole space occupied by the medium. If E (watts) is the total energy emitted from the source per second, the sound intensity I' at a concentric spherical surface of radius, r' (yards), is

I'=E/(4πr'2) watts/yd2.

At any other concentric surface of radius r" (yards), the sound intensity I" is similarly expressed-

  I"=E/(4πr"2) watts/yd2

or

(I'/I")=(r"2/r'2).   (1-6)

Thus the sound intensity at any surface varies inversely as the square of the distance of that surface from the sound source. This relation is commonly known as the inverse square law.

The inverse square law can be stated more simply by letting I be the intensity at any range, r, and I1 the intensity at unit range (source intensity). Equation (1-6) becomes

I=I1/r2.  (1-7)

Equation (1-7) expressed in the decibel system becomes

L=L1-20 log r,  (1-8)

 
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where "L=10 log I" is the sound level (db) at range r and "L1=10 log I1" is the sound level at unit range. The quantity, L1, is called the source level.

Because graphic presentation of data is often necessary for the interpretation of the principle, it is helpful to become familiar with the appearance of the foregoing equation plotted in different ways. The inverse square law, as expressed in equations (1-7) and (1-8), can be presented graphically in various ways.

In figure 1-1 the abscissa is proportional to r,

I/I<sub>1</sub> as a function of range.
Figure 1-1. - I/I1 as a function of range.

and the ordinate is I/I1 or 1/r2. This method of presentation is not useful because the graph approaches too close to the horizontal axis to be visible beyond about 10 yards.

This objection is overcome by plotting

L-L1 = 10 log (I/I1) = -20 log r  (1-9)

as ordinate against r as abscissa. Such a graph is shown in figure 1-2. The expansion of the scale for small values of I/I1 into large negative values of L-L1, makes such a graph useful over a wider interval of ranges.

A third type of graph also uses L-L1 as the ordinate but uses log r instead of r as abscissa. Figure 1-3 is the graph of equation (1-8) plotted in this way. This graph has two advantages-a much greater interval of ranges can be presented, and the graph of the inverse square law is a straight line.

The foregoing discussion of the inverse square law was based upon the assumption of a point

 

L-L1 as a function of range.
Figure 1-2. - L-L1 as a function of range.

source of sound. It must be recognized that the character of the sound field is altered materially with a departure from a point source. In the ocean, the sound sources that come into consideration differ widely. Hulls of ships are large sources, emitting noise with a complicated spectrum. Sound projectors are moderately large sources, emitting relatively pure tones or sound of controlled frequency bands. Very small bubbles of air may become secondary sources of sound.

Any source can be considered to be divided into elementary areas, each of which acts as a point source of sound. If the linear dimensions of the source are small compared to the wavelength of the sound, the differences in the distances from a remote point in the sound field to any two elementary areas on the surface are small compared to the wavelength. Thus waves from the two elementary areas arrive at the remote point substantially at the same time. Under this condition the waves from the elementary areas add. The sound, moreover,

L-L1 as a function of range with the range being
plotted on a logarithmic scale.
Figure 1-3. - L-L1 as a function of range with the range being plotted on a logarithmic scale.

 
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is radiated uniformly in all directions. Under this condition the source can be called small.

If a source of simple harmonic waves is large compared to 1 wavelength, the waves do not arrive at a given point at the same time. Hence, there are interference effects, and the intensity radiated in some directions is greater than that radiated in others. It will be shown later that these interference effects are the basis for directional projectors.

If the source is large, for example a ship, and emits noise rather than single-frequency sound, the more obvious interference effects disappear. The intensity radiated in some directions, however, is still different from that radiated in others.

INFLUENCE OF DIRECTIVITY

A problem of basic consideration in sonar is the control of the distribution of sound energy radiating from a source. The reader is now familiar with the inverse square law and the general deviations from it when the sound source is not small. If the sound energy emitted by a source is confined to a cone or beam of small angle, the intensity is greater at a given distance than it would be in the case of a point source radiating uniformly in all directions. Such concentration of sound energy within a narrow beam is called directional transmission.

Equation (1-8) which gives the sound level in any direction at a range, r, may be altered to give the level in a directional sound field. This alteration could be accomplished by assigning a different value to the source intensity, I1, for each direction; however, a simpler procedure is to designate the intensity at 1 yard in an arbitrary direction as the source intensity. The intensity in any other direction can then be obtained by multiplying by an appropriate factor determined by the direction. In the case of a sound projector that concentrates most of its energy in a beam, the value of the intensity at 1 yard from the source in the direction of the axis of the beam is considered to be the source intensity. Let the source intensity be Ia and let the intensity at 1 yard from the source in a direction making an angle θ with the axis of the beam be equal to I1(θ) (read "I1 of θ"), and the ratio of I1(θ) to Ia be equal to b(θ). Thus-

  b(θ)=I1(θ)/Ia.  (1-10)

Then equation (1-7) becomes

I=I1(θ)/r2

I=Iab(θ)/r2  (1-11)

In converting to the logarithmic form, let La be 10 log Ia. La is called the axial source level. Because b(θ) is usually a proper fraction, its logarithm is negative and represents a reduction in sound level. To avoid confusion in use of signs, it is better to express this reduction as a positive number and subtract it than to add it as a negative number. It is therefore defined as B=-10 log b(θ). Thus equation (1-11) converted to logarithmic form becomes

L=La-B-20 log r db.  (1-12)

The quantity, B, is called the beam pattern, or directivity function. If equation (1-10) is expressed in logarithmic form, B is defined by

B=La - L1  (1-13)

At all points on the axis, B=0, because b(θ) is unity. Under this condition, equation (1-12) becomes equation (1-8).

Figures 1-4 and 1-5 show polar graphs of the function b(θ) and B for the same projector. These graphs have been calculated theoretically for a vibrating rectangular plate, the side of which is about 4 wavelengths long.

The graph of b(θ) (figure 1-4) shows that most of the sound is projected in directions which make

Beam pattern of a projector.
Figure 1-4. -Beam pattern of a projector.

 
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Beam pattern of figure 1-4, with the ratio
expressed in decibels.
Figure 1-5. -Beam pattern of figure 1-4, with the ratio of (I1(θ)/Ia) expressed in decibels.

angles of less than 10° with the perpendicular to the plate. The radii represent the value of b(θ). Note the difficulty of showing the side lobes.

However, the very weak radiation at greater angles is important in some cases. Consequently, the graph of B (figure 1-5) is useful, because the logarithmic scale emphasizes these small intensities.

In figure 1-5 the maxima M, 1, 2, and others not shown are called lobes. M is the main lobe. For sonar bearings to be accurate, the main lobe should be narrow. The side lobes, 1 and 2, are detrimental for many purposes, and in the design of modern projectors side lobe suppression is an important consideration. The dotted curves show the result of lobe suppression. With modern designs, the maxima of all side lobes are usually more than 20 db below the maximum of the main lobe. Graphs like those in figures 1-4 and 1-5 are drawn for projectors from actual measurements of sound level in different directions. They are called directivity patterns. Projectors and directivity are discussed in more detail in chapter 2.

DEVIATION FROM INVERSE SQUARE LAW

The method by which the inverse square law is modified to represent the intensity or level of a sound field when the source does not radiate uniformly in all directions, has just been described. Deviation from the inverse square law when the sound transmission is in an actual medium such as the water of the ocean will now be considered.

  The ocean, taken as a medium for the transmission of sound, is far different from the ideal condition assumed previously. The extent of the ocean is limited, being bounded by the surface and the bottom. It is not homogeneous-the upper layers are usually warmer than the lower ones and near the mouths of large rivers the salinity is greatly reduced. Because of both these facts the water is less dense in the upper layers. The temperature and salinity may change also in a horizontal direction. The pressure increases with depth. These changes in the physical character of the ocean cause variations in the velocity of sound waves being transmitted in the ocean.

Other less obvious acoustic properties of the ocean contribute to making the calculation of sound intensity difficult. As a sound wave travels outward from a source in the sea, some of the energy is converted into heat by friction because of the viscosity of the water. This process is called absorption. Another portion of the energy goes into the production of secondary wavelets which travel in directions other than that of the primary wave. This phenomenon is called scattering. A more general term, embracing both absorption and scattering, is attenuation.

It is possible to measure the total transmission loss and to observe how it deviates from the inverse square law value of the ideal medium. To measure transmission loss the axial source level, La, of the transmitting ship is kept constant and sound level L is measured at the receiving ship.

The difference

H=La-L  (1-14)

is the loss in level suffered by the sound in being transmitted from one ship to the other and is usually called the transmission loss. Except, for sign and the effect of attenuation, this transmission loss is the same quantity as that plotted in figures 1-1 and 1-2.

Such an experiment shows that equation (1-12) does not accurately represent the actual transmission loss. The difference between the observed value of H and that calculated from equation (1-12) is thus a measure of the departure of the ocean from an ideal medium. This departure is often called the transmission anomaly.

