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CHAPTER 4

ESSENTIALS OF ECHO-RANGING EQUIPMENT

Basic Components of Echo-Ranging Systems
 

So far this textbook has dealt primarily with the behavior of sound in sea water, the ocean climate and characteristics, and marine life, all of which affect the transmission and reception of sound. Listening systems were described briefly at the end of chapter 3. The present chapter gives particular attention to echo-ranging systems and problems associated with echo ranging.

The typical echo-ranging system to be discussed here is of the directional-beam (searchlight) type. Particular emphasis is given to transducers because of their importance in the development of sonar. The purpose of this chapter is to give the reader a broad understanding of the basic components needed in an echo-ranging system.

TRANSMITTER

The source of the signal is called the transmitter and can be compared directly with the transmitter of a radio station. There is a variable-frequency oscillator that generates a signal in the frequency range of from 17 to 27 kc. This signal is fed into a wide-band amplifier that increases the signal intensity to about 400 watts if a magnetostriction transducer is to be driven, and to about 150 watts if a crystal transducer is to be driven. This signal is then used to drive the transducer. The length of transmission is controlled by keying contacts on the timing device. The measurement of time is started at the same time as the transmission. The timing device closes the keying relay at zero indication on the range scale. This action connects the transducer to the output of the transmitter and at the same time keys the oscillator for a predetermined length of time. When this time is over the relay returns the transducer to the receiver, so that any echo may be heard. The signal in the echo-ranging equipment may be followed

  in figure 4-1, which is a block diagram of the signal circuits.

SIGNAL USED IN ECHO RANGING

Signal Frequency

Practical considerations set rather definite upper and lower limits to the frequencies that can be used in echo ranging. The use of sonic frequencies (less than 10 kc) has not been considered practicable because of directivity requirements. A second reason for the use of ultrasonic sound is provided by considerations of the detectability of echoes. Echoes must always be detected against a background of interfering noises. Although these noises include sound of ultrasonic frequencies, the greater part of their energy is in the sonic region. Hence, ultrasonic echoes are masked less than sonic ones.

An upper limit to the practicable frequency is set by the attenuation of the sound in the sea. The attenuation coefficient increases very markedly with frequency. Hence, for search purposes, where long ranges are required, a frequency higher than about 25 to 30 kc is not suitable. When the range is being closed, and great accuracy of bearing is needed rather than a long range, the greater directivity associated with higher frequencies is the determining factor, and thus frequencies of from 50 to 100 kc may be useful. These frequencies ate especially useful for depth determination, where an extremely narrow beam is required; and because accurate depth determination is practicable only at comparatively short ranges, the high attenuation consequent to using high frequencies is not significant.

The United States Navy at first adopted a compromise value of 24 kc. This frequency allowed fair directivity to be achieved while the size of

 
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the transducer could be kept within practical limits. The attenuation was moderate. equipment using only 24-kc frequency is now being   replaced by more elaborate equipment that can emit various frequencies, for the reasons just mentioned.
Block diagram of an echo-ranging equipment.
Power Amp
Oscillator
Transducer
Filter Junction Box
Reciever - Amplifier - Speaker
Oscilltor
Keying Relay Range Indicator or Recorder
Figure 4-1 -Block diagram of an echo-ranging equipment.
 
Receiver-Amplifier
 
When the signal has been emitted, the transducer connections in the filter junction box are changed so that the transducer can act as a receiver of sound waves. An echo from a target or other sounds of proper frequency incident on the transducer plate produce oscillations that represent the various signals, but the frequency of these sounds is too great to be perceptible to the human ear. The mechanical pressure of the sound waves is converted into alternating currents by the magnetostrictive or piezoelectric effect. These signals must be amplified and changed to frequencies in the audible range, or they may be portrayed so as to be interpreted by the eye. The purpose of the receiver is to amplify the signal and present it in a suitable form.

The receiver is a superheterodyne type similar to the one described for ultrasonic listening equipment. The usual method is to amplify the signal at ultrasonic frequency, then mix it with a frequency from a local oscillator to obtain an intermediate frequency that is the sum of the oscillator

239276°-53-6

  and signal frequencies. This intermediate frequency might be 60 kc.

The i-f currents which represent the signal are amplified further and utilized to render the incident sound perceptible in various ways. Note in figure 4-1 that the signal may leave the amplifier by two different paths. The customary methods of portrayal are:

1. The amplified i-f currents are heterodyned to sonic frequencies, which are converted to airborne sound waves (made audible) by means of a loudspeaker or headphones.

2. The amplified signal voltage may be rectified and delivered to a "chemical range recorder," which utilizes the chemical effect of the current to record the range on a specially treated paper. The density of the trace is determined by the magnitude of the current. Thus the pulse of current representing the echo signal leaves a spot that is much denser than that part of the trace which represents the reverberation and noise. The range can be read from a scale opposite the

 
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spot. An advantage of this method, not possessed by the others is that it provides a comparatively permanent record of incident sounds.

3. A range indicator used with earlier models of echo-ranging equipment has a rotating neon light which flashes when the signal voltage is applied to it. The range is read from an adjacent scale at the time of the flash.

4. The amplified voltage may be rectified and applied to a cathode-ray oscilloscope by the following methods:

a. The spot of this indicator is usually made to move along a vertical y axis to indicate range. The rectified voltage may be applied so as to

  cause the spot to deviate from straight-line motion (deflection in the direction of the x axis). The echo is then recognized by a greater x-axis deflection than that produced by reverberation.

b. In a second method of portrayal involving the use of a cathode-ray oscilloscope, the spot always moves in a straight line to indicate range. Its brightness is controlled by the rectified voltage from the receiver, so that the echo appears as a bright spot on the relatively dim, or invisible, line traced by the spot in the absence of an echo. This method is called z-axis portrayal.

It is possible to combine x- and z-axis portrayal.

 
Training Device and Bearing Indicator
 
Because the transducer is directional it is necessary to provide some method of rotating it in the horizontal plane, so that sound may be projected in any direction. The device for rotating the transducer is called the training device. The training device is power-operated by one of two methods-(1) an amplidyne system or (2) a thyratron system. Regardless of the method, the   operator is able to control the position of the transducer.

To be able to train the transducer is not enough. The operator must be able also to determine the actual direction in which the transducer is trained and to read accurately either its true or its relative bearing. Bearing information is obtained from a device known as the bearing indicator.

 
Range Indicator
 
The timing device for measuring the time between transmission and echo is basically nothing more than an elaborate stop watch. It is called a range indicator or range recorder as the case may be.

The range indicator is a large motor-driven contactor that rotates a light behind a translucent scale. The scale is calibrated for various maximum ranges. When the light passes zero on the range scale the keying contacts are closed and the transmitter sends out a signal through the transducer. The transmission may last about one twenty-fifth of a second. When the keying contacts are broken the receiver is connected to the transducer and any echo received causes the rotating light to flash opposite the range corresponding to that from which the echo was received.

This procedure is exactly the same as that used in the first measurement of range when an observer held a stop watch, which he started when the transmission was made. He stopped it when he heard the echo. Because an ordinary stop watch gave very poor results, the British made a

  special one that speeded up the motion of the second hand. The United States Navy then improved on this method by developing the automatic type just described.

If the maximum range desired is 1,000 yards, the rotating light must make one revolution in the time required for the transmitted sound to travel 1,000 yards and return to the ship. If the velocity of sound is 4,800 feet per second, or 1,600 yards per second, the time for the sound to travel a total of 2,000 yards can be found from the equation

r=vt, or t=r/v,  (4-1)

where r is the range in yards, v the velocity in yards per second, and t the time in seconds. If the values given in the example are substituted in equation (4-1),

t=2,000/16,000= 1.25 seconds.

The time for a 1,000-yard range is 1.25 seconds. This is a convenient time unit to remember because it is the time per thousand yards of range.

 
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If the time for a 5,000-yard range is desired it is necessary only to multiply 1.25 by 5 and get 6.25 seconds.

The foregoing calculations indicate that the rotating light must make one revolution every 1.25 seconds for a maximum of 1,000 yards and that it must key the transmitter each time the light passes zero (once each revolution).

The range recorder was developed by the British still later than the range indicator and was used in World War II. The version used by the United States Navy was an improvement over the British model, but the principle was exactly the same. A recording paper was treated so that an electric current passing through it would make a mark on it. This paper was made to move at a uniform rate by a motor. This uniform rate of paper motion gave a time axis for the range plot. A stylus

  was caused to move perpendicular to the motion of the paper at a rate proportional to the echo-ranging velocity of sound, or 800 yards per second. The motion of the stylus gave a time-range plot of the echoes. When received, the echo causes a current to pass from the stylus through the paper and leave a mark on it opposite a range scale placed over the paper. The range recorder has the advantage of "memory" over the range indicator. With the range indicator, the flash of light is gone once it is made, and if the observer misses it he cannot get a second look. The recorder, however, gives a permanent record of each echo.

