US3364461A - Transducer array with constant pressure, plane wave near-field - Google Patents

Description

w. J. TRo'r-r 3,364,461

TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELD Jan. 16, 1968 '7 Sheets-Sheet l Filed July 30, 1965 q" El Jan. 16, 1968 Wl J, TROTT 3,364,461

TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEARFIELD Filed July 30, 1965 '7 Sheets-Sheet 2 fa ea da ca ba au fb eb db cb bb ab fc ec dc cc bc ac fd ed dd cd bd ad fe ee de ce be ne ff ef df cf bf af INVENTOR w//vF/ELD J. TROTT ATTORNEY Jan. 16, 1968 W. J. TROTT Filed July 30, 1965 7 Sheets-Sheet 5 sI-IADING ELEMENT Isa a sHADED LINE ARRAY 52\ SHADING ELEMENT I5b b SHADED LINE ARRAY 53N SRADING ELEMENT Isc c sHADED LINE ARRAY sHAoING ELEMENT Isd d SHADED LINE ARRAY I 55X I SHADING ELEMENT Ise e sHADI-:o LINE ARRAY 56`\ sHADING ELEMENT Isf f SHADED LINE ARRAY j 57\ sHAoING ELEMENT |59 sHADED LINE ARRAY i: 585 I sRAoING ELEMENT` Ish e sHADED LINE ARRAY 59\ I SHADING ELEMENT I5) SHADED LINE ARRAY GO sHAoING r ELEMENT I5k c SHADED LINE ARRAY 6l\ a sHADINs ELEMENT Ism b SHADED LINE ARRAY 62\ sHAnING ELEMENT Isn c I sRADED LINE ARRAY G6 SIGNAL souRcE 67 INVENTOR W/NF/ELD J. TROTT ATTORNEY W. J. TROTT Jan. 16, 1968 TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELD Filed July zo, 1965 mm .w-m mmm '7 Sheets-Sheet 4 F m, l

INVENT OR W//VF/ELD J. TROTT ATTORNEY W. J. TROTT Jan. 16, 1968 TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEARFIELD 7 Sheets-Sheet 5 Filed July 30, 1965 ATTORNEY Jan. 16, 1968 w. J. TRoT'r 3,364,461

TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELD Filed July 50, 1965 '7 Sheets-Sheet 6 2O log P O 0.5 I.O

W/NFIELD J. TROTT Y/ff/wiwg,

ATTORNEY W. J. TROTT Jan. 16, 1968 TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELD '7 Sheets-Sheet '7 Filed July 30, 1965 ATTORNEY United States Patent Oiice 3,354,461 Patented Jan. 16, 1968 3,364,461 TRANSDUCER ARRAY WITH CONSTANT PRES- SURE, PLANE WAVE NEAR-FIELD Winfield J. Trott, Orlando, Fla., assignor to the United States of America as represented by the Secretary of the Navy Filed July 30, 1965, Ser. N0. 476,214 28 Claims. (Cl. 340-6) ABSTRACT F THE DISCLOSURE A shaded transducer array wherein the individual elements of the array are shaded to produce a constant, plane wave near-field extending over the aperture of said array. The shading is such that the sensitivities of the elements increase from the extremities toward the center of the array according to the coefficients of a summed binomial probability distribution function.

The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.

This invention relates to a shaded transducer array and more particularly to a transducer array that lies in the Y-Z plane of a rectangular coordinate system and radiates a wave in the direction of the X axis wherein the lndividual elements of the array are shaded soas to produce a plane wave with constant Y and Z components in the near field of the array.

Where the elements of the array are electroacoustic transducers, the array produces a constant pressure, plane wave near-field. Where the elements are antennae, the array produces a constant electric field, plane wave near-field.

The present invention finds particular utility in the calibration of electroacoustic transducers from data obtained by measurements made in the near field of the unknown transducer and Will be described with particular reference to such use. However its utility is not limited thereto. Rather, the shading taught by the present invention can be used anytime that it is desired to obtain an electroacoustic array having a constant pressure, plane Wave nearfield or an electromagnetic array having a constant electric field, pl-ane wave near-field.

In the calibration of electroacoustic transducers, particularly those for use underwater, much attention has recently been focused on obtaining the calibration data from measurements made in the near field of the unknown transducer so that the inadequate dimensions and nonideal boundaries of existing calibration facilities do not prohibit the use thereof for the calibration of large, low frequency transducers which are becoming increasingly important for both civilian and military purposes. To date, the methods used to obtain the calibration data from measurements made in the near field have required that numerous delicate measurements be made thus requiring that considerable skill be employed and allowing significant room for error.

A general purpose of this invention is to provide an electroacoustic transducer array which can be used in the calibration of electroacoustic transducers in such a manner that all of the known ladvantages of Calibrating electroacoustic transducers from measurements made in the near field are retained while the aforedescribed disadvantages of so Calibrating transducers are eliminated. To attain this, the present invention contemplates shading the individual elements of an electroacoustic transducer array so that the array has a constant pressure, plane wave near-field and making the measurements necessary to calibrate the unknown electroacoustic transducer while it is maintained within this constant pressure, plane wave near-field.

An object of the present invention is to provide an array having a unique near-field.

Another object is to provide a transducer array which lies in the Y-Z plane of a rectangular coordinate system and which produces a plane wave radiating in the X-direction, which wave has constant Y and Z components in the near field.