It is sometimes difficult to isolate the effects of the beam pattern from other factors affecting transmission loss. Consequently, a more practicable

 
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definition of transmission anomaly is the difference between the observed transmission loss and the transmission calculated from the inverse square law alone, without taking into account the directivity effect. The directivity effect is thus included in the transmission anomaly defined by

A=H-20 log r  (1-15)

whence the actual sound level can be calculated from the equation

L=La-A-20 log r.  (1-16)

The usefulness of this concept of transmission anomaly is illustrated by figures 1-6 and 1-7.

Figure 1-6. -Comparison of transmission loss observed in an
experiment with that calculated from the inverse square law.
Figure 1-6. -Comparison of transmission loss observed in an experiment with that calculated from the inverse square law.

These figures are based on experimental data obtained under special conditions.

The solid curve of figure 1-6 is a graph of observed transmission loss H and, for comparison, the transmission loss calculated from the inverse square law is plotted as a dotted curve. The difference between these two curves does not seem very great, and would hardly be noticed if the dotted curve were omitted; yet the difference is very important in echo ranging.

Suppose the echo from a certain submarine can just be detected by a certain sonar equipment when the transmission loss is 70 db. If the inverse square law were valid, it could be detected out to 3,000 yards, but under actual transmission conditions it could not be detected beyond 1,250 yards, unless some other factor happened to be especially favorable at moderate ranges.

In figure 1-7 the increasing departure of the transmission loss from the inverse square law, as

  Figure 1-7. -The same experimental date as in figure 1-6,
plotted as transmission anomaly.
Figure 1-7. -The same experimental date as in figure 1-6, plotted as transmission anomaly.

range increases, is immediately apparent. Furthermore, a simple law is also obvious-the transmission anomaly, A, is proportional to range. Under favorable conditions, the transmission anomaly can be represented by the simple equation

A=ar  (1-17)

where a is an empirical constant called the attenuation coefficient.

Defined in this way, the transmission anomaly measures the difference in the transmission loss of sound from an actual source in the ocean and the loss of sound transmitted to the same range by a small source in an ideal medium. Besides the effect of directivity, other components of the transmission anomaly are:

1. Sound energy is converted into heat because of the viscosity of the water. This process is called absorption.

2. Variation in temperature and salinity cause changes in density, which, as the hydrostatic pressure increases with depth, result in variation of the velocity of the sound and consequent refraction of the sound rays.

3. The scattering of sound by reflection from the surface, by the bottom, and by particles suspended in the body of the ocean is a very important factor. A distinction should be made between specular, or regular, reflections-as from the surface and from the bottom-and the diffused reflections from the particles-ordinarily designated by the term "scattering."

4. Other factors about which little is known may contribute to the transmission anomaly.

 
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Refraction of Sound
 
VELOCITY OF SOUND IN SEA WATER

In the foregoing discussion the refraction of sound in sea water was mentioned as an important factor in the transmission of sound in the ocean. In a homogeneous medium sound would travel in straight lines. As in the analogous case of light, the path of a sound wave is curved if the velocity of propagation is not the same at all points. A plane wave that enters another medium obliquely undergoes a change in direction, if the velocity of the wave in the second medium is different from that in the first. One part of the wave travels faster than the other and the wavefront is bent toward the medium of lower velocity. This phenomenon is called refraction. The ordinary laws of geometrical optics can be applied to the refraction of sound, although they are strictly true only for sounds of very high frequency, and do not take into account such phenomena as scattering, diffraction, reflection, and absorption. Although these phenomena cannot be ignored, it is simplest to omit them in initial discussions.

The velocity of sound in a liquid medium may be computed from the elasticity modulus, E, and density, ρ, of that medium-

v=square root(E/ρ).  (1-18)

If E and p are in the British system of units, the velocity is in feet per second. As indicated by

Variation of the density of sea water with temperature and salinity.
Figure 1-8. -Variation of the density of sea water with temperature and salinity.

  equation (1-18) the ratio of elasticity modulus to density of any transmitting medium determines the sound velocity in that medium. Any influence which changes either factor to give a change in the E/ρ ratio has a corresponding effect on the velocity.

The E/ρ ratio is governed by temperature, pressure, and salinity and the velocity must be evaluated for any given set of conditions. An increase in any of these factors will increase the sound velocity, although this increase is not directly proportional. Temperature, for example, ordinarily affects density to a greater degree than it affects the elasticity modulus. Thus, the higher the temperature of the medium the lower the density and the higher the velocity. Of the three factors (temperature, pressure, and salinity) that control the variables, E and ρ, in equation (1-18), temperature is by far the most important in sound transmission in sonar practice.

Note in figure 1-8 that the density changes at a variable rate with temperature. Thus, at constant salinity, the velocity increases with the temperature at a variable rate.

Figure 1-9 shows the variation with temperature for three salinities. Changes of 20° F, in the upper layer of the ocean are not uncommon. An increase in salinity of 1 part in 1,000 increases the velocity of sound 4.27 ft./sec. In most cases, however, the effect of salinity can be neglected

Variation of the velocity of sound in sea water
with temperature at three values of salinity.
Figure 1-9. -Variation of the velocity of sound in sea water with temperature at three values of salinity.

 
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Variation of temperature with depth. A, Typical
slide; B, temperature-depth graph.
Figure 1-10. -Variation of temperature with depth. A, Typical slide; B, temperature-depth graph.

because salinity is comparatively constant except at the mouths of large rivers.

Increase of pressure with depth causes an increase in the speed of sound of 1.82 feet per second per 100 feet of depth. Figure 1-10, A, shows a typical bathythermograph slide with grid superimposed. The pressure effect is important only if both the temperature and the salinity are constant. This effect is shown in figure 1-10, B, in which the solid line shows how the temperature varies with depth in a particular case and the dotted line indicates the change in the velocity of sound with depth corresponding to this temperature distribution. The salinity effect is negligible. The effect of pressure on the velocity of sound in the isothermal layer of the upper 180 feet is evident from the velocity graph which shows a slight increase in the velocity with depth. Elsewhere

  the velocity curve parallels the temperature curve quite closely.

At greater depths, temperature and salinity change only slightly, and the pressure effect dominates. The average temperature decreases with depth, as shown in figure 1-11, and down to about 2,500 feet, this decrease is sufficiently great to neutralize the effect of increasing salinity and pressure, so that the velocity of sound also decreases. At greater depths, the pressure effect begins to outweigh the temperature effect, and the sound velocity is seen to increase with depth. This minimum velocity at great depths has interesting acoustic consequences.

Horizontal and Vertical Changes

In considering temperature changes in the sea, it can be assumed that only variations in a vertical direction are significant. On this thesis the ocean may be considered as consisting of strata, in any one of which the same temperature exists over a large horizontal distance. Compared with vertical variations of temperature, the horizontal variations actually observed are very small. Changes in temperature over a horizontal distance of 100 feet are rarely as much as 0.5° F and usually less than 0.1° F. Furthermore, they are not systematic

Figure 1-11. -Variation of temperature, salinity, and sound
velocity with depth in the ocean.
Figure 1-11. -Variation of temperature, salinity, and sound velocity with depth in the ocean.

 
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On the other hand, over a vertical distance of 100 feet the temperature may vary as much as 10° F, as figure 1-10 shows.

It is now evident that temperature distribution with depth is the dominant factor in determining conditions for sound transmission in sea water. The bathythermograph was developed to determine this distribution. The bathythermograph is frequently referred to by the abbreviation BT. It is rugged and convenient in size, and can be lowered over the side for use while the vessel is underway. Furthermore, as it is lowered into the sea, the bathythermograph automatically draws a graph showing the temperature as a function of depth.

A functional schematic of the bathythermograph is shown in figure 1-12. As the instrument is lowered, a stylus is moved by the thermal expansion

Schematic of the bathythermograph.
Figure 1-12. -Schematic of the bathythermograph.

or contraction of a liquid in the copper thermometer tube (thermal element). The increasing hydrostatic pressure compresses a bellows, which drives a smoked glass slide at right angles to the stylus which is driven by the thermal element. Thus a permanent graphical record of temperature against depth is obtained as the instrument is lowered and raised in the ocean. Figure 1-10, A, shows a typical slide with a coordinate grid superimposed; figure 1-10, B, is the temperature-depth graph made from the trace on the slide. Such temperature-depth graphs are called bathythermograms.

Twelve typical bathythermograms are shown in figure 1-13. These bathythermograms illustrate the variable character of the temperature distribution in the surface layers of the ocean. Examination of these charts shows that the temperature-depth curve can be subdivided usually into segments having different temperature gradients. Layers in which the temperature is uniform are called isothermal layers (figure 1-13, A). Negative

 

Typical bathythermograms
Figure 1-13. -Typical bathythermograms corresponding to various gradients. A, Isothermal surface layer; 13, negative temperature gradient in surface layer; C, positive temperature gradients.

gradients (figure 1-13, B) describe conditions in layers in which the temperature decreases with depth. Positive gradients (figure 1-13, C) describe conditions in layers in which the temperature increases with depth.

 
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A layer in which the temperature decreases very rapidly-particularly if it is immediately beneath an isothermal layer or a layer of smaller gradient-is commonly called a thermocline. The decrease in temperature which always occurs at great depth is sometimes called a permanent thermocline.