The recorder also operates the keying relay. The transmitter is keyed just as the stylus of the recorder starts to move across the paper. The position of the stylus at any time is then proportional to the range.

 
Transducers
 
It has been pointed out in chapter 1 that a vibrating body with dimensions that are small compared to the wavelength of the sound, radiates sound energy in all directions. If, however, the dimensions are large compared to the wavelength of the sound, the propagation becomes directional. Radiation from emitters of the first type is called spherical radiation; that from emitters of the second type is called directional radiation.

In underwater sound, the important consideration is the production of directional transducers. It is important to know the direction from which a sound is heard or received. In some systems, the sound emitted is from a cylindrical source and is transmitted horizontally in all directions at the same time. The vertical dimension of the beam, however, is made narrow and sharp. Even with this type of transmitter, it is important to receive with a directional transducer.

In sonar a transducer is a device which may convert electric energy into acoustic energy in the surrounding water. Such a transducer may also convert acoustic energy from the surrounding water into electric energy. It is both a transmitting and a receiving device. The term "projector" is commonly used for the device the specific function of which is to transmit sound energy into the water. The hydrophone is a device used specifically for converting sound

  energy in the water into electric impulses in the receiver system.

ULTRASONIC SOUND SOURCES

Sound waves having frequencies above the audible range have been produced and used in laboratories for many years, but only in relatively recent years has it been possible to generate such waves of sufficient energy for practical use. The most common method of producing ultrasonic sound at the present time is by causing formed bodies to vibrate at a high natural frequency (resonance) by the application of rapidly oscillating electric voltages.

Magnetostriction Effect

The change in length of a rod or tube of ferromagnetic material when it is placed in a magnetic field parallel to its length is called magnetostriction. Nickel, annealed cobalt, and a few alloys of nickel possess a more pronounced magnetostriction effect than other metals.

The phenomenon is not related in any simple manner to other magnetic properties. Figure 4-2, A, shows the relative change in length dL/L, as a function of field strength in gausses, for several materials. The change in length, although small,

 
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Magnetostriction transducer.
Figure 4-2 -Magnetostriction transducer. A, Magnetostriction in iron, nickel, and cobalt; B, construction of a magnetostriction transducer head.

depends upon the strength of the magnetic field, but is independent of the direction of the field. In addition, its magnitude depends on (1) the material, (2) its heat treatment and present temperature, and (3) the degree to which it was previously magnetized.

Figure 4-2, A, shows that nickel possesses the property of magnetostriction to a much greater degree than any other metal. It decreases in a fairly linear manner for an increasing field strength up to about 200 gausses. If the field is increased beyond this value, the additional change becomes extremely small. The maximum relative change in length is about 40 parts in a million. However, because Young's modulus for nickel is high (30 X 106 lb/in2), a large force is exerted against anything that resists this small change in length.

  Magnetostriction is reversible. If a previously magnetized rod of nickel is stretched, the magnetization of the rod is decreased; if it is compressed (in the direction of its length), the magnetization is increased.

Magnetostriction Sound Sources

Magnetostriction becomes a source of sound waves when a nickel rod is subjected to an alternating magnetic field by winding a coil of wire around it and sending an alternating current through the coil. The rod is shortened periodically in response to the changing field. Because the shortening of length is independent of the direction of the field, the rod is shortened when the current goes through the positive half of its cycle, regains its length as the current becomes zero, and is shortened again when the current goes through the negative half of its cycle. Thus, the rods goes through two cycles of motion while the current completes one oscillation. The doubling of the rod vibration frequency can be prevented by subjecting the rod to a constant magnetic field with properly arranged permanent magnets or by sending a constant direct current (polarizing current) through the coil. Figure 4-2, A, shows that if a nickel rod is initially shortened to some point on the steep portion of the curve by placing it in a constant magnetic field and is further subjected to the magnetic strains imposed by an alternating current, it can be made to shorten and lengthen in step with the alternating current. The polarizing magnetic field not only prevents a doubling of the frequency of vibration of the rod, but also allows operation on the steeper and more linear portion of the curve. This characteristic is an important advantage.

The natural fundamental frequency of vibration, F, of a rod of length L is

F=(½L) (M/σ)½  (4-2)

where M is the modulus of elasticity and σ is the density of the material. If a current of this frequency is sent through the coil, the amplitude of oscillation is a maximum; relative changes in length may be of the order of 1 in 10,000. Calculations using equation (4-2) show that a rod of nickel 5 inches long has a fundamental frequency of vibration of about 20 kc; and that one 1.6 inches long resonates at 60 kc.

 
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If a nickel rod is set in vibration in the manner just described, sound waves, with a frequency determined by the frequency of the magnetizing current, are emitted from the end of the rod. To obtain the maximum possible intensity, a practical transducer is constructed by embedding the end of several hundred small nickel rods into a steel diaphragm of dimensions which ensure that its resonance frequency is the same as that of the rods. Each rod is excited by its own coil.

A typical magnetostriction transducer is shown in figure 4-2, B.

Because of the reversibility of the magnetostriction effect, the transducer acts also as a receiver. Sound waves impinging on the diaphragm compress or extend the rods; corresponding changes in the magnetization of the rods induce alternating currents in the coils, which after amplification can activate a portrayal device.

Piezoelectric Effect

When subjected to a mechanical stress, some crystals-such as quartz, Rochelle salt (RS), ammonium dihydrogen phosphate (ADP)- exhibit electric charges on certain surfaces. This phenomenon, called the piezoelectric effect, was discovered by the Curie brothers in 1880. The electric charges developed are proportional to the stress applied to the crystal, and the charges are of opposite sign for compressions and tensions. Shortly after this discovery, the Curies found the inverse effect to be equally true-that is, when a

Quartz crystal, showing X-cut and Y-cut plates.
Figure 4-3 -Quartz crystal, showing X-cut and Y-cut plates.

 

Crystals, showing orientation of rectangular plates.
Figure 4-4 -Crystals, showing orientation of rectangular plates for: A, 45° X-cut and Y-cut RS (top) and B, 45° Z-cut ADP (bottom).

crystal is subjected to an electric field, mechanical strains occur in the crystal. Thus, these two effects are exactly reversible and a direct proportionality exists between cause and effect, in both magnitude and sign. The fact that magnetostriction is a nonlinear effect, where as the piezoelectric effect is linear, serves as an important distinction between these two phenomena.

If a piezoelectric crystal is placed between two electrodes and an oscillating electric voltage is applied to the electrodes, the crystal vibrates. Because the elastic properties of such crystals differ in different directions, the vibrations occur in different ways, depending on the orientation of the crystal relative to the electrodes. In any case, the natural frequency of vibration is given by an equation similar to equation (4-2), where the value of the elasticity modulus differs for different orientations of the crystal.

A crystal as used in this book indicates a properly oriented piece cut from the mother bar. If such a crystal is equipped with suitable electrodes and properly mounted and protected it serves to generate or receive sound signals.

 
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Rectangular plates cut from the mother bar at various angles of orientation are shown for quartz (figure 4-3), RS (figure 4-4, A), and ADP (figure 4-4, B).

Those crystals designated as 45° X-cut and 45° Y-cut RS and 45° Z-cut ADP are the only types of cut crystals that have so far found extensive practical application in underwater sound transducers in the United States. The use of 45° X-cut RS is now limited to special and rare circumstances, such as for small hydrophones on long cables where a preamplifier cannot be used. The use of 45° Y-cut RS has declined greatly. On the whole 45° Z-cut ADP is preferable unless some particular reason (such as low frequency) indicates otherwise. Quartz has been used effectively in England, but only because of an inadequate supply of RS and ADP crystals.

Quartz has the advantage of being strong and insoluble in water, whereas RS and ADP are fragile and soluble. Solubility is a disadvantage in all seagoing applications, although it can be overcome by suitable precautions in the design and construction of transducers. In the laboratory, on the other hand, solubility is an advantage

Mounting of transducer crystals. A, Asdic transducer; B, RS and ADP crystals.
Figure 4-5 -Mounting of transducer crystals. A, Asdic transducer; B, RS and ADP crystals.

  in that it makes possible the production of good artificial crystals, whereas quartz must be mined, and only a small fraction of quartz crystals found are large enough and perfect enough for acoustic purposes. Quartz also has the disadvantage of being very hard and consequently difficult to cut and polish. Both RS and ADP crystals are soft enough to be cut with a band saw and shaped by ordinary metal-working power tools, if care is exercised to prevent chipping.