A further object of the invention is the provision of a shaded electroacoustic transducer array which produces a constant pressure, plane wave near-field.

Still another object is to provide a shaded antenna array which produces a constant electric-field, plane Wave nearfield.

A still further object is to provide a simple and accurate method of obtaining the data necessary to calibrate an electroacoustic transducer from measurements made in the near field of the transducer.

Other objects and many of the attendant advantages of this invention will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings in which like reference numerals designate like parts throughout the figures thereof and wherein:

FIGS. 1A and 1B, when taken together, illustrate an electroacoustic line transducer shaded according to the present invention;

FIG. 2 illustrates the shading of one quadrant of a plane array having approximate circular symmetry;

FIG. 3 represents a plane array having approximate circular symmetry and being shaded according to the pres ent invention;

FIG. 4 illustrates the shading of one quadrant of a plane array having square symmetry;

FIGS. 5a and 5b, when viewed together, show one quadrant of a plane array having square symmetry;

FIG. 6 is an optimum plot of the relative sound pressure at the center of an array having approximate circular symmetry vs. frequency with the plot normalized for any size array;

FIG. 7 illustrates the method of using the array of the present invention to calibrate an unknown transducer; and

FIG. 8 shows an antenna line array shaded according to the instant invention.

Turning now t0 FIGS. 1A and 1B which, when viewed together, show a twelve element basic electroacoustic line array l5 shaded according to the present invention. Shading elements 19-30 are respectively coupled in series with parallel energized electroacoustic transducers 34-45. It should be understood that transducers 34-45 can be piezoelectric, magnetostrictive, or variable reluctance types and that shading elements 19-30 can be any type of shading means known in the art to be appropriate to shade the particular type transducer used. For instance, if transducers 34-45 are piezoelectric, shading can be obtained by placing appropriate size capacitors in series with each of the transducers or by etching away part of one or more of the electrodes making electrical contact with the respective transducer. If the transducers are magnetostrictive, shading can be obtained by properly selecting the number of turns in the winding on each ofthe cores or by varying the amount of magnetostrictive material in eachV of the cores. These examples are not exhaustive but, rather, illustrate that the shading may be obtained by any means appropriate to control the sensitivity of the individual transducers.

lt should be understood that the line array of FIG. 1 could be used as a receiver as well as a transmitter. When operating as a receiver, the output voltage would appear across a load (not shown) connected between terminals 46 and 47.

It is known that if shading in the form of a Gaussian function is used to shade an array, the pressure along the beam axis of the array will be constant in the near iield and all minor lobes in the far eld directivity of the array will be suppressed. But, shading of this form does not provide the constant pressure, plane wave near-eld desired because the pressure falls off rapidly for positions in the aperture of the array that are 01T the beam axis.

Rather, the instant invention contemplates the use of a shading function that will cause the first minor lobe in the far eld directivity to be attenuated only slightly and all successive minor lobes to be suppressed to a much greater extent. An array so shaded will have a pressure maximum on the beam axis at the far extreme of the near field and a maximum at the arrays periphery. Therefore, the pressure will be kept constant over a greater portion of the aperture than when all minor lobes are suppressed as in shading in the form of a Gaussian function.

It has been found that if a line array is symmetrically shaded about the center with the sensitivity of the individual transducers thereof decreasing from the center toward the respective ends thereof according to the values of a binomial probability distribution function, the desired slight attenuation in the rst minor lobe and much greater suppression of all subsequent minor lobes in the far field directivity will be obtained. The values of the binomial probability distribution can be determined from the formula:

where S=the transducer sensitivity;

This formula will be recognized as the formula for determining the probability of at least r occurrences in n independent binomial trials, when the probability in any single trial is 0.5.

When the line array of FIG. l is shaded in this manner, the shading elements 19-30 are chosen so that transducers 34--45 have sensitivities of 0.03, 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19, and 0.03 respectively. These cO- efficients are obtained by setting n in Equation l equal to 5. Of course, n would not have to be equal to 5 but could be any value appropriate to the number of transducers in the array. One of many published tables in which these coefficients are available is; National Bureau of Standards, Applied Mathematics Series No. 6, Tables of the Binomial Probability Distribution, U.S. Government Printing Oice, Washington, DC., 1950. This table gives coeflicients for values of n from 2-49.

It should be understood that there is quite a bit of latitude available in adding unshaded transducers (that is, transducers unacected by any shading so that their normalized sensitivity coeicients are equal to 1.0) in the center of the array without destroying the constant pressure, plane wave near-field. For example, if a fourteen element line array were desired, the individual transducers thereof could have sensitivities from the left to right ends thereof of 0.03, 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19 and 0.03 respectively.

It is also possible to derive the coeicients of the shading function of the present invention by taking the coeiiicients of a binomial series having a power equal to n and replicating the coeicients of the series m times where m=n+c and c equals the number of unshaded transducers in the center of the array. For example, the coelicients of the shading function for the line array of FIGS. lA and 1B are from left to right, 0.03, 0.19, 0.5,

d 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19 and 0.03. These coetiicients represent the normalized result of taking the coefficients of a binomial series having a power rt=5 and replicating them m=11lc=7 times.