SNELL'S LAW OF REFRACTION

It has been pointed out how a sound beam is bent or curved from a straight path if it passes obliquely from one layer of sea water to a second layer where the velocity is different from that in the first layer. With a method of determining the velocity of sound at each point in the sea, it is theoretically possible to calculate the sound rays, or paths, along which the sound travels. If, for simplicity, the ocean is assumed to be stratified so that the temperature at all points having the same depth is the same, the calculation becomes quite simple.

No attempt is made here to give a detailed explanation of the computational methods. The computation is based on the familiar Snell's law of refraction that is discussed in all textbooks of physics as it applies to light rays. Figure 1-14 shows an especially simple case of three layers, or strata, in each of which the sound velocity is constant.

If a plane wave is considered to be passing through these three layers, Snell's law is

v1/cos(θ1) = v2/cos(θ2) = v3/cos(θ3)   (1-19)

Diagram illustrating Snell's law
Figure 1-14. -Diagram illustrating Snell's law

  where v1 and θ1 are the velocity and inclination of the ray in the first layer, and so on. Note that the angle of inclination, θ, is the complement of the angle usually given with Snell's law. The ray in each layer is a segment of a straight line; but if the layers are allowed to become very thin, the ray approaches a smooth curve. At each point along the ray, however, the relation between the inclination of the ray and the velocity of sound is still given by equation (1-19).

TYPICAL RAY DIAGRAMS

Because most velocity distributions can be approximated for series of layers from bathythermograms, an approximate ray construction can be carried out with the aid of Snell's law as indicated. Such a ray diagram represents the sound field produced by sound energy transmitted from a sonar projector. If an underwater target is located within the bounds of the ray diagram a return echo may be received at the sonar vessel.

Marked Downward Refraction

A ray diagram for typical conditions of sharp downward refraction is shown in figure 1-15. It should always be borne in mind that the curvature of the rays is greatly exaggerated because of the necessary contraction of the horizontal scale. In figure 1-15 the ratio of horizontal to vertical scale is 75 to 1.

Figure 1-15. -Ray diagram with sharp downward refraction.
Figure 1-15. -Ray diagram with sharp downward refraction.

 
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Diagram of part of figure 1-15 drawn with undistorted scale.
Figure 1-16. -Diagram of part of figure 1-15 drawn with undistorted scale.
Figure 1-16 shows a portion of the same diagram drawn on an undistorted scale.

The contracted horizontal scale also exaggerates the inclination of the rays with the horizontal. This inclination is shown in figure 1-17, the numbers being the true angles in degrees and the lines showing the angles as plotted on the diagram. The part of the beam above the axis is considered to have positive inclination; the part below the axis, negative inclination. In a directional transducer, nearly all of the energy is concentrated in a cone of about 10° opening. Hence a judicious selection of rays with initial inclinations of 5° or 6° on either side of the axis provides a sufficiently complete picture of the paths followed by the sound rays.

The velocity-depth graph of figure 1-15 shows three layers in which the velocity gradient is constant. The projector is at a depth of 16 feet. The following three rays are drawn:

1. The ray that leaves the projector at -6°, and which may be considered as the lower boundary of the main lobe of the projected beam of sound. The dimensions of the diagram do not permit the inclusion of the +6° (upper bounding) ray.

2. The ray that leaves the projector horizontally-the axial or 0° ray. This ray is shown bent sharply downward.

3. The ray that leaves the projector at +1.4°. This angle was chosen because this ray is tangent to the surface.

These three rays are also shown on figure 1-16 with an undistorted horizontal scale. The most striking feature of this ray diagram is that all the sound is confined to a very limited region and beyond about 500 yards from the projector the surface casts a shadow. The explanation of this shadow follows.

The outer rays of the upper half of the sound beam fall on the surface and are reflected there. A ray of a certain critical inclination is refracted

  downward so that its inclination when it reaches the surface is zero. Any rays with inclinations greater than this critical value are reflected back by the surface inside the region bounded by the ray tangent to the surface.

A ray with less initial inclination does not reach the surface but curves down inside the critical ray; the 0° ray illustrates this point. The critical ray in the present example is the 1.4° ray. It bounds the direct sound field and for this reason is called the limiting ray.

Except for sound scattered or diffracted from the direct sound field, the shadow should be a region of silence. This picture is approximately a true one; observations made under conditions of strong downward refraction show a sharp drop of from 30 to 40 db in the sound level near the range indicated by the limiting ray.

Diagram showing how the inclination of the rays
is distorted in the conventional ray diagram.
Figure 1-17. -Diagram showing how the inclination of the rays is distorted in the conventional ray diagram.

 
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Isothermal Layer and Thermocline

Another common type of thermal distribution is shown in figure 1-10. This figure shows an isothermal layer at the surface, below which a sharp negative gradient occurs. In the isothermal layer, the velocity gradient is positive because of the pressure effect, as shown in figure 1-10, B. About 90 percent of the bathythermograph records taken all over the world show this type of thermal structure. The sound-velocity graph and ray diagram corresponding to this example are shown in figure 1-18.

Ray diagram for an isothermal surface layer.
Figure 1-18. -Ray diagram for an isothermal surface layer.

Theory predicts a shadow, limited by the ray which is horizontal at the level of maximum velocity. The rays above the limiting rays are refracted upward and are ultimately reflected at the surface. Those below the limiting ray enter the thermocline and are there refracted downward. The sound beam is split along the limiting ray into an upper and a lower section; hence the term "split-beam pattern" is commonly applied to this type of ray diagram.

The shadow beyond the limiting ray might be expected to be a region of relative silence, as in the previous case. Actually the shadow in figure 1-18 differs from that in figure 1-16 in that it is penetrated by surface-reflected rays such as those designated by A and B. Because the surface reflects approximately all the incident sound energy, it is obvious that the shadow in figure 1-18 is not so complete as the one in figure 1-15. In the second velocity graph, the corner at the point of maximum velocity, C, is actually round instead of being sharp as shown. When this rounding is properly introduced to the diagram the "shadow" is found to be a region into which few rays, rather than none at all, penetrate.

Experiments show that there is no noticeable shadow under these conditions. The intensity at

229276°-53-2

  a given depth decreases gradually with increasing range and shows no abrupt drop as the limiting ray is crossed. The intensity gradient is much greater below the "splitting" point than above.

Other thermal structures result in the sound field conditions illustrated by the ray diagrams in figures 1-19 and 1-20.

Sound field bounded by two limiting rays.
Figure 1-19. -Sound field bounded by two limiting rays.

Figure 1-19 illustrates the case in which two limiting rays bound the field.

Figure 1-20 shows a velocity distribution resulting in what is called a sound channel. All rays leaving the projector between rays A and B are alternately refracted up and down. The rays are thus confined to a certain layer, to which the term "sound channel" is applied. Transmission losses in sound channels are exceptionally low, and extremely long ranges are possible.

In the open sea, sound channels are rare and transitory in the upper layers, because the thermal conditions causing them are unstable. Near the mouths of large rivers, where salinity conditions cause changes in sound velocity, it is possible to have stable sound channels in the surface layers.

Formation of a sound channel.
Figure 1-20 -Formation of a sound channel.

At great depths, where the temperature is practically constant, the pressure effect causes the sound velocity to increase with depth and there is a permanent sound channel. The extremely long ranges that are possible with low-frequency sound signals in this permanent sound channel are utilized in a long-range position-fixing system that uses signals from explosions set off at the depth of the sound channel. A full description of this system is given in chapter 16.

 
13

Calculation of theoretical intensities for typical ray diagram. A, Bathythermogram; B, ray diagram; C, intensity
contours; D, anomaly graph for several depths.
Figure 1-21 -Calculation of theoretical intensities for typical ray diagram. A, Bathythermogram; B, ray diagram; C, intensity contours; D, anomaly graph for several depths.
Ray Divergence

The effects of refraction have been presented in black-and-white pictures of silent shadows and regions of direct or reflected sound. This concept comes from the earliest form of theory on which echo-range predictions were based. However, it has since been found that the shadows are not silent and that there are marked variations within the field of direct sound.

Even before this experimental knowledge was obtained, attempts had been made to enlarge the ray theory to enable the calculation of intensity changes in the direct field. This intermediate theory is still useful for some purposes even though it also predicts completely silent shadows that are not observed.

The results of these theoretical calculations can be presented graphically in several ways, as illustrated in figure 1-21. Figure 1-21, A, is a typical bathythermogram showing an isothermal layer and thermocline. The corresponding ray diagram is shown in figure 1-21, B. Figure 1-21, C, shows a series of contours on which the sound level is constant. These contours are identified by the

  values of transmission loss in db. Above the thermocline they represent the loss calculated from the inverse square law. In general, above the thermocline, these contours are farther from the projector than they are below the thermocline, and they are more widely spaced above than below. Throughout the whole shadow (shaded area) the calculated intensity is zero, and the transmission loss is consequently infinite.