The British Asdic, the forerunner of our sonar, utilized X-cut quartz crystals. These crystals were laid flat on a steel plate, as shown in figure 4-5, A, and arranged in a mosaic so that the plate was adequately covered. An identical plate (not shown in the figure) was then laid on top of the crystals, thus forming a sandwich. The assembly was made mechanically rigid by means of clamps at the edges of the plates. Insulating washers made it possible to connect the plates to the terminals of the a-c source.

The deformation of the crystal when the voltage is applied is shown in figure 4-5, A, by the arrows. When the potential of the upper face is positive, the thickness increases. Simultaneously, the other two dimensions shrink. The changes which occur in the length, width, and thickness are such that the volume of the plate remains the same.

When the potential is reversed, the deformations are in the opposite direction. The two faces are not equivalent; hence, care must be taken to arrange all the plates in a mosaic so that they expand and contract "in step." Because the plate is compressed during one-half of the cycle of the a-c field and extended the same amount during the other half, it vibrates with the same period as that of the field. If this is the natural frequency of the crystal, the amplitude of vibration is a maximum. The natural frequency of the thickness vibrations, the one used in the Asdic transducer, calculated from equation (4-2) is

F=285.5/t kc,  (4-3)

where t is the thickness of the plate in centimeters. However, experiments showed that this relation is only approximately true, because the plates generally execute vibrations in other modes than the ones mentioned; moreover, besides compressional vibrations, vibration due to shear may

 
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Types of crystal stacks for transducers.
Figure 4-6 -Types of crystal stacks for transducers.

also be present. Such vibrations, coupled to the primary ones, tend to change the primary frequency of vibration.

Rochelle Salt and ADP Transducers

Sonar transducers using plates of Rochelle salt and ADP crystals are mounted so as to utilize the length vibrations instead of the thickness vibrations, as shown in figure 4-5, B. The two large faces are coated with a metal foil, and the a-c voltage is applied to the foil. The arrows indicate the

  deformation resulting from the indicated charge. There are many designs and methods used in assembling crystals into a transducer; however, in general, the crystals are cemented to a single heavy plate. To prevent short-circuiting, the surface of the backing plate must be enameled.

Many crystals are mounted on a single backing plate, as shown in figure 4-6, and sound is radiated from the free ends of the crystals. They are protected from the sea water by a "window." The window may serve to separate two liquid media, as sea water and castor oil, or the crystals may be attached directly to the inside of the window, the window not only protecting the crystals from the action of the sea water but also serving as a means of support. The space not occupied by crystals, between the backing plate and window is filled with carefully purified castor oil. Rubber has been widely adopted for acoustic windows in crystal transducers, primarily because of the good impedance match obtainable with sea water but in part due to its elastic properties, abrasion resistance, and its electric resistivity.

The resonant frequency of the length vibrations of the crystal plates, as shown in figure 4-5, is a function of both the length, L, and the width, w, of the plate; it is generally multiplied by the length to form a term called the frequency constant-

FL=64.7-(13.6) (w/L)2 kc.  (4-4)

DIRECTIVITY

One important property of a transducer, when acting as a projector, is the manner in which transmitted energy is distributed in direction; or when acting as a receiver, the dependence of its sensitivity on the direction of the incident sound. energy.

Let us now look into some of the principles involved in the design of a directional transducer. Brief consideration has already been given to the directional properties of a large source-one with linear dimensions several times as great as the wavelength of the emitted sound, as compared to the omnidirectional properties of a "point" source. Directional transmission of sound results from the interference of waves spreading out from two or more poi at sources or from several points on a large source.

 
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Interference of waves from two sources
Figure 4-7 -Interference of waves from two sources, for d=λ/2

Consider two point sound sources, P1 and P2, in figure 4-7, located a distance d apart equal to one-half wavelength, vibrating in phase with the same frequency and amplitude. Along CD, the perpendicular bisector of the line that joins the two points, condensations from the sources arrive at C at the same time, as do rarefactions, and the interference is constructive. Thus the sound pressure at C is the sum of the pressures from each source. The transmitted sound energy is a maximum along line CD. At point B, on the line joining the two points, each source again exerts a pressure. In this case condensations produced by one source arrive with rarefactions due to the other and destructive interference results. The sound pressure at B is also the sum of the pressure from each source; however, because the waves

  from the two sources are arriving at B in phase opposition, the sound pressure at B is zero.

This special case of two point sources located one-half wavelength apart constitutes the basis for a directional transducer, with a maximum output along the perpendicular bisector of the line joining the two sources and zero output along the line joining the two sources. In directions between AB and CD, the sound pressure resulting from the combined waves varies with direction. The polar diagram, B, of figure 4-8, shows a complete picture of the distribution of sound pressure resulting from the interference of waves from two point sources spaced one-half wavelength apart.

If the two sources are separated by some other fraction of a wavelength, the difference in the pressures at points C and B depends on the amount of the separation. For example, if the separation is λ/10, the difference in pressure at C and B is about 5 percent. The smaller the separation of the sources-that is, the smaller the dimensions of the whole source relative to the wavelength-the smaller is the difference in pressure between points on the two lines under discussion.

If the point, X, under observation lies in a direction making an angle θ with the perpendicular bisector of the line joining the two sources (figure 4-7), the wave from one source lags behind the one from the other by a distance, d sin θ, where d is the distance between the two sources. The phase lag between the two waves arriving at the point X is (2πd / λ) sin θ radians. The ratio of the resultant pressure, p, at point X to the pressure,

Graphs of equation (4-5) for various values of d.
Figure 4-8. -Graphs of equation (4-5) for various values of d.
 
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po, at the corresponding point, C, on the normal (θ=0) may be obtained by vector addition and is

p/po=cos(πd / λ sin θ).  (4-5)

Graphs of this function (figure 4-8) normalized to a maximum value of unity show a series of maxima and minima for four values of d as θ is made to vary through 360°.

Practical sources of sound can be considered to be composed of a number of point sources. By reasoning similar to that just used, the pressure at any point in the field surrounding the source can be calculated. The calculations become extremely complicated for all but the simplest possible arrangements; however, they have been made for several simple geometrical configurations and are found in standard works on sound. A brief discussion and a few equations will be given to illustrate the problem involved.

There are three types of wavefronts that can be handled rather simply-that is, waves that are (1) plane, (2) cylindrical, and (3) spherical. Because the spherical wave represents a point source and gives a nondirective field, it is of little use in transducer design. The plane and cylindrical cases, however, are useful in that most transducers have either plane or cylindrical radiating faces.

Mathematical calculations show that the sound field of any shaped-plane radiator should, close to the surface, exhibit many maxima and minima distributed in space both along the axis perpendicular to the surface and in planes parallel to the surface. Furthermore, such a sound field should at great distances, exhibit a central maximum with side lobes of decreasing amplitude with distance from the central axis. The field at a great distance is of course important in echo ranging, and the field close in is important in the coupling between two or more transducers that must be operated close together.

Most plane radiators in use are bounded by squares or circles and the chief interest is in the distant part of their sound fields. Under these conditions, the directivities are quite easily calculated. Although a mathematical analysis is beyond the scope of this book, a simple statement of the types of functions that represent the variation of pressure with the angle, θ, is of interest.

  Sinx/x as a function of x square plane radiator.
Figure 4-9. -Sin(x)/x as a function of x where x=(πd/λ)sinθ (square plane radiator).

The function that represents the square or rectangular radiator is

p(x)=(ab/v1) (sin x/x),  (4-6)

where x= ka/2 sin θ in the plane perpendicular to the side a, or x=kb/2 sin θ in the plane perpendicular to the side b. If only the variation with angle θ is needed the first term, ab/v1 may be omitted because (sin x)/x=1 when x=0. A graph of the function (sin x)/x normalized to unity, is shown in figure 4-9.

The case of the circular radiator was first solved by Rayleigh. A mathematical expression of the circular case involves Bessel functions and is not given here. The graphical representation, however, is very similar to the square radiator pattern shown in figure 4-9.

The zeros of the (sin x)/x function corresponding to the nulls between lobes come at x=Nπ, N=1, 2, ..., while the maxima, side lobe peaks, occur at x=4.5, 7.7, 10.9, 14.1, 17.1, 20.3, ...

 
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Other useful facts about these functions are given in table 6.

Using the (sin x)/x function for the case of a plane radiator bounded by a square, the expression for p/po becomes

p/po = sin(ka/2 sin θ) / (ka/2 sin θ),  (4-7)

where k is 2π/λ, and a is the length of the side of the square.