The general equation for the directivity of a line array shaded according to the present invention is:

sin me D um singb s (2) where P: pressure;

di= sin 0,

d=element spacing; t=the wavelength at which the line is operating;

and

zthe angle in the plane containing the line and the beam axis of the line between the plane normal to the line which also contains the beam axis and the plane that contains a distant measuring point and intersects the line at the beam axis.

Applying Equation 2 to the line array shown in FIG.

l, it is seen that the directivity thereof is:

sin 7b p: 7 sintb COSS qs n to have the plane array have the same far-eld directivity as that of the line array.

Consider a plane array lying in the Y-Z plane of a rectangular coordinate system with each row of individual transducers thereof being parallel to the Y axis and containing a given number of transducers spaced from one another a distance d1 and with a given number of columns of individual transducers spaced from one another a distance d2 and each column being parallel to the Z axis. Further, consider a line array superimposed over each row and column of the plane array with the line arrays superimposed over the rows containing the same number of transducers as contained in the rows with the transducers in the line arrays spaced from one another the same distance d1 as the transducer in the rows and with the line arrays superimposed over the columns containing the same number of transducers as contained in the columns with the transducers in the line arrays spaced from one another the same distance d2 as the transducers in the columns. Then, it is known that if the transducers of the plane array are shaded so that their respective sensitivities are equal to the product of the sensitivities of the two transducers superimposed thereover, the plane array will have approximate circular symmetry and a far-field directivity the same as that of the shaded line arrays.

Thus, if the shading function for the line array of FIGS. 1A and 1B is represented, from left to right, by

a, b, c, d, e, f, f, e, d, c, b, a, the individual transducers for a twelve by twelve plane array having approximate circular symmetry and the same directivity as that of the line array would have sensitivities as shown for the upper right quadrant of such as array in FIG. 2. As is obvious from the above, if the identically shaded line arrays used to derive the shading function for the plane array have constant pressure, plane wave near-fields the plane array will also have a constant pressure, plane wave near-held.

FIG. 3 represents a typical embodiment of a twelve by twelve plane array having approximate circular symmetry of shading elements and being shaded according to the.

present invention. Line arrays a-15h, 15j, 15k, 15m; and 15m are all shaded identically to each other and to line array 15 of FIGS. 1A and 1B. For convenience of expression, the 0.03, 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5 0.19, 0.03 shading coecients of line array 15 of FIGS. 1A and 1B and 15a-15h, 15j, 15k, 15m, and 15n and of FIG. 3 are respectively represented by a, b, c, d, e, f, f, e,d,c,b,a.

Shading elements 51-62 are connected in series with the inputs to line arrays 15a-15h, 15j, 15k, 15m, and 1511 respectively and have shading coefficients of a, b, c, d, e, f, j", e, d, c, b, a respectively. Thus, the effective shading of each of the transducers will be the product of the shading coeflicient of the shading element connected in series with the input to the line array containing the transducer and of the shading coefficient of the shading element within the line array that is connected in series with the input to the transducer. For example, the effective shading of the individual transducers within line array 15a will be, from left to right, aa, ab, ac, ad, ae, af, af, ae, ad, ac, ab, aa.

When the plane array of FIG. 3 is operating in a transmitting mode, signal source 65 will impress an input signal across terminals 66 and 67. When the array is operating in a receiving mode, the output voltage will appear across a load (not shown) connected between terminals 66 and 67.

It should be clear that FIG. 3 does not :represent the only possible embodiment of a plane array having approximate circular symmetry. Rather, such an array could be composed of any type of electroacoustic transducer `and by Shading appropriate to the transducer used. However, it will be noted that the sensitivities of the individual elements in the plane array all satisfy the equation:

S'---Slsa (3) where S=The sensitivity of a transducer in the plane array;

S1=the sensitivity of the transducer in a first line array which occupies the same position with respect to the other transducers in said first line array as said transducer of the plane array occupies with respect to the other transducers of the row of said plane array in which it is included, said first line array containing the same number of transducers as contained within said row being spaced identically to said transducers in said row, lthe sensitivities of the transducers in said first line array being shaded so as to increase from the extremities toward the center thereof according to the coefficients of a binomial probability distribution function; and

S2=the sensitivity of the 4transducer in a second line array which occupies the same position with respect to the other transducers in said second line array as said transducer of said plane array occupies with respect to the other transducers of the column of the plane array in which it is included, said second line array containing the same number of transducers as contained within said column being spaced identically to the transducers in said column, the sensitivities of the transducers in said second line array being shaded so as to increase from the extremities toward the center thereof according to the coefficients of a binomial probability distribution function.

FIG. 4 represents the sensitivities of the transducers of a plane array having square symmetry wherein the plane array is shaded according to the present invention. Only the upper-right quadrant is shown. However, it can be seen from FIG. 4 that the full array has families of rectangular, or more particularly square, groups of transducers having the same sensitivities with the groups so arranged that each transducer lies in a horizontal row and a vertical column of transducers. Furthermore, it can be seen from FIG. 4 that the groups are so arranged that the perpendicular bisectors of the sides of the rectangles that are defined by the transducers having the same sensitivity all meet at a common point.

In order that Equation 2 represent the far-field directransducers in the plane array satisfies the equation:

1 th 1 S s. s.