Another method of presenting the results is shown by figure 1-21, D. The transmission anomaly was calculated for various points. If the depth is held constant-for example at 70 feet-and its distance from the source is allowed to vary, the series of values obtained can be plotted as a curve. See the curve marked "70 feet" in figure 1-21, D. These graphs are smooth curves when the depth is greater than that of the thermocline. When the point is above the thermocline, the transmission anomaly is practically zero until the point reaches the shadow zone where it suddenly becomes infinite. The discontinuous change in the transmission anomaly is due partly to the approximate velocity-depth curve used in the

 
14

calculation. If these approximations were eliminated from the calculation, the change at the shadow would not be so abrupt.

The very marked increase in the transmission anomaly in the thermocline has important operational implications. From figure 1-21, C, it appears that if, for example, at a range of 1,000 yards a hydrophone is lowered to a depth of from 80 to 90 feet, it enters a region where the sound transmission is poorer by nearly 10 db than it is at from 20 to 30 feet higher. The sudden increase of the transmission anomaly is called the layer effect. The importance of the layer effect is enhanced by the prevalence of this type of thermal pattern in the ocean all over the world.

Figure 1-22 shows corresponding diagrams for a case of downward refraction.

INADEQUACY OF THE RAY THEORY

There has been much speculation about the reasons for the differences between the ray diagram theory and experiment-that is, the absence of sharp silence shadows and the presence of marked

  variations of intensity within the field of direct sound.

The failure to observe the sharply bounded silent shadow predicted by the ray-diagram theory should not be surprising. It is well known that even in the case of light, shadow boundaries are not sharp. The encroachment of a wave motion into the geometric shadow of an obstacle is known as diffraction.

Calculations of theoretical intensities have been made of sound fields for various sound waves around corners in air. As explained in textbooks on physics these diffraction effects increase with the wavelength of the wave disturbance, so that the ray theory becomes less and less correct as the wavelength increases. The wavelength of 24-kc sound in sea water is several inches and much longer than the wavelength of light, so that considerable diffraction of sound may be expected. Calculations have been made which show that the predicted effect due to diffraction is large enough to explain some of the irregularities in the transmission anomaly. However, the quantitative agreement between these diffraction calculations and measurements are not exact.

Calculation of theoretical intensities for downward refraction.
Figure 1-22. - Calculation of theoretical intensities for downward refraction.
 
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Another possible explanation of the sound energy observed in the shadow is the scattering by small obstacles and particles suspended in the sea. The scattering of light by particles such as dust, snow flakes, and rain drops in the atmosphere is a familiar phenomenon and is known to be responsible for the many changes in the color of the sky and in the visibility of objects: The scattering of sound corresponding to this phenomenon occurs in the sea. For particles that are small compared with the wavelength, the relative amount of energy scattered depends surprisingly upon the wavelength.   This dependence is expressed quantitatively in Rayleigh's law of scattering: The relative amount of sound energy scattered by small particles in a medium is inversely proportional to the fourth power of the wavelength; or, qualitatively, the shorter the wavelength, the greater is the scattering. For example, the wavelength of 5-kc sound is 10 times that of 50-kc sound-that is, a small particle scatters 10,000 times more sound of 50-kc frequency than of 5-kc frequency. It is probable that scattering is the explanation for some of the variations in the ray theory.
 
Reflection and Scattering
 
The mechanism of scattering, with its resulting reverberation, and the mechanism of echo formation from underwater targets are very similar. They can be discussed conveniently at the same time.

When a short-tone pulse is sounded in a large, empty room, the sound echoes and re-echoes from the walls, ceiling, and floor for a considerable time. This phenomenon is called reverberation. It has been studied extensively by acoustic engineers, because it interferes with the understanding of speech and the enjoyment of music. A suitable wall covering deadens sound and eliminates reverberation.

When an echo-ranging pulse of sound is emitted into the ocean a phenomenon called reverberation is observed. Although the ocean has a floor and a ceiling, it lacks the four walls of a room, and neither the laws nor the causes of underwater reverberation should be confused with those of reverberation in acoustic engineering.

Theoretically, if the surface and bottom of the sea were mirror-flat and if there were no suspended matter (including fish) in the water, there would be no reverberation. Every departure from these ideal conditions results in an echo, usually a very weak echo. There are many irregularities on the ocean bottom, each wavelet on the surface and each suspended particle in the water probably contribute their individual echoes. The combined result is a scattering of sound in all directions. Some of this scattered sound comes back to the transducer and is heard in the sonar loudspeaker.

  This reverberation has very important connections with echo ranging.

Reverberation is therefore to be considered as the resultant of a large number of very weak echoes. Some of the targets producing these echoes are not very obvious, nor is much known concerning them. They may be air bubbles, suspended solid matter, organic matter such as plankton and the fish feeding on plankton, or minute inhomogeneities in the thermal structure. Minor irregularities of the ocean bed are very effective scatterers, and reverberation is very high when the sound beam strikes the bottom. The surface waves undoubtedly contribute appreciably to it.

Reverberation is easily distinguished from extraneous noise because reverberation is a tone of fairly definite pitch, whereas noise has a wide band of frequencies. The individual echoes mentioned as forming reverberations are not perceptible as such; they overlap one another in time, causing marked fluctuations in the intensity. If the signal is of constant frequency, transmitted horizontally, it is succeeded by a quavering, ringing tone of rapidly decreasing loudness, interspersed with occasional bursts of sound that might be mistaken for echoes by an inexperienced observer. In shallow water a crescendo, effect may be perceived after a certain interval because of sound that is scattered backward by the bottom.

If relatively long pings (transmissions of sound with a duration of about 200 milliseconds) of constant frequency are used, reverberation has a

 
16

musical sound. With shorter pings the musical character disappears; although the pitch can still be distinguished, the tone becomes rough and grating.

When a frequency-modulated signal is used, the reverberation loses its musical character. Some frequency modulation may occur because of improper functioning of the sonar oscillator. If the reverberation from long pings of supposedly constant frequency is not musical, the oscillator should be examined for frequency instability.

ECHO FORMATION

When a sound wave passes over an obstacle suspended in a medium, the medium is set into vibration and becomes a secondary source of sound. The amplitude of the vibration is proportional to the amplitude of the primary sound, and consequently the intensity of the secondary sound is also proportional to the intensity of the primary sound.

The simplest example is that of an object like a submarine or a large fish, with dimensions that are large compared to the wavelength of the sound. Such an object intercepts a certain amount of sound and casts an acoustic shadow. The intercepted power is reradiated as the secondary sound, or, as it is more usually called, the echo.

The amount of power intercepted is determined by the target area of the obstacle. For the present, the target area may be defined in a simplified manner by imagining a shadow cast by the obstacle to fall on a plane perpendicular to the sound rays. The shaded area is the target area, σ. In a sphere with a diameter, d, for example, it follows that the target area would be a circle of area

σ=¼πd2.  (1-20)

In irregular objects, the target area depends on the direction from which the sound is incident.

If F is the energy flow (in watts per unit area) at the obstacle and W is the total power intercepted,

W=Fσ.  (1-21)

If the target is perfectly reflecting, all this energy is reradiated as sound. If the target is not perfectly reflecting, only a fraction, α, of this

  energy is reradiated as sound. Thus, the secondary sound power is

Ws=Fασ.  (1-22)

The effect of absorption is thus the same as if the target area were reduced. This secondary sound is radiated in all directions, though not necessarily equally in all directions.

A sphere reradiates the sound equally in all directions and is thus the simplest example to treat. It may seem that the existence of a shadow is in contradiction to this statement; however, at great distances from the sphere, diffraction causes the shadow to disappear. Consequently, the statement is strictly correct only at a considerable distance from the spherical target.

At a great distance, r, the power, Ws, that is reradiated from the target flows through the whole area, 4πr2, of an imaginary spherical surface centered at the target. Hence, the energy flow of the secondary sound is

Fs=Fασ / 4πr2.  (1-23)

If the target is not spherical, it radiates more sound in some directions and less in others than is predicted by equation (1-23). But this equation nevertheless still is valid on the average. The target area already depends on the direction of the incident sound, and may also be considered to depend on the direction in which the sound is scattered and on the reflecting properties of the target. If target area is adjusted to account for these factors, an effective target area, σ', may be used in expressing the secondary energy in the field surrounding the target

Fs=Fσ' / 4πr2.  (1-24)

INTENSITY OF SCATTERED SOUND

Because the energy flow, F, is defined as the intensity, I, equation (1-24) may be written also as

Is=(Iσ')/(4πr2).   (1-25)

Note that, in this equation, r is the distance from the target to the point at which the scattered intensity is being calculated. Is represents the secondary intensity. The primary intensity itself,

 
17

I, depends on r', the distance from source to target, and in general r' does not equal r. Neglecting refraction, which has been implicit in all of the previous equations, the following equation is applicable:

I=I1 / (r')2.

Therefore,

Is=I1σ' / (4πr2(r')2.  (1-26)

If the echo is received at the source of the sound as in practical echo ranging, r=r' and hence

Is=I1σ' / 4πr2r4.

The phenomenon of scattering or reverberation differs from echo formation only in that it results from the action of many relatively small targets rather than from one large target. The action of a single scatterer can still be described by equation (1-27).