If values of θ are substituted in equation (4-7), the pressure in all directions relative to pressure on the normal to the surface can be plotted for arbitrary values of a and λ. A directivity pattern is obtained if the results are plotted on polar coordinate paper.

To achieve a higher degree of directivity, the linear dimensions of the transducer must be several times as great as the wavelength of the sound energy. Sound of 10 kc in sea water has a wavelength of about 6 inches. To get a minimum degree of directivity at that or a lower frequency obviously would require a larger transducer surface than is practicable.

Directivity Patterns

It is customary to plot the directivity function B, or-10 log b (θ), rather than b(θ) itself; but this means the importance of the side lobes is stressed, as can be seen from figures 1-4 and 1-5. In echo ranging, the side lobes are important because an echo may be received along one of them and considered to be due to the sound of the main beam. Such a misinterpretation would result in a large bearing error. Thus the suppression of side lobes plays an important part in the design of transducers. For example, if the velocity of vibration over the surface of a plane transducer is not constant, but is less around the edges of the transducer than in the center, the side lobes are always reduced in magnitude. However, the main lobe is generally broader. Several methods of calculating the sound field from transducers of variable surface velocities and phases have been used.

For practical reasons transducers are not designed with velocities continuously variable but

  with step variation over their surfaces. however, the continuously variable velocity method has given patterns for a wide variety of distributions which are a valuable guide for design and which also give a perspective to the lobe-suppression problem.

Several cases using two velocity distributions have been calculated for both circular and square surfaces. Experimentally, the velocity ratio of 3

TABLE 6 - Useful Facts About Radiation Function

Db down Formula
  Circular Radiator
α=radius of circular radiator
λ=wavelength
3 θ=sin-1 0.258 λ/α
6 θ=sin-1 0.305 λ/α
θ=sin-1 0.595 λ/α 1st zero
17.8 θ=sin-1 0.818 λ/α max first lobe
θ=sin-1 1.111 λ/α 2nd zero
23.8 θ=sin-1 1.34 λ/α max 2nd lobe
θ=sin-1 1.62 λ/α3rd zero
  Square Radiator
α=side of square radiator
3 θ=sin-1 0.446 λ/α
6 θ=sin-1 0.605 λ/α
θ=sin-1 1.00 λ/α 1st zero
13.47 θ=sin-1 1.43 λ/α max first lobe
θ=sin-1 2.00 λ/α 2nd zero
18.24 θ=sin-1 2.36 λ/α max 2nd lobe
θ=sin-1 3.00 λ/α 3rd zero
 
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Theoretical and experimental directivity patterns
of a crystal transducer.
Figure 4-10 -Theoretical and experimental directivity patterns of a crystal transducer.

to 1 and a diameter ratio of 0.6 to 1 have given the greatest suppression of side lobes so far in the circular type, as shown in figure 4-10, which includes the theoretical pattern (broken line) as well as the experimental pattern (solid line). The relatively small size of transducers usually limits the number of velocity steps to two or three.

Directivities are usually calculated in some plane which is normal to the face of the transducer; and because the beam is three-dimensional, the plane in which a directivity pattern is measured must be specified. If the transducer is a circular type, the beam may have symmetry about the normal to the transducer face, as shown in figure 4-11, where the frequency is 25 kc and the diameter is 15 inches. However, if the transducer is nonsymmetrical, there exists a directivity pattern for each possible axis of rotation, and in general these various patterns are different.

The effects of variations in surface velocities have been discussed. Phase variations also are important and both phase and velocity variations may be used simultaneously. If the radiating

  surface is uniform in velocity, the phases in adjacent lobes differ by 180°. (See figure 4-12, A.) By a reciprocal theorem, if the radiating surface is divided into zones the amplitudes of vibration of which decrease in magnitude and alternate 180° in phase in a manner similar to the lobe pattern of a uniform radiator, the pattern should be uniform over a certain arc and have no side lobes. Such a pattern is shown in figure 4-12 B. If a linear phase shift across the radiating surface is used, the main lobe is shifted in direction as shown by figure 4-13, in which the phase is shifted 30° per point radiator. Phasing of this type can be used to train the main lobe electrically while the transducer is fixed.

Theoretically it is possible to fashion the directivity of a transducer into any desired form. Success in such fashioning, however, requires the radiating surfaces to perform according to

Three-dimensional directivity pattern for a circular
plate.
Figure 4-11 -Three-dimensional directivity pattern for a circular plate.

 
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Reciprocal relation between the surface velocity function and the corresponding directivity function in a
square-plane radiator.
Figure 4-12 -Reciprocal relation between the surface velocity function and the corresponding directivity function in a square-plane radiator. A, Uniform velocity in phase; B, surface velocity and phase distribution.
prescribed conditions, leading to one of the most difficult problems in the construction of transducers. Wide variations in the agreement between theory and experiment are encountered in transducers of different design, and often are

Shifting of the main lobe by a linear phase
variation over the length of a line of point radiators.
Figure 4-13 -Shifting of the main lobe by a linear phase variation over the length of a line of point radiators.

  encountered in particular units at different frequencies. These departures from theory vary in magnitude all the way from negligible departures to those large enough to render the unit useless for its intended purpose. The analysis of these eccentricities can be divided into two parts, one treating the main, or central, lobe and the other treating the side lobes. The most important feature of the main lobe aside from its absolute intensity is its width, which can be defined by two points on each side of the center that are 6 db down in intensity from the maximum. These theoretical beam widths for the square- and circular-plane radiators with this definition are

θ=2 sin-1 0.605 λ/α (square)

and

θ=2 sin-1 0.305 λ/α (circular).

(See table 6.)

 
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In general the experimental beam widths are in good agreement with theory even when the accompanying side lobes are in very poor agreement. The width of the central lobe can thus, usually with good approximation, be predicted from the over-all dimensions of the transducer.

Detecting submarines under various conditions establishes requirements for echo ranging that can be met only by using several transducers. For general long-range search purposes, it is desirable to have a relatively wide beam with circular symmetry and small attenuation. For this purpose a circular transducer driven at 15 kc is suitable. For close ranges, a narrower beam can be achieved by using a transducer driven at 30 kc; the loss in range due to increased attenuation at the higher frequency is compensated for by the greater concentration of the beam and the greater accuracy in obtaining bearings on a target.

The QGA echo-ranging equipment is designed along these lines. The two transducers are mounted in a single dome, although they are operated independently. The system consists of two complete equipments, which are practically identical except that one operates at 15 kc and the other at 30 kc. Both transducers may be trained through 360° in azimuth. The 30-kc transducer may be tilted to 45° for maintaining contact with submarines that approach close enough to pass under the horizontal beam. The directivity patterns for the two frequencies of the QGA are shown in figure 4-14. The solid curve is the pattern for the 15-kc transducer; the dotted curve that for the 30-kc transducer. The numbers

Directivity patterns of QGA echo-ranging transducer.
Figure 4-14 -Directivity patterns of QGA echo-ranging transducer.

  Directivity pattern of magnetostriction 24-kc
echo-ranging transducer.
Figure 4-15. -Directivity pattern of magnetostriction 24-kc echo-ranging transducer.

on the axis indicate db below the maximum. The directional characteristics of the transducer are described by the directivity index which is a measure of the fraction of the sound energy that is sent out in the desired direction. The directivity index is expressed by a negative number. The larger the number numerically the more directional the transducer. The directivity index is described in more detail later in this chapter. The directivity index at 15 kc is -18.1 db; at 30 kc, -23.2 db.

Directivity patterns of transducers used in some of the older sonar equipments are shown in figures 4-15 to 4-18, inclusive. Figure 4-15 shows the pattern of the standard QC transducer, which consists of 608 hollow nickel tubes arranged on a circular diaphragm. Numbers on the axis indicate decibels below maximum. In this gear the tubes are arranged in circular form, and are pre-magnetized by a polarizing current. The directivity index is -21.4 db.

Another form of QC gear, the QCU, has the directivity pattern shown in figure 4-16. Numbers on the axis indicate decibels below maximum. The directivity index is -22.5 db. This unit consists of 182 nickel tubes spaced in an equilateral triangle. The tubes are premagnetized by permanent magnets.

Directivity patterns of two types of QB transducers are shown in figures 4-17 and 4-18. Figure 4-17 is the pattern of the QBF, an echo-ranging transducer consisting of 450 Y-cut Rochelle-salt crystals mounted on a steel plate. Numbers on the axis indicate decibels below maximum. The directivity index is -25.2 db.