Jp 7LiL ,L X=jLI1 lolx-l) x where jp identifies any one of the groups by the number of groups said one of said groups is removed from said common point and :1, 2, 3, tp;

tpzthe total number of groups;

Sjp=the sensitivity of the transducers in the jp group;

jL identifies any one of the transducers in said line array by the number of transducers said one of said transducers is removed from the center of said line array and :jpg

njL=the number of transducers the jL transducer is removed from the center of said line array;

SjL=the sensitivity of the jL transducer;

tL=onehalf the total number of transducers in said line array;

X identifies any one of the transducers in said line array;

nx=the number of transducers the x Itransducer is removed from the center of said line array; and

SX=the sensitivity of the x transducer.

FIGS. Sa and 5b, when viewed together, show the right quadrant of a l2 by 12 plane array having square symmetry and being shaded according to the present invention. As shown, the shading is achieved by placing appropriate sized shading elements in series with each of the transducers. Such an arrangement would obtain where the shading elements are capacitors and the transducers are piezoelectric elements. However, it should be understood that the present invention is not limited to the use of piezoelectric transducers or series capacitor shading but, rather, may be used in conjunction with any type transducer and appropriate means to control the sensitivity of the transdncer used.

As shown in FIGS. 5a and 5b, the array is connected so as to operate in a transmitting mode with signal source 75 impressing an input voltage across terminals 76 and '77. However, the array could also be used in a receiving mode in which case signal source 75 would be removed and the output voltage would appear across a load (not shown) connected between terminals 76 and 77.

There are a set of conditions pertaining to the upper and lower operating frequency limitations that the line array, plane array having circular symmetry, and plane array having square symmetry all must satisfy if they are to produce a constant pressure, plane wave nearfield. To express this set of conditions, it is necessary to define a radius R as being 1/2 the distance between the points each side of center of the line array around which a symmetry exists with respect to the sensitivities of the transducers on the respective sides of the center Whose sensitivities are affected by the shading. For example, the shading of the basic shaded array of FIGS. 1A and 1B is, from left to right, 0.03, 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19, 0.03. It can be seen that the shading does not affect the transducers having sensitivities of 1.0 and that there is a symmetry of shaded elements on the respective sides of the center around the elements shaded to 0.5. Thus, if each element is considered to be spaced from the adjacent element a distance d, 1/2 the distance between the shaded elements around which symmetry of shaded elements exists would be equal to 1/2 the distance between the elements shaded to 0.5 which would be equal to 75l/2.

Then, the upper and lower operating frequency conditions can be expressed as:

where d=spacing between individual transducers in the line array;

)t the wavelength of the operating frequency; and

R=1/2 the distance between the points each side of the center of the line array around which a symmetry exists with respect to the sensitivities of the transducers on the respective sides of said center whose sensitivities are affected by said shading.

A computation of the relative pressure vs. frequency at the center of an array having circular symmetry shows the greatest pressure amplitude variation that will exist whether the array has approximate circular or square symmetry provided that both arrays are derived from the same basic shaded line.

An array having approximate circular symmetry can be considered to be a piston of unity source strength density with a radius equal to R (which was defined above to be 1/2 the distance between the shaded elements of the basic line array around which symmetry of shaded elements exists) with shading superimposed on this piston by the addition of ring sources.

The relative near-held pressure at the center of an unshaded piston of unity source strength density is given p=2sin1/2kRexpi(wt-l1/z1r1/2 R) (6) where C=21r/)\; \=the wavelength of the operating frequency of the piston;

R=the radius of the piston; and w=the operating frequency of the piston in rad/sec.

The relative near-field pressure at the center of a ring source is given by:

p=2wi sin l/zkd exp [(wt-l-l/zn-*kRQ (7) where W=the source strength density of the ring; k=21r/;

From Equations 6 and 7 it is possible to compute the relative pressure at the center of a plane array having approximate circular symmetry. For example, the plane array of FIG. 3, which was derived from the basic shaded line array of FIGS. 1A and 1B, can be represented by a piston having unity source strength density and a radius equal to the distance between the center Of the line array and the element of 0.5 source strength density with tive ring sources, each of a width d equal to the distance between adjacent elements in the basic line array, superimposed thereon. The rst ring has a source strength density of 0.03 and an average radius Ri equal to the distance from the center of the line array to the element of source strength density equal to 0.97 in the line array; the second has a source strength density of 0.19 and an average radius Ri equal to the distance from the center of the line array to the element of source strength density equal to 0.81; the third has a source strength density of 0.5 and an average radius R1 equal to the distance from the center of the line array to the element of source strength density equal to 0.5; the fourth has a source strength of 0.19 and an average radius Ri equal to the distance from the center of the line array to the element of source strength density equal to 0.19; and the fifth has a source strength density of 0.03 and an average radius Ri equal to the distance from the center of the line array to the element of source strength density equal to 0.03.

When ring sources of source strength densities of 1/2, iWl, iWg, iWa, etc., are superimposed on a piston with a radius R and unity source strength density, the pressure at the center for half of the ring sources, n odd, including the ring with a source strength density of 1/2 and radius R, is given by:

The remaining half of the ring sources n even, will include no ring of source strength density 1/2 and radius R. For this remaining half, R is defined as the distance from the common center of the rings to a point half way between the ring with a source strength density greater than 1/2 and the ring with a source strength density less than 1/2 and the relative pressure at the center of these rings is given by:

It will be observed that Equations 8 and 9 are based on a stepped density distribution rather than a point density distribution. However, this distribution is valid for a square array having approximate circular symmetry.