The simplified definition of a target area fails completely when the scatterer has dimensions that are less than the wavelength of the sound. The target area, or the effective cross section, of small solid or liquid particles is much less than their actual section in a ratio that is roughly (πd/λ)4 where d is the diameter of the particle and λ is the wavelength of the sound.

There are occasions when air or vapor bubbles might be expected to exert an appreciable influence on the transmission of sound. It is difficult to understand how bubbles can exist permanently in the sea, because sea water is not saturated with air except very near the surface. There are several obvious sources of intermittent bubble formation: (1) Whitecaps; (2) the breaking of the bow wave, which causes bubbles to be washed under a ship and into its wake; and (3) the rotation of the propellers of ships or even submerged submarines.

An air bubble is much more compressible than the surrounding water. Under the influence of a sound wave, it therefore pulsates with a relatively large amplitude. If the pulsation is to be followed, the water immediately surrounding the bubble must oscillate with an amplitude considerably greater than that of the water at a distance. The mass of this surrounding water, coupled with the compressibility of the air, results in resonance at a frequency, FR, which depends on the diameter of

  the bubble, d, and on the average pressure, p, of the gas in the bubble. The dependence on p arises because the compressibility of a gas depends on its pressure.

The sharpness of the resonance peak of the bubble is determined by a parameter, Q, analogous to that of electric circuits. The value of this parameter cannot be calculated readily but is certainly less than λR/d, where λR is the wavelength corresponding to FR.

It is difficult to calculate the exact value of the effective cross section of an air bubble compressed in water. However, when excited by sound frequencies near resonance, the effective cross section or target area becomes very large and may approach λR. For example, at a depth of 66 feet where a bubble 0.02 inch in diameter has a resonant frequency of 20 kc, the target area may be several square inches.

For frequencies more than 1 octave below resonance, the target area is considerably less than the actual cross section and approximate calculations show that gas bubbles scatter low-frequency sound considerably more effectively than do solid particle s of the same size.

The mathematical investigations on which the preceding discussion of air bubbles is based have been confined to spheres. Their extension to non-spherical objects is not simple, but has been carried out for some objects. It is clear that the same general laws govern the more general shapes. For example, a fish that is not too flat or elongated casts a shadow roughly equal in area to that of a sphere of the same volume.

Our ignorance of the reflection coefficient causes some uncertainty in these calculations. The reflection coefficient depends largely on the compressibility of the fish. If the fish has a swim bladder (air cavity), it probably is the most effective portion in reflecting sound. Similar principles apply to kelp and other forms of marine life. These plants have gas-filled floats and are therefore very good reflectors of sound.

The bottom is especially important in the production of reverberations. Such objects as boulders, pebbles, shells, and coral are all potential scatterers of sound. A smooth sand or mud bottom theoretically behaves more or less like a mirror and scatters little sound back to the source.

 
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The waves on the sea surface also act like separate targets. The large surfaces reflect ultrasonic waves somewhat like curved mirrors. The effect of the smaller ripples is not clearly understood, but such ripples probably scatter the sound about equally in all directions.

THEORY OF REVERBERATION

None of the small scatterers just discussed returns an appreciable echo by itself. The simultaneous reception of the echoes from a large number of the scatterers constitutes what we call reverberation.

To understand the manner in which the scatterers cooperate in producing reverberation, consideration must be given to the manner in which a pulse of sound (a "ping") is propagated. If the duration of the pulse is to seconds, it consists of a train of waves the total length of which is vto, where v is the velocity of sound. This distance is called the train length of the pulse. Because v is 1,600 yd/sec, approximately, a pulse of duration 0.1 second (100 msec) results in a wave train 160 yards long. If the frequency is 24 kc, there are 24,000 X 0.1 = 2,400 complete waves in the train.

One-half the train length is called the ping length; a pulse lasting 0.1 second thus has a ping length of 80 yards. The ping length is a more useful concept than the train length, for two reasons.

In the first place, in echo ranging, the time required for the pulse to travel from projector to target and back to the receiver is measured. The clock is the range dial and is calibrated in terms of the range of the target that returned the echo- not in terms of time. If a target is at range r, the travel time is 2r/v. Therefore, if the echo is a pulse of duration to, the range indication increases by the amount ro=-vto/2 during the reception of the echo. This amount equals exactly the ping length as just defined.

In the second place, if there are many targets or scatterers, the echoes that are heard simultaneously come from those scatterers for which distance s from the sonar differ by less than ro. At a given instant, therefore, echoes are received from all scatterers that lie in a spherical shell, with a thickness ro, as shown in figure 1-23. At this

  Instantaneous relation between the region (A)
from which echoes are being heard and the volume (B)
occupied by the wave train for a beam whose angular half
width is alpha radians.
Figure 1-23 -Instantaneous relation between the region (A) from which echoes are being heard and the volume (B) occupied by the wave train for a beam whose angular half width is α radians.

instant, the actual train of waves no longer passes over this particular lot of scatterers; it has moved onward during the time the echoes were returning to the sonar. The instantaneous relation between volume A (from which the echoes are being heard), and volume B (which is occupied by the wave train), is shown in figure 1-23.

Figure 1-23 also shows graphically how the ping length and train length are related. Very little further reference is made to the train length, as almost no interest centers on region B. On the contrary, frequent reference to region A and the ping length is necessary.

The effect of scatterers suspended in the volume of the sea can now be calculated. Consider the simplest possible case:

1. There are N scatterers per unit volume.
2. Each scatterer has the effective target area σ'.
3. The sonar has a sharply defined beam of half width α. Its directivity pattern is shown in figure 1-24. The dotted line represents the axis of the beam.

Ideal beam pattern of half width alpha.
Figure 1-24 -Ideal beam pattern of half width α.

4. The sonar is in such a location that all effects of surface and bottom can be ignored.

The intensity of the echo from a single scatterer is given by equation (1-27), provided it is in the beam; otherwise, it is zero. There are many scatterers in the active shell (region A, figure 1-23) at

 
19

any instant. If V is the volume of this region, the number of scatterers whose echoes are being received is NV. If this number is combined with equation (1-27), the intensity of the reverberation is

Ir=I1NVσ' / 4πr4  (1-28)

Now the volume V is easily calculated. It is given approximately by

V=2πr2ro(1-cos α),  (1-29)

where r is the range to the center of region A. Hence,

IR=I1 ((Nσ'ro)(1-cos α)) / 2r2.  (1-30)

Several conclusions can be drawn from equation (1-30). A brief list of the simpler conclusions follows:

1. Because the reverberation intensity, IR, is proportional to the source intensity, I1 increased sound output increases the reverberation.

Relation between ping length and reverberation
intensity.
Figure 1-25 -Relation between ping length and reverberation intensity.

2. Because the reverberation intensity is proportional to the ping length, ro, a long ping causes more reverberation than a short one. (See figure 1-25.) If the reverberation intensity were strictly proportional to the ping length, the dots would lie on the solid graph.

3. Because (1-cos α) increases as a increases, a broad beam causes more reverberation than a narrow one. In general, doubling the width of the beam causes IR to increase about fourfold.

4. The (volume) reverberation intensity varies inversely as the square of the range, r; this relation should be compared with equation (1-27),

  Active areas on surface and bottom for two
different positions of the wave train.
Figure 1-26 -Active areas on surface and bottom for two different positions of the wave train.

which shows that the echo from a single target varies inversely as the fourth power of r. The reason for the difference is the increase in the active volume, V (region A, figure 1-23), as r increases.

The theory of volume reverberation, as presented in the previous paragraph, requires only slight modification when the scatterers are located on either the surface or the bottom. These two cases are, in many ways, identical. Instead of an active volume, V, an active area, A, must be dealt with, namely, the area of the intersection of the surface (or bottom) with region A of figure 1-23, already discussed. In figure 1-26, which is similar to figure 1-23, two successive locations of active volume are shown. Until the beam intersects the bottom, there is no active area on the

Variation of active areas on surface and bottom
as a function of range, when the projector is very close to the
surface.
Figure 1-27 -Variation of active areas on surface and bottom as a function of range, when the projector is very close to the surface.

 
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bottom; at position 1, there is an active area on the surface, but none on the bottom. After some time, position 2 is reached and there is an active area on the bottom as well as on the surface. Figure 1-26 is drawn for a sonar mounted on a surface vessel; if the sonar were on a submarine near the bottom, the situation would be reversed. Note that at very short range there is no active area on either bottom or surface; this condition is shown in greater detail in figure 1-27.

The mathematical expression for the active areas is rather complicated, except in the special case in which the projector is very close to the surface. In such a case

A=2αror,  (1-31)

where α is to be expressed in radians. The graph of this equation is shown as a dotted line in figure 1-27. The departures at short ranges are obvious.

For simplicity it will be assumed that there are N' scatterers per unit of active area and that each scatterer has the target area σ'. The intensity of reverberation is (compare with equation 1-28)

IR=(I1N'Aσ') / (4πr4).  (1-32)

If the range r is great enough so that equation (1-31) can be used for A,

IR=(I1N'σ'roα) / (2πr3).  (1-33)

Conclusions (1) and (2) drawn from equation (1-30) apply to equation (1-33) also. Conclusion (3) requires only slight modification, because (1-cos α) is replaced by α. Consequently, doubling the width of the beam increases surface reverberation by a factor of only two rather than

Dependence of surface and bottom reverberation
on range.
Figure 1-28 -Dependence of surface and bottom reverberation on range.