 
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Figure 4-18 shows the pattern of the QBG transducer taken in the horizontal and vertical planes at 22 kc. Numbers on the axis indicate decibels below maximum. The directivity index for the horizontal pattern at 22.5 kc is -17.3 db. The QBG is a small Rochelle-salt gear intended for small ships.

When a transducer is used as a hydrophone, the directivity is generally found to be nearly identical with its pattern when used as a projector, provided the electric connections are equivalent for both sending and receiving. The beam pattern or directivity function B (equation (1-13) expressed in decibels) gives information concerning the response of the transducer to sound coming from a specified direction. Even if the sources of sound are uniformly distributed in all directions, the directivity

Figure 4-16 -Directivity pattern of magnetostriction 25-kc
echo-ranging transducer (QCU).
Directivity pattern of magnetostriction 25-kc echo-ranging transducer (QCU).

function gives information about the transducer to such multidirectional sound fields, because the response is caused largely by those sources in the direction of the main lobe. Sources in other directions do not contribute appreciably.

These multidirectional sounds very often interfere with the reception of echoes. The response of a transducer to these extraneous sound sources, and their previous measurement under various sea conditions and at various locations all become important.

The magnitude of a multidirectional sound field is most readily specified in terms of its rms sound pressure, p. This pressure can be directly

  Directivity pattern of Rochelle-salt (Y-cut) crystal
echo-ranging transducer (QBF) at 30 kc.
Figure 4-17 -Directivity pattern of Rochelle-salt (Y-cut) crystal echo-ranging transducer (QBF) at 30 kc.

measured by means of a nondirectional hydrophone-that is, one for which b=1 in every direction.

To provide a more accurate bearing determination, the electric connections to the acoustic elements of the transducer may be altered when its function is changed from projector to receiver. One method is to split the transducer elements electrically into two halves and connect them so that through one amplifier the transducer is most sensitive to sounds coming from slightly to the right of the transducer bearing. Simultaneously through another amplifier, the transducer is most sensitive to sounds coming from slightly to the left. The transducer, as a hydrophone, thus has two directivity patterns, which are not the same as the pattern when the electric connections are not altered.

Directivity patterns of Rochelle-salt crystal (45 degrees
Z-cut) echo-ranging transducer (QBG) taken in both vertical
and horizontal planes.
Figure 4-18. -Directivity patterns of Rochelle-salt crystal (45° Z-cut) echo-ranging transducer (QBG) taken in both vertical and horizontal planes.

 
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Directivity Index

The directional characteristics of a transducer can be described by stating the fraction of the sound energy that is sent out in the desired direction. This fraction is found essentially by computing the directivity index.

The directivity factor, K, is the ratio of the total energy radiated by a transducer to the energy that would be radiated if the transducer radiated its maximum intensity in all directions. The directivity factor is also the ratio of the average of intensities taken in all directions to the maximum intensity. This ratio evidently provides quantitative information on the directivity. The directivity factor may be useful also in computing the total acoustic power from an absolute-intensity calibration made upon the principal lobe. If K is unity, the transducer is entirely non-directional, whereas if it is a small fraction, a large proportion of the energy is concentrated near the direction of maximum emission, the "acoustic axis."

If the average intensity is bar(I), and the maximum or axial intensity is Ia, the directivity index, D, is defined by

D=10 log K=10 log bar(I)/Ia.  (4-8)

For a nondirectional transducer, D is zero; for a directional one, D is a negative number. The directivity indices of the various highly directional transducers mentioned in the preceding paragraphs range from -20 to -26 db.

Measurement of the directivity index is required in order to obtain the efficiency of a transducer. It is unfortunate that these measurements are the most difficult and least accurate of all calibration tasks. In the present state of the art very great care is required to obtain an accuracy of ±1 db, and errors of ±2 or ±3 db are much more usual. For this reason and for theoretical reasons it is desirable to obtain an expression for the directivity factor and index.

If simplifying assumptions are made-such as uniform loading, uniform phase and amplitude distributions, and infinite baffle-the directivity index for a transducer of a given size and shape can be calculated theoretically from the constants of the apparatus without involving excessively unwieldy mathematical treatment. For example,

  a circular plate with a diameter, d, that is greater than 2 wavelengths can be shown to have a directivity factor of

K=(λ/πd)2,  (4-9)

or the directivity index is

D=20 log (λ/πd),   (4-10)

where λ is the wavelength in units the same as those of d.

Generally, D is calculated from the beam pattern, or directivity function, b, which was defined by

b=I/Ia,  (4-11)

where I is the intensity at a given point and Ia is the intensity at a point equally distant from the source but located on the axis. If b is averaged over all directions, this average evidently gives K, and hence D.

When used as a hydrophone, the directivity index of a transducer is defined as follows:

Sound incident on the hydrophone from a standard source located at a point in any direction at a distance r from the hydrophone generates electric power W2. The same source placed on the acoustic axis at the same distance generates electric power Wa. The ratio W2/Wa can be called b', the directivity function of the receiver. The values of b and b' are equal for a given transducer; unless, the transducer is split for accurate bearing determination.

As with the projector, b' can be averaged over the directivity pattern and the value of D calculated as before.

The directivity index of a hydrophone also determines its response to a multidirectional sound source. Consider two sound fields, one caused by a single source located on the axis of the hydrophone, and another by sources distributed equally in all directions from the hydrophone. Let (1) both sets of sources result in the same sound pressure at the hydrophone, (2) Ea be the emf generated by the single source, and (3) Ei be the emf generated by the isotropically distributed sources. Thus

20 log Ei=20 log Ea+D.   (4-12)

Because D is a negative number, Ei is less than

 
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Ea. This relation has practical importance in the calibration of hydrophones.

IMPEDANCE OF TRANSDUCERS

Mechanical Impedance

Transducers designed for the generation of ultrasonic waves in water have a construction that is markedly different from that of the familiar loudspeakers for the generation of sound in air. It is not possible to do justice here to all the factors entering into these designs, but some of the basic principles are summarized.

The objective, in both constructions, is to set the medium into periodic motion. To set it into motion a force must be applied to the medium. This operation is accomplished most readily by means of a plate, or diaphragm, to which the force is applied almost directly. This plate is often a circular disk. Suppose it is desired to give a point on its surface the velocity

v=vo cos ωt cm/sec,  (4-13)

where vo is the maximum value of the velocity, ω is 2πf, and f is the frequency of the sound to be produced. If this velocity is attained, the water or air in immediate contact with the plate probably moves with this same velocity when vo, is not too great. Later in this chapter, the possibility is considered that the medium does not follow the motion of the plate, but for the present such lost motion is ignored.

The first problem is the calculation of the force required to produce the motion. This force is proportional to vo and to a quantity Z. This relation is analogous to that between voltage and current in an electric circuit, and Z is, by analogy, called the mechanical impedance of the plate. The resistance and the inductive and capacitive reactance that make up the electric impedance have their mechanical analogies. The value of Z depends on (1) the mass, size, and shape of the plate; (2) the mechanical properties of the plate, such as stiffness; (3) the density of the medium; (4) the velocity of sound in the medium; and (5) ω.

The force required to drive the diaphragm or plate at the velocity vo is supplied by an electro-mechanical device called the motor. It is similar to the ordinary motor in that it converts electric power into mechanical motion, but, because the motion is oscillatory rather than rotatory, the analogy is not very close.

  A closer analogy is obtained by considering the motor as a transformer. Then the velocity Vo, is analogous to the output current, and the force F to the output voltage of the transformer. The radiation impedance is directly analogous to the impedance of the output circuit of the transformer. This analogy can be used to describe the effect of taking a transducer out of water and into air. Suppose the transducer has been designed to work under water. Then the lower radiation impedance of air effectively short circuits it. The transducer heats up, just as an ordinary transformer when it is short-circuited. Very little power is usefully transformed.

Conversely, a transducer designed to work efficiently in air is analogous to a transformer with a low-voltage, high-current secondary. Such a transducer is not efficient under water, where the requirements correspond to a high-voltage, low-current secondary.

The physical differences between a loudspeaker designed to work in air and a transducer designed to work in water can be understood by means of this analogy. The loudspeaker always has a thin diaphragm of small mass-one that is easily movable. The motor usually applies the necessary force by magnetic means. In principle, a small bit of magnetized steel attached to the diaphragm might be attracted and repelled by a stationary electromagnet through which an alternating current is passed. Even if such a device could be immersed in water without physical damage, the force obtainable in this way would not be sufficient to move the mass of water in contact with the diaphragm, and the device would "stall."