Using Equations 8 or 9 it is possible to compute the relative sound pressure vs. frequency at the center of an array having approximate circular symmetry. In the example involving the array having approximate circular symmetry of FIGS. 2 and 3, which was derived from the basic shaded line array of FIGS. 1A and 1B, w1 and wz of Equations 8 and 9 are equal to 0.19 and 0.03 respectively.

FIG. 6 is an optimum plot of Equations 8 and 9. The abscissa is normalized in terms of R so that the plot applies to any size array having approximate circular symmetry which is shaded according to the present invention. FIG. 6 can be used to determine the optimum shading function for any array Whether the array be a line array or a plane array having approximate circular or square symmetry. This can be done by shading a plane array having approximate circular symmetry according to a binominal probability distribution function and then adding or taking away unshaded elements from the center of the array until a plot of Equations 8 or 9 approximately yields the curve of FIG. 6. The line array can then be obtained from the plane array having approximate circular symmetry through the application of Equation 3 and the plane array having square symmetry can be derived from the line array through the application of Equation 4.

Measurements show that the diameter of the aperture for a constant pressure, plane wave near-held is approximately the distance between, the transducers having normalized sensitivities of approximately 0.8 and that the axial depth of `the constant pressure, plane wave near field is approximately RZ/, where R equals the distance from the center of the basic line array to the shaded element one side of said center around which symmetry of shaded elements the same side of center exists and =the yvavelength of the operating frequency of the a array.

When it is desired to apply the present invention in the field of electroacoustics, the line array, plane array having circular' symmetry, and plane array having square symmetry can all he constructed in any o-f the manners commonly used in the electroacoustic transducer art. For example, the individual elements may be mounted in oillled tubing or molded into rubber or plastic jackets and mounted according to conventional practice. Often it is desirable to obtain electrical shielding. This can be done in any way Well known in the art such as by mounting the array between wire screens.

It should be understood that the showing of 12 by 12 plane arrays in FIGS. 3 and 4 is not meant to constitute a limitation upon the scope of this invention. There is no necessity that the number of transducers in the rows equal the number of transducers in the columns for an array could be rectangular in shape with one set of binomial probability distribution coet'icients along the rows and another set along the columns.

If an electroacoustic transducer array shaded according to the present invention is to be used as a measuring array to calibrate unknown electroacoustic transducers, the array must be acoustically transparent, This is accomplished by maintaining the operating `frequency at least one octave below the resonant frequency of the individual transducers of the array and by making each of these transducers small in comparison to the Wavelength of the operating frequency.

Also, the array must be calibrated, meaning that the free-field voltage sensitivity M, the near-field transmitting current response Sp, and the ratio of the free-field voltage sensitivity to the effective area of the array M/A must be determined.

Within the range of frequencies, 4 kc.-l2 kc., in which the array of the present invention is of particular interest for calibration, the free-held voltage sensitivity M is the same as the free-field voltage sensitivity of the individual elements of the array.

The near-field transmitting current response of the array, Sp, can be measured by placing a calibrated transducer in the near field of the array and operating the transducer in its receiving mode and the array in its transmitting mode. Then:

Where A :the effective area vof the array, and pc=the characteristic impedance of the environment.

Then,

Sp=M/Jp=M(pc/2A) (12) Ms can be expressed in terms of the far-field transmitting current response, Ss, of the calibrated transducer and the spherical wave reciprocity parameter JS. By denition,

J,=2D t/pc (13) where D==the reference distance for the far-field transmitting current response; )\=the wavelength of the operating frequency;

and

pc--the characteristic impedance of the environment.

Substituting Equations 12 and 14 into Equation l0 yields,

Espa 2SSDM (15) The ratio Es/I is the transfer impedance Ibetween the calibrated transducer when operating as a receiver and the Mpc/2A) Cil l@ array when operating as a transmitter. When the transducer is operating as a transmitter and the array is operating as a receiver, the transfer impedance is E/,Is, Where E equals the open circuit output voltage of the array and Is equals the current driving the calibrated transducer.

Since the acoustical reciprocity theorem applies,

Substituting Equation 16 into Equation l5 and solving for the ratio M/A yields:

Xios. E 17) The sound pressure represented by the product ISSS can be measured by means of a calibrated hydrophone placed in the far iield of the small calibrated transducer when operating in its transmitting mode since where Eh=the open circuit output voltage of the calibrated hydrophone, and

Mh=the free-eld sensitivity of the calibrated hydrophone.

In practice, only the measurements represented by Equations 17 and 18 need, in fact, be made. These measurements provide all the information necessary to determine the near-field transmitting current response, Sp, and the ratio of the free-field voltage sensitivity M to the elective area A.

When an array having a constant pressure, plane wave near-field is calibrated, it can be used to calibrate any unknown transducer whose volume is smaller than that `of the constant pressure, plane Wave near-field of the array.

FIG. 7 illustrates a method of obtaining the data to calibrate an unknown transducer 81 through the use of a calibrated array 32 incorporating the present invention so as to have a constant pressure, plane wave near field. The only limitations upon the use of array 82 to calibrate an unknown transducer 81 are that the Volume of transducer 81 be no greater than the volume 83 of the 4constant pressure, plane wave near-field of array 82, that the unknown 81 be substantially within this constant pressure, plane wvave near-field, and that the operating frequency of unknown 81 be within the range of operating frequencies of array 82. Thus, it is clear that an array shaded according to the instant invention may be designed to calibrate most any size transducer.