  Comparative levels of (A) echo from a single
target, (B) surface (or bottom) reverberation, and (C) volume
reverberation.
Figure 1-29 -Comparative levels of (A) echo from a single target, (B) surface (or bottom) reverberation, and (C) volume reverberation.

four. Finally, surface reverberation varies inversely as the third power of the range, while volume reverberation varies as the inverse second power of the range.

If the range is not great enough so that equation (1-31) can be used, somewhat more elaborate calculations are needed. The first three conclusions concerning volume reverberation apply without appreciable change, however, and only the dependence on range is changed. The graphs of figure 1-28 show this dependence on range for surface and bottom reverberations. In this figure it has been assumed that N', the number of scatterers per unit of active area, has the same value for both surface and bottom. Actually N' has a much greater value for the bottom than for the surface. This condition results in shifting the graph of bottom reverberation upward relative to the surface graph.

Figure 1-29 shows comparative levels of (1) an echo from a single target, (2) volume reverberation, and (3) surface (or bottom) reverberation, as calculated from equations (1-27, 1-30, and 1-33) respectively. To give a standard of comparison, it is assumed that all three factors have the same level at 1,000 yards, although this assumption is not necessarily the case in practice. Note that, at ranges of less than 1,000 yards, the levels increase in the following order: (1) Volume reverberation, (2) surface (or bottom) reverberation, and (3) echo. At longer ranges they decrease in this same order. The graphs diverge 10 db from their neighbors for each tenfold increase or decrease in range.

 
21

Figure 1-30 -Oscillograms of reverberation and echo. Note that figure 1-29 does not show the dependence of surface and bottom reverberation upon range; to show this dependence it should be modified in accordance with figure 1-28.

REVERBERATION IN NONIDEAL CONDITIONS

All of the preceding calculations have been based on a number of simplifying assumptions that are not correct under actual conditions but are useful in presenting the basic ideas. The complications introduced by departures from the ideal cases just examined will now be considered.

The first simplification was that the scatterers all have the same target area, σ', and that there are N of them in each unit volume (or N' on each unit area). Obviously, the scatterers are not all the same, but because only the combination Nσ' enters the final equation, this assumption does not cause any particular trouble. It is seen the m=Nσ' is the total target area of all the scatterers in a unit volume. This quantity is called the volume-scattering coefficient. Because N is measured in yd-3 and σ' in yd2, m is measured in yd-1; that is, 1/m is a length. It is essentially the distance a wave train can travel before much of its energy is scattered.

In the same way, n=N'σ' is the total target area of all the scatterers located on a unit area; it is called the surface- or bottom-scattering coefficient. Because N' is measured in yd-2 and σ' in yd2, n is dimensionless; that is, it has the same numerical value whether yards or feet are used as units.

The second simplification is the assumption that the projector emits the sound in a sharply defined beam, with no side lobes. When actual projectors are involved, the factor (1-cos α) in equation (1-29) and the factor α in equation (1-31) must be replaced by others, the exact values of which depend on the beam patterns of the projector. If these factors are called Kv and Ks, respectively; equation (1-30) and (1-33) then become respectively

IR=(I1Kvmro) / 2r2 (volume reverberation)  (1-34)

and

IR=(I1Ksnro) / 2πr3 (surface reverberation)   (1-35)

The two factors, Kv and Ks, like the ones they replace, bear a simple relation to the half-width

  of the main lobe of the transducer. Let a be redefined as the angle (in degrees) at which the beam pattern has a value 6 db below the maximum (or axial) level. Then the values of Ks and Kv are given approximately by the equations,

Ks=4.2 X 10-3α  (1-36)

and

Kv=4πKs2=5.5 X 10-5α2.  (1-37)

Note that the scattering coefficients are independent of the projector, whereas Ks, and Kv are independent of the ocean.

Finally, it has been assumed implicitly that the sound rays are straight lines and that the inverse square law determines the whole transmission loss. In actual cases the departures from ideal laws introduce marked effects, which can be ascribed to departures from the inverse square law of transmission loss.

In order to deal with these complications in as simple a manner as possible it is convenient to define the reverberation level, RL, by

RL=10 log(IR/I1)  db.  (1-38)

Note that RL is independent of the sound output of the sonar.

The volume- and surface-reverberation indices, Jv and Js, are defined by

Jv=10 log Kv  (1-39)

and

Js=10 log Ks  (1-40)

respectively and, with these substitutions, equations (1-34) and (1-35) become respectively

RLv=Jv+10 log(mro/2)-20 log r (volume)  (1-41)

and

RLs=Js+10 log(nro/2π)-20 log r (surface)   (1-42)

Equations (1-41) and (1-42) are correct only if the transmission of sound is accurately given by the inverse square law. It can be shown that the departures from the inverse square law are in most cases properly taken into account in the following equations:

RLv=Jv+10 log(mro/2)-2Hv+20 log r   (1-43)

and

RLs=Js+10 log(nro/2π)-2Hs+20 log r  (surface)  (1-44)

where Hv and Hs are the actual transmission losses

 
22

Oscillograms of reverberation and echo.
Figure 1-30. -Oscillograms of reverberation and echo.
from the sonar to the active regions responsible for the reverberation. It is easily seen that if Hv=Hs=20 log r, equations (1-43) and (1-44) reduce to equations (1-41) and (1-42).

The form of equations (1-41) and (1-42) suggests that the reverberation decreases steadily with time from an initial high level. This is not true. The ringing sound mentioned earlier in the discussion indicates that rapid changes in the intensity occur, that are not predicted by these equations. The oscillograms of recorded reverberation show these changes, as in figure 1-30.

These oscillograms are typical of the experimental data in this field and will be discussed in some detail. The three oscillograms were taken in rapid succession with different ping lengths of 0.8 yard, 8.0 yards, and 24 yards. Range marks are spaced 40 yards apart at the upper edge. The electric input to the transducer was coupled to the oscillograph and is recorded at the extreme left. This recording of the electric input is followed by a blank interval of about 0.025 second, during which the connections were changed from send to receive. The portions of the trace to the right of this interval are reverberation, except for the echo, which is clearly visible in each. The early reverberation is so intense that it is off scale in the two right-hand examples. The ordinates of the three oscillograms are comparable, except that the electric circuit for recording the outgoing ping did not respond fully to the very short 0.8-yard ping. The receiving circuits, however, responded fully to its echo. This echo is rather weak, but the other two echoes have the same amplitude.

  The theory presented above asserts that the intensity of the reverberation should be proportional to the ping length, ro. Consequently, the amplitudes of reverberation should be proportional to ro½ so that the three oscillograms should show amplitude of approximately 1:3.2:5.5. It is obviously difficult to verify this by a single measurement, because of the rapid and irregular fluctuations in the amplitude of the reverberation. On the average, these ratios are quite close.

A more detailed study of the problem shows that the theory described here refers only to such average values, and that there is a good explanation of the rapid changes in amplitude. Two possible causes immediately suggest themselves:

1. The number of scatterers in the active region varies as the active region moves outward.
2. The echoes from the different scatterers interfere.

The first of these possible causes is easily seen to bring about some fluctuation, but it is often relatively unimportant as compared to the second. If there are many small scatterers, only the second cause need be considered. As the number of scatterers in the active region decreases, the relative importance of the first cause increases.

Thus the second cause would dominate in long pings (large active regions) and the first cause would dominate in exceedingly short pings (small active regions). An inspection suggests, however, that even for the 0.8-yard oscillogram, the second cause of fluctuation is important, although some of the long "spines" may be caused by single scatterers.

 
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REVERBERATION WITH A HORIZONTALLY DIRECTED BEAM

In the usual echo-ranging condition, the transducer is directed horizontally in deep water, and both surface and volume reverberation are generally observed. The intensity of the resulting reverberation at each range therefore depends on which of these two types of reverberation is dominant. Thus, as shown by the following explanation, volume reverberation always dominates at long ranges, whereas surface reverberation usually dominates at short ranges.

It is convenient to begin the discussion with average reverberation-range curves obtained under practical echo-ranging conditions. Surface and volume reverberation then are considered separately in more detail; finally, average values of the scattering coefficients are given.

Two reverberation curves are shown in figure 1-31; they are averages of observed reverberations

Effect of wind speed on average reverberation
level.
Figure 1-31 -Effect of wind speed on average reverberation level.

  Comparison of reverberation at wind speed of
17 miles per hour with horizontal and vertical beam.
Figure 1-32 -Comparison of reverberation at wind speed of 17 miles per hour with horizontal and vertical beam.

at high and low wind speeds. The measurements were made at 24 kc, using echo-ranging equipment with the transducer mounted at a depth of 16 feet and a standard ping length of 80 yards.

The two curves exhibit the following features:

1. At short ranges (less than 500 yards) the average reverberation level depends strongly on the roughness of the sea surface as measured by wind speed.
2. At long ranges (beyond 1,000 yards the average reverberation is independent of wind speed.
3. With high wind speed the reverberation level drops rapidly.
4. With low wind speeds, the reverberation drops more slowly.