Underwater transducers usually have more massive diaphragms, which are appropriately described as plates. The moving part of the motor is in rigid physical contact with the plate. The large force necessary to move the plate and adjacent water is produced by any one of several methods. It is possible to design electromagnets to furnish this force, but more motors in use at the present time depend on the magnetostrictive of the piezoelectric effect for this purpose. These effects are capable of producing large forces without the complications that would result from the use of large electromagnets.

 
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SOUND OUTPUT

Electric Power Input and Acoustic Power Output

In rating a transducer, it is essential to know how much of the applied electric power is available as acoustic power, and how much of the available acoustic power is concentrated in a narrow beam.

The electric power input can be measured either from the applied voltage and the impedance of the transducer or from the current and impedance of the transducer.

The acoustic power output can be computed from measured pressure levels. The total power is given by the energy flow per second through the surface of a sphere surrounding the transducer. The average intensity bar(I), over a sphere of radius r multiplied by the surface area of the sphere, 4πr2, therefore is a measure of the acoustic output of the transducer. Because bar(I)=KIa, where K is the directivity factor and Ia is the axial intensity, the acoustic power is 4πr2KIa.

The axial intensity is commonly measured by mounting a hydrophone at a convenient distance on the acoustic axis of the transducer and transmitting continuous sound by use of a constant signal current.

Efficiency and Response of a Transducer

Only that portion of the electric power that is converted into acoustic power is available for echo ranging. The efficiency of a transducer is defined in decibels by 10 log (Po/Pi), where Pa is the acoustic power output and Pi is the electric power input. If a system is, for example, 50 percent efficient, the efficiency is 10 log ½, or -3 db. An efficiency of 10 percent would be -10 db, and so on. The efficiency of a standard echo-ranging transducer ranges from -2 db to -15 db.

A convenient method of rating a transducer is to state the axial sound level reduced to 1 yard1 (the axial source level) per volt or ampere of the impressed voltage or current. This value is called the response of the transducer.

The acoustic power output P, the axial source level Sa, and the directivity index D, are related by the empirical equation

Sa=71.6+10 log P-D.  (4-14)


1 The standard unit distance for calibration adopted by the Navy is 1 meter. One yard and 1 meter are not sensibly different for this purpose.
239276°-53-7
  The performance of a given transducer is completely described by the response, the directivity index, and the efficiency. The characteristics of some standard echo-ranging transducers are listed in table 7.

TABLE 7 - Characteristics of Some Standard Transducers

Code Type Resonant
freq.
(kc)
D Source level (Sa) Efficiency
(db)
QGA-942 MS1 30 -23.2db 85db -6
QGA-941 MS 15 -18.1 77 -7.5
QBF RS
Y-cut
24 -21.1
(20kc)
-23.5
(26kc)
-25.2
(30kc)
88.5 -3.6
QBG RS
X-cut
22 -17.3
(22 kc)
33
(22kc)
39
(45kc)
?
QCU MS 25 -22.5 84 -3.8
QCL MS 20 -21.4 43 -9.5
QCJ MS 24 -22.1 46.5 -9.5
Asdic Quartz 15 -22.0 56 -3.1

1 Magnetostriction.

Limitation of Power Output by Electric Characteristics

It would appear that very long echo ranges might be achieved by increasing the power input into the transducer system, and that the only limit on the available power would be imposed by the permissible size and weight of the gear. This supposition is not true. There are two limiting factors in determining the power output, aside from structural requirements.

The first of these factors results from electrical characteristics. The voltage across the face of a crystal cannot be increased indefinitely, for at a certain critical voltage a spark passes. This action is referred to as voltage break-down. Some idea of the magnitude of the maximum voltage that can be applied may be gained from the fact that the specifications for ADP crystals for echo-ranging transducers require that each crystal must withstand a voltage gradient of 20,000 volts per inch at a frequency approximately one-half the resonant frequency for at least 30 seconds.

In magnetostriction transducers a limitation to the power input is set by the fact that the magnetostriction effect becomes negligible when a certain critical value of the magnetic field strength

 
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is reached. Nickel, for example, exhibits practically no magnetostriction for field strengths greater than from 200 to 250 gausses. (See figure 4-2.)

Limitation of Power Output by Cavitation

The second factor that limits the power output of transducers is cavitation.

An acoustic transducer consists essentially of a vibrating face or piston. The motion of the face is imparted to the water, in which the disturbance is propagated as a wave. This process can proceed efficiently only as long as the water follows the motion of the transducer face. When this motion becomes too violent, the face tears away from the water, with a marked loss of efficiency in the process of sound production.

This limitation on the output of a transducer is thus closely related to the phenomenon of cavitation. Let p be the rms acoustic pressure at a point where the normal hydrostatic pressure is po. Then once each cycle of the sound wave the total pressure changes from po-1.41p to po+1.41p and back again. Cavitation may occur whenever the total pressure tends to become negative. Accordingly, the greatest rms acoustic pressure that can be transmitted through a region where the hydrostatic pressure is po is p=po/1.41. In terms of sound level,

critical level= 20 log po-3.  (4-15)

When po is 1 atmosphere (35 feet of water or 106 dynes/cm2), L is 117 db. When the sound level exceeds this critical value, cavitation bubbles may be formed and cause high transmission losses. Cavitation bubbles are described with wakes in chapter 2.

Because the acoustic pressure is highest at the face of the transducer cavitation occurs there before it occurs elsewhere. This action constitutes the process discussed earlier. As a result of the process, the power output of the transducer, for a given motion of the face, is reduced.

Aside from the reduction of power output of a transducer for a given motion of its face because the water does not follow the moving face, the power output may be reduced for other reasons. Thus, it has been observed in experimental tanks at the Naval Research Laboratory that small air bubbles may form on the transducer when it is

  warmer than the water. This formation may occur also under other conditions. Accompanying the formation of these almost invisible bubbles, the sound output of the transducer for a given electric input is much reduced. Under similar circumstances its sensitivity as a hydrophone also diminishes.

Three Standard domes.
Figure 4-19 -Standard domes.

 
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EFFECT OF DOMES

The transducer unit, consisting of the transducer and the shaft that supports it, is usually installed near the bow of the ship. Because the housings in which transducers are encased are usually spherical, they cause excessive turbulence, and sometimes cavitation, even at moderate speeds. This action causes excessive background noise. For this reason, transducers are generally enclosed in streamlined metal shells called domes.

Several types of domes are in current use by the Navy. They are all made of corrosion-resistant steel; the front is very thin so as to form a "window" to transmit the sound; the back is heavy so as to damp unwanted noise from the propellers. One type is equipped with a bulkhead just aft of the transducer, which supports a sound-absorbing baffle on the forward side and a sound-reflecting pad on the after side. Both these devices reduce sound reception through the stern section of the dome, and the baffle also aids in reducing multiple reflection within the dome. Some domes are retractable and when not in use are withdrawn into a sea chest built into the hull.

Some standard domes are illustrated in figure 4-19.

The acoustic effects of the use of domes are two-fold. In the first place, the axial source level of a dome-enclosed transducer is less than that of the same transducer without a dome. In the second

  place, the directivity pattern of the dome-enclosed transducer differs from that of the same transducer without a dome.

The two effects are closely related. It is not possible to construct domes of materials that are entirely transparent acoustically. Thus, a certain amount of multiple reflection occurs inside the dome. As a result, some of the sound energy that is emitted by the transducer into the main lobe of the sound beam is diverted from it. This action reduces the axial source level.

Any energy diverted from the main lobe, however, must be redistributed in some manner. It is quite possible, therefore, that new side lobes may be added to the directivity pattern, for the regular shape of the dome would preclude a mere random redistribution of the diverted energy. Moreover, it is obvious that multiple reflections inside the dome may affect the original side lobes of the bare transducer pattern.

The decrease in the axial source level due to the distortion of the directivity pattern is equal to the change in the directivity index that ensues when the transducer is placed in a dome.

In echo ranging, a loss in the transmission reduces the effective range; and the distortion of the directivity, especially if accompanied by the formation of prominent side lobes, tends to confuse the determination of bearings. Hence, the various factors that have been adduced must be considered in the design of a dome.

 
Receiver Sensitivity and Background Noise
 
RECEPTION

Although the several methods of echo portrayal are quite different, the general principles that govern them are similar. The echo is only one of many sounds picked up by the sonar. Each sound, whether wanted or unwanted, actuates the portrayal device. The echo must be heard in spite of the unwanted sounds that are being heard at the same time, or it must be seen among the records of these other sounds.