As an approximation, the volume represented .by 83 can be considered to have a diameter equal to the distance between the transducers in the basic shaded line array having sensitivities of approximately 0.8 and to extend to a distance from the face of array 82 of RZ/k where R equals the distance from the center of the basic line array to the shaded element one side of center of the basic line array around which symmetry of shaded elements on that side of center exists and )t equals the Wavelength of the operating frequency of the array.

The free-field voltage sensitivity MX of unknown transducer 81 is given by MX=fEx/SPI (19) where Ex=the open circuit output voltage of the unknown;

SP1-the near-field transmitting current response of the a1'- ray; and

I=the current driving the array.

The measurements necessary to determine MX can be made in the embodiment shown in FIG. 7 by putting switch 84 in contact with terminal 85 so that voltmeter 86 will read EX and by putting switch 91 in contact with terminal 92 so that ammeter 93 will read the driving `current I supplied by source 94, Sp is determined in the calibration of array 82.

1i il.

The far-iield transmitting current response of the unknown transducer l, SSX, can be mathematically derived from the near-field transmitting current of the unknown SPX, the plane wave reciprocity parameter Jp, and the spherical wave reciprocity parameter JS.

With the unknown Si operating as a so-urce and the calibrated array S2 operating as a receiver, the near-field transmitting current response of the unknown is given by SML-PHx VJ here p=the average sound pressure produced by the unknown which is measured by the array; and lxzthe current driving the unknown.

Since the average pressure p exists over an eective area AX which is less than the effective area A of the array, the pressure measured by the array is P=(E/M) (f1/Ax) where Ezthe output voltage of the array; and M=the free-field voltage sensitivity of the array.

Therefore,

5px: (E/MIX)(A/AX) (20) The far-field transmitting current response Ssx of the unknown is related to the near-field transmitting current response Spx by the ratio of the plane-wave reciprocity parameter 1p to the spherical-wave reciprocity parameter J5 Thus,

SsX=EA/MIXD (21) where E :the open circuit output voltage of the array;

A :the effective area of the array;

M- ethe free-field voltage sensitivity of the array; Ix=the driving current of the unknown;

D=the reference distance (generally 1M); and Azthe wavelength of the operating frequency.

The measurements necessary to determine Ssx can be made in the embodiment shown in FIG. 7 by putting switch 91 in contact with terminal 96 so that voltmeter 97 will read E and by putting switch 84 in contact with terminal 98 so that ammeter 99 will read the driving current ix supplied by source 100. M and A are determined in the calibration of the array and D is generally 1M.

The directivity of the unknown transducer 81 can also be determined oy measurements made in the near field. Since the acoustical reciprocity principle applies, the directivity of the unknown can be measured by operating it as a source and the array as a receiver and by making point by point measurements of the sound pressure incident upon the calibrated array as the unknown transducer is rotated in such a manner that its beam axis is rotated through 360 while being maintained in the horizontal plane which contains the beam axis of the array.

Other acoustic parameters of the unknown 81, such as source level, transmitting voltage response, and transmitting power response, can also be determined from measurements made in the constant pressure, plane wave near-field of the array 82.

FIG, 8 illustrates how the present invention may be applied to an antenna array. A line array comprising dipoles 111-122 is shown. However, it should be understood that the present invention could be applied to a plane antenna array since the antenna array is analogous to the transducer array.

Dipoles 1111-1 22 are respectively fed in parallel from source 137 through shading elements 12S-136. Shading elements 12S-136 may be coils or any of the other known means of controlling the sensitivity of a dipole. As shown, the shading function for the 12 element antenna line array is, from left to right, 0.03. 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19, 0.03 which will be recognized as the same shading function as that for the 12 element transducer line array shown in FIGS. 1A and 1B. Similarly, a plane antenna array having approximate circular symmetry could be constructed as shown in FIG. 3 by substituting dipoles for the transducers shown and by using appropriate shading means and, also, a plane antenna array having square symmetry could be constructed as shown in FG. 5 by the substitution of dipoles and the use of appropriate shading means.

lith the substitution of `dipoles for electroacoustic transducers and of shading means appropriate to dipoles, an antenna array shaded according to the present invention is subject to the same design limitations, such as requisite distance between elements and operating frequency, as a transducer array so shaded.

An antenna array shaded according to the present invention has a constant electric field, plane wave near-field which is analogous to the constant pressure, plane wave near-eld that an electroacoustic transducer array has when shaded according to the present invention.

Thus, the present invention provides an antenna array which has a constant electric field plane wave near-field and an analogous electroacoustic transducer array which has a constant pressure, plane wave near-iield. Furthermore, the present invention provides a method of calibrating an electroacoustic transducer from measurements made in its near-field.

Obviously many modifications and variations of the present invention are possible in the light ot the above teachings. It is therefore to be understood that, within the scope of the appended claims, the invention may be practiced otherwise than as specifically described.