The dependence of the short-range reverberation on wind speed clearly indicates that at ranges shorter than 500 yards and at high wind speeds, surface reverberation completely dominates volume reverberation. This conclusion is supported by observations made at nearly the same time with horizontal and vertical beams. At high wind speeds and at short ranges the reverberation levels obtained with a horizontal beam are much higher than those obtained with a tilted beam. Figure 1-32 shows data of this type taken at a wind speed of 17 miles per hour. Points A and B represent deep scattering layers. Comparison of the two curves shows that in the first 100 yards the horizontal reverberation is about 20 db above the vertical reverberation. Two scattering layers (A and B) are also shown in figure 1-32 at depths of 80 and 400 yards.

 
24

At low wind speeds (curve 2 of figure 1-31) the short-range reverberation is volume reverberation. The evidence for this statement is afforded by experiments of the type described in the previous paragraph. When such measurements are made at very low wind speeds, with the sea dead calm, the horizontal reverberation is much lower than in figure 1-32 and agrees well with the vertical reverberation. These measurements indicate that at very low wind speeds, volume reverberation is dominant and surface reverberation is negligible.

Finally, at long ranges (figure 1-31) the reverberation is independent of wind speed. This fact is taken as evidence that at these ranges, volume reverberation always dominates surface reverberation.

Thus, three main conclusions may be drawn regarding deep-water reverberation with a horizontal beam.

1. At short ranges and high wind speeds, surface reverberation is high and dominates volume reverberation (curve 1, figure 1-31).
2. At short ranges and low wind speeds, surface reverberation is negligible and volume reverberation is dominant (curve 2, figure 1-31).
3. At long ranges (beyond 1,000 yards), volume reverberation dominates at all wind speeds and is independent of wind speed.

Surface and volume reverberation will now be considered in more detail.

SURFACE REVERBERATION

The discussion of surface reverberation given earlier in this chapter predicts an inverse third-power dependence on range. An example of this dependence is shown in figure 1-33. The data shown in figure 1-33, A, were taken with a transducer, almost nondirectional in the vertical plane and mounted at a depth of 20 feet. Short pings, 6.4 yards long, were used. Wind speed was about 15 miles per hour, so that the resulting reverberation could be identified as surface reverberation. The observed points agree well with the theoretical inverse cube law.

Figure 1-33, B, is a reverberation curve taken the same day under similar conditions. However, the transducer had a pattern in the vertical plane which was highly directional. The axis of the beam was horizontal in both cases. The theoretical curve takes account of the beam pattern.

  At very short ranges the active surface area is energized by the outer portions of the beam, beyond the angle α; these portions emit sound of a lower intensity than the main beam, and the receiver has a lower response at large angles than on the axis; thus, there is a noticeable drop in the reverberation level. This drop can be calculated from the beam pattern, as shown by the solid curve. At ranges greater than 80 yards the active area is energized by the main beam only, and the measured reverberation levels fit the inverse third-power line closely.

The curves in figure 1-33 must not be regarded as universal. Examples of reverberation curves that show an inverse fifth-power variation of the reverberation with range are frequent. Curve 1 of figure 1-31 is an example. The reason for this rapid decay is not understood.

There is a thermal condition in which the surface reverberation is frequently observed to drop off more rapidly than the inverse cube. This can be explained by conditions of strong downward refraction. The reverberation would be expected to decrease at that range where the sound beam is bent away from the layer of surface scatterers.

Comparison of observed and calculated surface
reverberation.
Figure 1-33 -Comparison of observed and calculated surface reverberation. A, Measurements made with a transducer that was almost nondirectional in the vertical plane; B, data taken the same day under similar conditions, but with the transducer turned so that its directivity in the vertical plane was high.

 
25

Effect of downward refraction on reverberation.
Figure 1-34 -Effect of downward refraction on reverberation. A, Bathythermograph and corresponding ray diagram; B, comparison of observed results with those calculated from simple theory.

Figure 1-34, B, shows an example of this drop, indicated by the arrow. It occurs at about 300 yards; at greater ranges the reverberation level is about 20 db below the value as given by equation (1-42). The data were taken with short (9-yard) pings. The transducer depth was 20 feet and the wind speed 12 miles per hour. The ray diagram based on the bathythermogram in figure 1-34, A, shows that the limiting ray leaves the surface near the 300-yard range (indicated by an arrow), thus affording support for the preceding suggested explanation.

Dependence of reverberation level at short rang
(100 yards ) on wind speed.
Figure 1-35 -Dependence of reverberation level at short rang (100 yards ) on wind speed.

  The dependence of surface reverberation on wind speed is very marked at short ranges, as can be seen in figure 1-31. At 100 yards the reverberation level at high wind speeds is some 35 db above that for low speeds, but at 500 yards the difference is only 10 db. The rapid increase of reverberation at 100 yards is seen more clearly in figure 1-35, in which the average reverberation level is plotted against wind speed.

The reverberation level is constant for wind speeds up to 6 miles per hour. This confirms the conclusion that volume reverberation is dominant at these low wind speeds. For wind speeds of 6 to 20 miles per hour the reverberation increases as much as 35 db; the curve then levels off, and above 20 miles per hour there is little further dependence on wind speed. This dependence on wind speed is closely correlated with the roughness of the sea. At 6 miles per hour the wind is strong enough to roughen the surface appreciably; occasionally, wavelets may slough over, but no well-developed whitecaps begin to appear, and when the wind has reached 20 miles per hour the sea is liberally covered with them. When this stage is reached, further increase in whitecaps has no effect on the reverberation.

It has been pointed out that surface reverberation in the ocean rarely exhibits the inverse third-power dependence on range which is predicted by the simple theory. This lack of agreement is found even at short ranges (100 to 500 yards), as shown by the steep slope of curve 1 in figure 1-31. Thus, it is clear that even the average surface reverberation cannot be fitted by equation (1-42) at all ranges. It is possible, however, to apply the equation to the observed reverberation level at one range to obtain the scattering coefficient as a function of wind speed.

VOLUME REVERBERATION

The simple theory of volume reverberation that was developed earlier and that assumed inverse square transmission and uniform distribution of volume scatterers predicts an inverse square dependence on the range. The first assumption is approximately valid at short ranges but breaks down at long ranges because of refraction and attenuation effects. The second assumption is sometimes valid over a limited region; in general,

 
26

however, the horizontal stratification of the scatterers invalidates the assumption that they are uniformly distributed. Thus, it is clear that the two basic assumptions made in the simple theory are usually not satisfied.

In order to take account of refraction and the uneven distribution of scatterers, it would be necessary to carry out a volume integration over the active scattering volume at each range. There is insufficient data to warrant such a complex theory. The correction for attenuation, however, is easily made in connection with the calculation of the volume-scattering coefficients. Finally, surface reflection, on the average, raises the reverberation by an additional 3 db. Thus, for a horizontal beam, the theoretical volume reverberation, corrected for surface reflection and attenuation, is

RLv=Jv+10 log (mro/2)-20 log r-2ar+3  (1-45)

Comparison of calculated and observed volume
reverberation, showing the close agreement.
Figure 1-36 -Comparison of calculated and observed volume reverberation, showing the close agreement.

The observed volume reverberation beyond 1,000 yards agrees closely with the theoretical reverberation given by equation (1-45) for typical values of a and m. This is shown in figure 1-36. Curves 1 and 2 are the observed averages at high and low wind speeds. Curves 3 and 4 were calculated from equation (1-45) using attenuation coefficients of a=0 and a=0.0045 db/yard. The latter is typical of good transmission at 24 kc.

For the remaining parameters the following values were used:

Jv=-25 db
ro=80 yards
m=10-6 yards-1.
  The importance of attenuation at long ranges is shown strikingly by the large differences between curves 3 and 4. Thus, at 5,000 yards the attenuation reduces the reverberation level by about 45 db below the inverse square value of curve 3. Note that the shape of the theoretical curve 4 beyond 1,000 yards is determined largely by the particular value of the attenuation coefficient a.

To return to the fit between equation (1-45) and the average curve at long ranges-not only does curve 4 fit the average volume reverberation, but it also fits most individual reverberation curves fairly well. These results indicate that the long-range volume reverberation is due largely to a deep scattering layer. These scattering layers may be colonies of plankton, or fish feeding on it, or bubbles generated by it. Further evidence for this conclusion is afforded by the fact that at short ranges, where the sound beam has not yet reached the deep scattering layer, the observed volume reverberation (curve 2) falls below the theoretical curve.

It has been remarked that beyond 1,000 yards most individual reverberation curves fit curve 4 closely. This is true over a wide range of oceanographic conditions, with one exception: no significant dependence has been found on wind speed, sea state, location, season, or thermal structure of the ocean.

The exception occurs under conditions of extremely sharp downward refraction and provides an interesting check on the importance of the deep scattering layer.

The effect of sharp downward refraction is to concentrate the sound beam into a relatively narrow cone. This produces a maximum in the reverberation curve at the range where the sound beam reaches the layer. An example of this effect is shown in figure 1-37, where refraction and reverberation are compared for 2 days.