An ideal sonar would respond only to the echo and not to any other sound. This ideal is unattainable, but steps can be taken to approach it. For example, in listening to the radio we wish to hear the broadcast of only one station at a time;

  so we tune our set, with the result that it responds only to the electric waves of the relatively narrow range of frequencies emitted by the particular station and not to those of any other. In the same way, because the echo has a definite frequency, it obviously is desirable to tune the receiver to this frequency, thus excluding much of the unwanted sound. Such tuning is more important with visual than with aural methods of portrayal, for the ear has the ability to ignore unwanted sounds and to hear a note of definite pitch even in the presence of noise.

The sonar receiver can be tuned at various stages. The first is the so-called radio-frequency (r-f) stage, in Which the receiver can be tuned to the frequency of the incoming echo. In the

 
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second stage, the receiver is tuned to the intermediate frequency (i-f), which is the first heterodyne stage. Finally it is possible also to tune the receiver in the audio-frequency (a-f) stage, where the once-heterodyned signal is heterodyned a second time to an audible frequency. The tuning is under the control of the operator, and can be accomplished at any one stage or in several stages at once.

Another approach is found in the fact that the echo is sound coming from a particular direction, whereas background noises may come from all possible directions. The unwanted noise can be reduced by using directional transducers. The obvious disadvantage of such a receiver is that it cannot then be alert in all directions simultaneously; but this drawback is offset by the consideration, that, if an echo is received on a directional sonar, the bearing of its source is known at once.

RESPONSE OF TRANSDUCERS AND AMPLIFIERS

Response of Transducers

The electromotive force generated by the transducer is a function of the sound pressure on its diaphragm. This response of the transducer partially determines the response of any system into which it may be connected. Transducer sensitivity at the frequency, F, is defined as the emf developed in the transducer when it is in a sound field of frequency, f, and in a rms pressure of 1 dyne/cm2. If e is the emf generated by the transducer when in a sound field of p dynes/cm2, the ratio

k=e/p  (4-16)

defines the sensitivity of the transducer. It is measured in volts/dyne/cm2. The quantity

K=10 log k2= 10 log e2/p2=20 log k,  (4-17)

is called the response of the transducer. The response is the decibel ratio of the power generated by the transducer per ohm resistance of the external circuit to the intensity of the sound field at the transducer. If P is power per ohm resistance and I is the intensity of the sound, then because P=e2 and, I is proportional to p2

K=10 log e2/p2=10 log P/I.  (4-18)

  Response Curves

The graph showing the response at each frequency is called the response curve of the system. The response curves of two QC magnetostriction transducers are shown in figure 4-20. They

Response curves of transducers. A, Type QCJ;
B, type QCL.
Figure 4-20 -Response curves of transducers. A, Type QCJ; B, type QCL.

respond well only to sounds in the neighborhood of the resonance frequency, Fo. In the case of the QCJ (figure 4-20, A), Fo is 24 kc-that is, the transducer is said to resonate at 24 kc. The width, w, of the resonance peak, shown in the figure, is usually defined as the frequency separation of the two points on the curve which are 3 db below the maximum. In the given curve, w is about 1 kc.

Another commonly specified quantity is the resonance parameter, Q=Fo/w. If Q is greater than about 20, the system is said to be highly resonant; if Q is less than 4 or 5, the system is nonresonant. The QCJ transducer has a Q of about 24. The QCL shown in figure 4-20, B, is more sharply.

 
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resonant than the QCJ; the w of the QCL is between about 200 and 300 cycles, and, because it resonates at 21 kc, its Q is between 70 and 105. Because of the resonance of these transducers they are tuned-that is, the echo frequency must be near the resonant frequency or they will not respond effectively.

Response of Amplifiers

The amplification ratio of an amplifier is similar to the transducer sensitivity. It is the ratio of the output voltage to the input voltage. The response is defined in terms of amplification ratio in exactly the same manner that the response of a transducer is defined in terms of its sensitivity. Response curves can be plotted for amplifiers as well as for transducers and the same terminology is applied to them.

SPECTRUM LEVEL AND RESPONSE TIME

Power Spectrum Level of Noise

The response curve shows the emf generated by a transducer in responding to a sound of a definite frequency. Most of the unwanted sounds encountered in echo ranging do not have a definite frequency, and it is necessary to consider the emf generated by the transducer in response to such a sound.

Consider an ideal transducer the response curve of which is rectangular, as illustrated in figure 4-21. Suppose that it is possible in some way to vary both Fo and w, while the transducer is exposed to a constant noise. The emf generated then depends on both Fo and w. If w is made successively smaller, the power P of the generated emf finally becomes proportional to w-

P=k2I(Fo)w.   (4-19)

The two other factors in this equation are k, the sensitivity of the transducer to a sound of the frequency Fo, as defined by equation (4-16) and a function I(Fo), which is characteristic of a particular noise. This function has not been given a simple name but is sometimes called the intensity of the noise in a 1-cycle band. The function

N(Fo)=10 log I(Fo)  (4-20)

is called the spectrum of the noise, or its spectrum level at Fo.

To distinguish the spectrum of a continuous noise from that of a pulse, it is often necessary to

  call that of a continuous noise a power spectrum and that of a pulse, an energy spectrum. Equation (4-19) then becomes

10 log P=K+N+ 10 log w.  (4-21)

For all wide-band noises encountered in echo ranging, equation (4-19) is sufficiently accurate even for resonant transducers like the QC, the response curves of which are far from ideal. Equation (4-19) indicates that the power generated by a wide-band noise is proportional to the width of the resonance peak of the transducer.

Energy Spectrum Level of a Pulse

Although the intensity of an uninterrupted, constant sound is most conveniently measured in terms of power (energy per second), the intensity of a pulse is better measured in terms of energy- that is, power times duration. The energy spectrum of a pulse can be defined in much the same manner as the power spectrum of an uninterrupted sound.

If the pulse consists of a train of sinusoidal waves, it will have a fairly definite pitch, say F cycles per second provided the train contains many complete waves. The definite pitch of such a pulse indicates that its energy spectrum has a sharp maximum at the frequency F. If the number of waves in the train is diminished, the height of this peak decreases, and its width w increases. The complete mathematical discussion of this effect can be given in elaborate form, but the essential result is simple.

Let the duration of the wave train be τ seconds. Each wave requires 1/F second to pass a given point. Therefore, if the wave train contains many complete waves, τ >> 1/F The duration of the pulse

1 -Response curve of an ideal transducer.
Figure 4-21 -Response curve of an ideal transducer.

 
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Response time of a filter in relation to its bandwidth.
Figure 4-22 -Response time of a filter in relation to its bandwidth.
τ and the width w of the resonance peak of the pulse are connected by the approximate equation

wr=1.  (4-22)

The greater width of the resonance peak associated with a shorter pulse duration makes it appear that a short pulse can be analyzed into a much wider group of frequencies than a long one. The human ear behaves in a manner consistent with this mathematical relation. If a listener hears pulses consisting of trains of sinusoidal waves, his sensations depend on the number of waves in the train. If the pulse contains many complete waves, the sensation is that of a short tone of well-defined pitch. As the number of complete waves diminishes and the pulse becomes shorter the listener finds it more and more difficult to be sure of the pitch. Finally, very short pulses consisting of only two or three waves lose all tonal characteristics and are best described as "clicks" or "pops."

  Response of Band Filters to Short Pings

Because the width of the spectrum peak of a pulse is inversely proportional to the pulse duration, it might be expected that in designing filters intended to pass only a restricted group of frequencies centered at F cycles per second the duration of the pulse would have to be considered. As a very short pulse has a wider peak it seems obvious that if a filter is to pass it with minimum diminution of intensity the width of the filter must be greater than is necessary for a longer pulse with its proportionally narrower peak.

This fact can be stated in another way. There is a relation between the width of the filter and the speed with which it responds to sudden changes of input. This relation is illustrated in figure 4-22. The two upper graphs, A (1) and A (2), represent the input of a long and a short pulse, respectively. The graph at the left in row B shows the frequency response curve of a wide

 
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filter; the one at the left in row C, that of a narrow filter. The remaining curves-B (1), B (2), C (1), and C (2)-represent the outputs of the filters when excited by the corresponding pulse shown in A (1) and A (2).

The input in each case begins and ends gradually. It requires a certain time interval, tR, to come anywhere near its maximum value. theoretically, it requires an infinite time to reach its maximum value; hence the rather vague working of this sentence. This time interval is indicated in each of the diagrams, which show that it is much shorter in the wide filter, B, than in the narrow one, C. The relation between the response time, tR seconds, and the width, w cycles per second, of the filter is given by the inequality

tR greater than or equal to 1/w.  (4-23)

In a well-designed filter the equality may be assumed to hold; but a poorly designed filter may have a response time considerably greater than 1/w.