What is claimed and desired to be secured by Letters Patent of the United States is:

1. An electroacoustic transducer array comprising:

a plurality of transducers;

means for shading the sensitivities of said transducers so that their sensitivities increase from the extremities toward the center of said array according to the coeiicients of a summed binomial probability distribution function having the general formula X (Dromwhere nzthe number 0E independent binomial trials and r=1, 2, 3 n; so that said array has a constant pressure, plane wave near-field extending across the aperture of Said array.

2. The electroacoustic transducer array of claim 1 wherein said array is a line array.

3. The electroacoustic transducer array of claim 1 wherein said array is a plane array.

4. An electroacoustic transducer line array comprising:

a plurality of electroacoustic transducers;

means for shading said transducers so that their sensitivities increase from the extremities toward the center of said line according to the coefficients of a summed binomial probability distribution function so that said line has a constant pressure, plane wave near-field extending along said line.

5. The line array of claim 4 wherein said transducers are electrically coupled in parallel.

6. An electroacoustic transducer line array having a constant pressure, plane wave near-field and an operating frequency comprising:

a plurality of electroacoustic transducers equi-distantly spaced forming said line array;

means for shading said transducers so that their sensitivities increase from the extremities toward the center of said line array according to the coe'icients of a binomial probability distribution function having the general formula (Drown n=the number of independent binomial trials, and r=1,2,3 ...n; said operating frequency being such that where where d=the distance between adjacent transducers in said line array, -:the wavelength of said operating frequency and R=the distance between the points each side of the center of said line array around which a symmetry exists with respect to the sensitivities of the transducers on the respective sides of said center whose sensitivities are affected by said shading.

'7. The line array of claim 6 wherein said transducers are electrically coupled in parallel.

8. The line array of claim 6 wherein:

each of said transducers has approximately the same resonant frequency;

said operating frequency is at least one octave below said resonant frequency; and

the size of said transducers is small in comparison to said distance between adjacent transducers;

whereby said array is acoustically transparent.

9. An electroacoustc transducer plane array comprisa plurality of horizontal rows of transducers lying in a common plane;

each of said rows containing a plurality of transducers:

the transducers in each of said rows being vertically aligned with the transducers in each of every other of said rows so that a plurality of vertical columns of transducers are formed;

means for shading each of said transducers so that its sensitivity satisfies the equation S=S1S2 where S--the sensitivity of the transducer in the plane array; S1=the sensitivity of the transducer in a first line array which occupies the same position with respect to the other transducers in said iirst line array as said transducer of said plane array occupies with respect to the other transducers of the row in which it is included, said first line array containing the same number of transducers as contained within said row being spaced identically to said transducers in said row, the sensitivities of the transducers in said first line array being shaded so as to increase from the extremities toward the center thereof according to the coefficients of a binomial probability distribution function; and S2=the sensitivity of the transducer in a second line array which occupies the same position with respect to the other transducers in said second line array as said transducer of said plane array occupies with respect to the other transducers of the column in which it is included, said second line array containing the same i number of transducers as contained within said column being spaced identically to the transducers in said column, the sensitivities of the transducers Ain said second line array being shaded so as to increase from the extremities toward the center thereof according to the coefficients of a binomial probability distribution function.

10. The plane array of claim 8 wherein:

said plane array has the same number of rows as columns;

the transducers'in said first and second line arrays are shaded so that their sensitivities increase from the extremities toward the centers of the respective line arrays according to the coefficients of the same binomial probability distribution function.

11. An electroacoustic transducer plane array having circular symmetry, a constant pressure, plane Wave neareld and an operating frequency comprising:

a plurality of equidistantly spaced horizontal rows of transducers lying in a common plane;

each of said rows containing the same number of trans` ducers and the transducers within each row being spaced the same distance apart;

said rows being vertically aligned so that each transducer in a row is aligned in a vertical column with a transducer from each of every other row;

means for shading each of said transducers so that its sensitivity satisfies the equation S=S1S2 where S=the sensitivity of the transducer in the plane array; S1=the sensitivity of the transducer in a first line array which occupies the same position with respect to the other transducers in said first line array as said transducer of said plane array occupies with respect to the other transducers of the row of said plane array in which it is included, said first line array having the same operating frequency as said plane array and containing the same number of transducers as contained within said row spaced the same distance apart as the transducers in said row, the sensitivities of the transducers in said iirst line array being shaded so as to increase from the extremities toward the center thereof according to the coeflicients of a binomial probability distribution function having the general formula (www where (wwwrun-ru n=the number of independent binomial trials, and r=l, 2, 3 n; and S2=the sensitivity of the transducers in a second line array which occupies the same position with respect to the other transducers in said second line array as said transducer of said plane array occupies with respect to the other transducers of the column of the plane array in which it is included said second line array having the same operating frequency as said plane array and containing the same number of transducers as contained within said column of said plane array spaced the same distance apart as the transducers in said column, the sensitivities of the transducers in said second line array being shaded so as to increase from the extremities toward the center thereof according to the coeliicients of a binomial probability distribution function having said general formula: said operating frequency being such that where d=the distance between adjacent transducers in one of said line arrays, ?\=the wavelength of said operating frequency, and R=the distance between the points each side of the center of said one of line arrays around which a symmetry exists wit-h respect to the sensitivities of the transducers on the respective side of said center whose sensitivities are affected by said shading.