On the first day there was a deep mixed layer extending from the surface to a depth of 40 yards. Figure 1-37, A, shows the ray diagram and the deep scattering layer indicated by the shaded portion. The angle shown on each ray is the angle of the ray at the projector, measured downward from the horizontal: the ray of 6° is the effective lower edge of the sound beam. Two days later, on March 17, the same deep layer was still present, but thermal conditions had changed

 
27

radically, producing the strong downward refraction shown in figure 1-37, B.

Typical reverberation curves for each day are shown in figure 1-37, C, together with the theoretical reverberation (curve 4), of figure 1-36. The reverberation observed when there was a mixed layer (curve 1) agrees well with the theoretical curve 3 between 1,000 yards and 2,500 yards; beyond 2,500 yards it reaches the noise level and flattens out. Curve 2, on the other hand. observed when there was sharp downward

Comparison of reverberation and refraction for
2 days.
Figure 1-37 -Comparison of reverberation and refraction for 2 days. A, Ray diagram for March 15, 1945; B, ray diagram for March 17,1945; C, observed reverberation for these 2 days compared with calculated values.

  refraction, shows a large maximum near 2,000 yards, corresponding to the range at which the central portion of the sound beam reached the depth of maximum scattering (figure 1-37, B).

At short ranges, curve 1 rises steeply with decreasing range. This rise is due to surface reverberation and is to be expected, because the data were taken at a wind speed of 20 miles per hour. The data of curve 2 were taken at a wind speed of 12 miles per hour; it shows a corresponding rise in the reverberation level at very short ranges (100 yards), but there is a minimum when the sound beam has left the surface and has not yet reached the scattering layer. When the lower edge of the beam reaches the deep layer (about 1,000 yards) the reverberation begins to increase with increasing range, culminating in the main maximum.

BOTTOM REVERBERATION

Because in echo ranging, the transducer is generally near the surface, the sound scattered back from the bottom provides an important contribution to the reverberation only in shallow water; here, however, it may well be the dominant factor in limiting the range from which detectable echoes can be obtained.

When the reverberation is caused by the surface, it is the state of the sea that determines the intensity of the scattered sound; bottom reverberation levels may be expected to depend on the character of the sea bed. In practical work four types of bottoms are recognized-(1) rock, (2) sand, (3) mud and sand, and (4) mud. The criterion of classification is the size of the particles constituting the sea bed, as determined by examining samples obtained by sounding with special devices. Recently, also, techniques of underwater photography have been perfected and have proved useful in studying the bottom. The difference in reverberation intensities among these types of bottoms are discussed as follows in connection with the discussion of bottom-scattering coefficients.

It is obvious, from the discussion of figure 1-26 that any bottom reverberation will be combined with volume reverberation and, when a horizontally directed beam is used, with surface reverberation. Thus, from the geometry of the experiment, it should not be expected that the measured levels of predominantly bottom reverberation have the levels predicted by the simple theory expressed

 
28

Schematic diagrams illustrating surface and bottom
reverberation.
Figure 1-38 -Schematic diagrams illustrating surface and bottom reverberation. A, No refraction; and B, strong downward refraction.

by equation (1-44). It will be recalled that equation (1-44) applies to bottom reverberation as well as to surface reverberation. In shallow water over a bottom that scatters strongly, such as rock, the bottom reverberation may be so much greater than either the surface or volume reverberation that a rough check of the theory is possible. If this is attempted, however, the simple inverse square loss will not provide a very reliable guide to the total transmission loss. Consideration must also be given to the loss due to attenuation (absorption and scattering).

In addition, the effect of the surface in reflecting the sound incident on it toward the bottom, from which it may be scattered back to the transducer either directly or by way of the surface a second time, must be taken into account. If the surface is considered to act as a perfect reflector, direct sound at the bottom evidently will be doubled, and thus the intensity of the scattered sound will be doubled. Moreover, the reflection of this scattered sound from the surface causes the intensity at the transducer to be doubled. Hence, the

239276°-53-3

  surface increases the intensity of the reverberation fourfold or, expressed in decibels, raises the reverberation level 6 db.

The effects of the refraction of the sound are more difficult to evaluate. The bending and distortion of the sound beam affects the intensity of the bottom reverberation in several ways. If the beam is bent sharply downward, the sound strikes the bottom at a shorter range and may be more concentrated; the surface reverberation decays very rapidly, And the bottom reverberation may be more intense. This is illustrated schematically in figure 1-38, A and B.

Data of an experiment illustrating the conditions
shown in figure 1-38.
Figure 1-39 -Data of an experiment illustrating the conditions shown in figure 1-38. A, Bathythermogram; B, ray diagram; C, observed reverberation at water depths of 87 yards (curve 1) and 210 yards (curve 2).

 
29

It is clear from figure 1-38, A, that, if the beam is not refracted downward, surface reverberation will be received continuously after the beam first strikes the surface. On the other hand, as seen from figure 1-38, B, a sharply refracted beam strikes the surface in a limited area only; the surface reverberation consists of a burst of sound that dies away very rapidly, to be followed by a second burst of sound as the bottom reverberation comes in.

These effects are shown in figure 1-39. The bathythermogram shows the thermal pattern of the sea when the reverberation shown by curves 1 and 2 in figure 1-39, C was measured. The two curves represent reverberation in two different depths of water (87 and 210) yards . The bending

Comparison of calculated and observed reverberation in shallow water over mud bottom.
Figure 1-40 -Comparison of calculated and observed reverberation in shallow water over mud bottom.

and distortion of the beam is seen in the ray diagram; a dotted line is drawn showing the path of the ray to be -6° in the absence of refraction. The refracted ray of -6° strikes the bottom at ranges shorter by 500 and 1,200 yards at the two depths. Moreover, the beam is concentrated between the ray of -1° and the upper limiting ray. The expected rapid decrease in reverberation as the upper half of the beam strikes the bottom is shown in curves 1 and 2 of figure 1-39, C. They also show the expected rapid decrease of the surface reverberation. The difference in levels between the two curves is due to the difference in ranges to the bottom in the two cases.

A large number of bottom-reverberation records have been plotted and compared with the graph of RL, as a function of r, as given by equation (1-44).

  Comparison of calculated and observed reverberation in shallow water over sand and mud bottom.
Figure 1-41 -Comparison of calculated and observed reverberation in shallow water over sand and mud bottom.

Of the terms in this equation, Js, and ro, are known; the value of Hs, must either be predicted or else obtained by making transmission measurements as nearly simultaneously with the reverberation runs as practicable. The magnitude of the scattering coefficient n is, of course, not known; hence that value of n which best fits the experimental points is considered to be the appropriate scattering coefficient. In determining the best fit give the greatest weight to the region of the graph between 500 and 1,000 yards.

The agreement between the calculated and observed reverberation is illustrated by figures 1-40 through 1-43.

Each of these figures exhibits results of measurements taken over a particular bottom type. Two theoretical curves are drawn for each bottom type except mud, in which transmission data were available only for poor transmission. The values

Comparison of calculated and observed reverberation in shallow water over sand bottom.
Figure 1-42 -Comparison of calculated and observed reverberation in shallow water over sand bottom.

 
30

Comparison of calculated and observed reverberation in shallow water over rock bottom.
Figure 1-43 -Comparison of calculated and observed reverberation in shallow water over rock bottom.

of H, were measured in conjunction with the reverberation measurements.

The experimental points represent averages of from 5 to 30 reverberation runs. The quartile deviation is about ± 5 db on the average. All measurements were made using 80-yard pings.

It should be stressed that these experimental data are not to be considered as representing bottom reverberation generally in shallow water. The measurements that yielded them were taken over particular small patches of particular bottom types. The curves serve, however, to convey a fairly realistic picture of the main features of reverberation in shallow water.

The four figures have certain features in common. The figures are good for all ranges less than 1,500 yards. Beyond this range the measured reverberation is consistently higher than the calculated one. Over mud bottom, the difference is 10 db at 2,500 yards over sand and mud; it is 20 db at 2,000 yards. The reason for this divergence

 
TABLE 1 -Bottom Scattering Coefficients

Bottom type 10 log n
Rock -22
Mud and mud sand -30
Sand -34

at longer ranges is not known. See table 1 for coefficients of bottom scattering.

FORWARD SCATTERING

The simple refraction theory predicts a "black" shadow zone under conditions of strong downward refraction. This is not confirmed by transmission runs of 24 kc made in deep water off the southern California coast. Sound of very low intensity is observed out to ranges of 5,000 yards (some 4,000 yards beyond the limit of the strong direct sound field), which can be explained neither as direct nor as bottom reflected sound. Instead there appears good reason to believe that it is sound scattered in the forward direction from a deep scattering layer.

The general good agreement between the theory and the observations, in the few cases that can be checked, indicates two main conclusions:

1. Sound observed in the shadow zone under conditions of sharp downward refraction is forward scattering from the deep scattering layer.
2. The scatterers in the deep scattering layer are approximately isotropic; that is, they scatter sound nearly equally in all directions.

 
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