The diagrams show that if the pulse is long enough, as illustrated by the curves B (1) and C (1), the response time is short enough that the pulse can come near its maximum value even in the narrow filter, C (1). If the pulse is short, however, as illustrated by the curves B (2) and C (2), the response time of the wide filter is short enough to permit the pulse to come up to maximum value B (2), but this principle does not apply to the narrow filter, C (2). In C (2) the input ceases before the response time has elapsed. Thus the output is always less than its maximum value.

It follows that if a receiver is to respond fully to pings of a length ro yards, the duration of which is thus

2ro/v = ro/800 seconds,

where v is 1,600 yd/sec, the pass band of the receiver must be at least 800/ro cycles per second.

LIMITATIONS ON THE USE OF SHARPLY TUNED RECEIVERS

Besides the limitation on filter width set by the ping length, there are several other factors that prevent full exploitation of tuning as a method of

  eliminating the unwanted sounds. Several of these factors will be discussed briefly.

Doppler Effect

Even if the frequency of the emitted signal remains constant, the frequency of the echo is not always the same, but depends on the rate at which the range of the target is changing. Therefore, the tuning cannot be made indefinitely sharp without endangering the reception of echoes from targets with a high range rate.

The change in frequency due to the Doppler effect can be very great.

In chapter 3 it was shown that if the emitted frequency in kilocycles is Fo, and the range rate in knots is dR, the frequency of the echo is changed by f cycles per second. Because the Doppler effect is to shift the frequency by 0.7 cycle per kilocycle per knot of range rate,

f=0.7 FodR cycles per second.  (4-24)

This change, f, is an increase if the range is closing and a decrease if it is opening. Because the sonar vessel has a usable speed of 20 knots and the possible targets have speeds of 20 knots, the range rate can be 40 knots opening or closing. If the sonar vessel is emitting 24-kc sound, the doppler shift is 672 cycles per second. This condition requires that the band pass be twice that width, or 1,344 cycles per second. To overcome this difficulty (as previously mentioned), a device called own doppler nullifier (ODN) and target doppler nullifier (TDN) is used.

This device automatically adjusts the output of the receiver to the desired frequency, for example, 800 cycles per second. These doppler nullifier circuits are shown in chapter 7.

Relation of the range rate to the relative bearing
of the target.
Figure 4-23 -Relation of the range rate to the relative bearing of the target.

 
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The range rate depends not only on the speeds of the sonar vessel and the target, but also on the relative bearing of the target. This principle is illustrated in figure 4-23, which shows three successive positions of two vessels passing on constant courses at constant speeds. At time 1 the target is off the port bow of the sonar vessel, and the range is closing rapidly. At time 2 the target is at the point of closest approach and the range rate is zero. At time 3 the target is on the port quarter and the range is opening rapidly. Thus in this case, the change from closing to opening range occurs continuously as the bearing changes.

Reverberation

Reverberation occupies a peculiar position, as it is in some ways an unwanted sound and in others a wanted sound. From the standpoint that reverberation can mask the echo, it is unwanted. Considered in this light, it unfortunately has a frequency that is very close to that of the echo, so that extremely sharp tuning is needed if the receiver is to respond to the echo and not to the reverberation.

Reverberation consists essentially of a large number of echoes; hence its frequency also is affected by doppler. Because the scatterers responsible for the echoes are presumably at rest in the water, the range rate at which the vessel is approaching these scatterers and the magnitude of the accompanying Doppler effect are determined solely by the speed of sonar vessel and the relative bearing of the sound beam. For this reason the Doppler effect of reverberation is called own doppler. If an echo is received from a moving target, there is a difference between the echo frequency and that of the reverberation. This difference is called target doppler. The magnitude of the target doppler is an important factor because it is a measure of the speed and approximate relative bearing of the target. From the standpoint that it enables the determination of target doppler, reverberation is a wanted sound.

When the difference in frequency between echo and reverberation is large and the echo is detected

  by ear, the masking power of reverberation is reduced. Reverberation then functions primarily as a wanted sound. On the other hand, when the speed of the target through the water is small or at right angles to the transducer heading, the unwanted masking effect of reverberation is dominant. This unwanted effect is always the dominant one with visual methods of portrayal, because the range recorder cannot distinguish between sounds of various frequencies.

Own Doppler Nullifier

The major limitation on the use of narrow-band receivers is the necessity for allowing echoes of many different frequencies to pass through the receiver. The limitation would not be so severe if the sonar vessel were at rest, or if its motion did not affect the frequency of the echo. The speed of most targets is relatively small, and the width of the receiver band could be reduced if target speed were the only cause of Doppler shifts.

It would be possible to operate with a narrow-band receiver if the operator could constantly change the frequency of the transmitted signal by an amount just sufficient to compensate for own doppler. The reverberation frequency would then remain constant, and the only frequency shift to be accommodated would be that of target doppler. Because of the necessity of sweeping the sound beam over a wide range of bearings, the own doppler changes rapidly, and it is not feasible for the operator to make this adjustment and still perform his other duties. The ODN automatically accomplishes this result to a high degree of approximation.

When the ODN is set for some one frequency, for example, 800 cps, the local oscillator frequency is automatically adjusted to give this value during the first fraction of a second after transmission; thereafter the adjustment remains constant. In this way the frequency of the reverberation is kept quite constant, and considerably narrower filters can be used in the receiver. The ODN is less useful against submarines capable of high underwater speeds. The TDN is employed to return any echo to 800 cycles at the receiver output.

 
Sonar Location
 
CONSIDERATION OF VESSEL HULL DESIGN

Sonar equipment is installed in many types of vessels ranging from small patrol craft to large

  aircraft carriers and submarines. Each type of hull presents different problems of sonar installation, depending on its width, length, draft, material, and
 
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construction. Although wooden hulls have a greater attenuation for hull-born noise, they have several disadvantages. They are usually very short, are of shallow draft, and are not a stable platform for sonar equipment. The best sonar platform is a long, deep-draft hull.

Selection of Transducer Location

There are no fast rules for the location of a sonar transducer. However, certain factors must be taken into consideration. Because the screws are the primary source of noise, the transducer should be located as far from them as possible. In most deep-draft vessels, the screws are high above the keel, so that if the transducer is located well forward, the hull acts as a shield or baffle and allow, the equipment to look in all directions with less interference from the screws.

Another reason for locating the transducer well forward and deep is because of the pressure gradients created by the bow wave.

The pressure of the water around the bow is increased and as the water moves aft the pressure decreases toward normal. When the pressure of sea water is reduced it begins to gas, causing the water to become bubbly. If the transducer were located in or aft of the point at which this condition occurs, a great number of bubbles would strike the face of the transducer with about the same effect as small steel pellets and thus create a very objectionable water noise.

The magnitude of the bow wave depends on the speed of the vessel. Therefore, its normal tactical employment must be considered when selecting the transducer location. Because the normal tactical speed of the vessel is known the normal magnitude of the bow wave is known, and the Neidemair formula can be used to locate the transducer in the high-pressure area. According to the Neidemair formula, the number of feet aft of the bow that the transducer should be located is determined by KV2, where V is the ship's speed in knots and K is a constant with an average value of 0.14.

  There are two reasons why the transducer is not located at the bow. First, the bow does not provide enough width for mounting a standard sonar dome, and second it is not stable. On small craft it does not require a heavy sea to cause the vessel to pound so that the keel section near the bow is "twixt wind and water." Even if this condition didn't cause mechanical damage to the transducer it would cause too many of the transmissions to be quenched.

Use of Fairing on Hull To Reduce Self-Noise

Any protuberance ahead of the transducer probably is a source of noise; therefore, such things as the pitometer log and water intakes should be aft of the transducer. Loose or badly formed rivets are a source of noise. From these statements it should be clear that the flow of water along the hull must be as smooth as possible and any possible point of turbulence must be smoothed.

Fairing is used on hull openings below the waterline to smooth the flow of water. It is made of an elevated,. half-round section of metal that surrounds the opening. It may be either circular or shaped like a tear drop.

Reduction of Noise from Local Machinery

All machinery aboard ship contributes to the over-all self-noise of the vessel. Because the machinery is mounted on the hull and because the hull is a good conductor of sound, the transducer must be insulated from the hull in order to reduce direct coupling of this noise into the sonar equipment. The transducer is generally insulated by a rubber gasket. A noise survey of the machinery may disclose that certain pumps, motors, or generators are excessively noisy. If after inspection the noise cannot be laid to mechanical trouble in the noisy equipment, it should be determined whether the machinery can be mounted so that the noise it generates is not directly coupled to the hull.

The problem of local noise differs from ship to ship and requires individual noise surveys.

 
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