12. The plane array of claim 11 wherein:

each of said transducers of said plane array has approximately the same resonant frequency;

said operating frequency is at least one octave below said resonant frequency;

the size of said transducers of said plane array is small in comparison to said distance between adjacent transducers of said plane array; and

whereby said plane array is acoustically transparent.

i 13. An electroacoustic transducer plane array comprising:

a plurality of rectangular groups of transducers lying in a common plane; the transducers in each of said groups being so arranged that each of said transducers lies in ia horizontal row and a vertical column of transducers; said rectangular groups being so arranged that the perpendicular bisectors of the sides of :the rectangles formed thereby meet at a common point; means for shading each of said groups so that the sensitivity of the transducers in the jp group is expressible as a function of the number and sensitivities of the transducers in a line array where said number of transducers in said line array equals twice the number of said groups in said plane array and the sensitivities of the transducers in seid line array are shaded so as to increase from the extremities toward the center thereof according to the coefficients `of ra binomial probability distribution function, the sensitivity of the transducers in said jp group being expressible as a function of the nurnber and sensitivities of the transducers in said line .array according to the equation tL SV sin S, nir z=jt|i Winx l) where jp identifies any one of said groups by the number of groups said one of said groups is removed from said common point and :1, 2, 3 tp, tpzthe total number of groups, Sju=the sensitivity of the transducers in the jp group, jL identies any one of the transducers in said line array by the number of transducers said one of said transducers is removed from the center of said line array and =jp, njLzthe number of transducers the jL transducer is removed from the center of said line array, SjL=the sensitivity of the jL transducer, tL=onehalf the total number of transducers in sald line array, x identifies any one of the transducers in said line array, nx=the number of ltransducers the x transducer is removed from the center of said linz array, and Sxzthe sensitivity of the x transducer. 14. The elcctroacoustic transducer plane array of claim 13 wherein adjacent transducers in said rows, said co1- umns, and said line array are `all the same distance apart. 15. The electroacoustic plane array of claim 13 wherein all the transducers in said plane yarray are electrically coupled in parallel.

t6. An electroacoustic transducer plane array having square symmetry, a constant pressure, plane wave nearield, and an operating frequency comprising:

a plurality of rectangular groups of transducers lying in a common plane; the transducers in each of said groups being so arranged that each of said transducers lies in a horizontal row and a vertical column of transducers; means for shading each of said groups so that the sensitivity of the transducers in the jp group is expressible as a function of the number and sensitivities of the transducers in a line array having the same operating frequency as said plane array and having the same distance between adjacent transducers as the distance between groups of adjacent transducers in said plane array, said number of transducers in said line array being equal to twice the number of said groups and 'the sensitivities of the transducers in said line array being shaded so as to increase from the extremities to the center thereof `according to the coefficients of a binomial probability distribution function having the general formuia l5 n=the number of independent binomial trials, and r=l, 2, 3 n; the sensitivity of the transducers in said jp group being eXpressible as a function of the number and sensitivities of the transducers in said line array according to the equation where jp identities any one of said groups by the number of groups said one of said lgroups is removed from said common point and :1, 2, 3 tp, tpzthe total number of groups, Sju=the sensitivity of the transducers in the jp group, jL identifies any one of the transducers in said line array by the number of transducers said one of said transducers is removed from the center of said line array and =jp, nj=the number of transducers the jL transducer is removed from the center of said line array, SjL=the sensitivity of the jL transducer, tL=oneha1f the total number of transducers in said line array, x identifies 4any one of the transducers in said line array, nx=the number of Itransducers the x transducer is removed from the `center of said line array, and Sxzthe sensitivity of the x transducer; said operating frequency being such that where dzthe distance between adjacent transducers in said line array, t=the wavelength of said operating frequency, and R=1/ 2 the distance between the points each side of the center of said line array around which a symmetry exists with respect to the sensitivities of the transducers on the respective sides of said center whose sensitivities are affected by said shading.

i7. The plane array of claim 16 wherein:

each of said transducers of said plane array has approximately the same resonant frequency;

said operating frequency is at least one octave below said resonant frequency;

the size of said transducers of said plane array is small in comparison to said distance between adjacent transducers of said plane array; and

whereby said plane array is acousticaily transparent.

1S. An antenna array comprising:

a plurality of antennae;

means for shading the sensitivities of said antennae so that their sensitivities increase from the extremities toward the center of said array according to the coefficients of a summed binomial probability distribution function having the general formula t?, @yorin where ftzthe number of independent binomial trials and r- -1, 2, 3 n; so that said array has a constant electric ield, plane wave near-field extending over the aperture of said array.

19. The antenna array of claim 1S wherein said array a line array.

20. The antenna array of claim 18 wherein said array a plane array.

21. An antenna line array comprising:

a plurality of antennae;

means for shading said antennae so that their sensitivities increase from the extremities toward the center of said line according to the coeicients of a summed binomial probability distribution function so that said array has a constant electric field, plane wave near-field extending along said line.

22. An antenna line array having a constant electric field, plane wave near-field and an operating frequency comprising:

a plurality of antennae equidistantly spaced forming said line array;

-means rfor shading said antennae so that their sensitivi- !ties increase from the extremities toward the center of said line array according to the coeiiicients of a binomial probability distribution function having the general formula (wwwrun-rn n=the number of independent binomial trials, and r=1, 2,3...n; said operating frequency being such that @man where 23. An antenna plane array comprising:

a plurality of rows of antennae lying in a common plane;

each of said rows containing a plurality of antennae;

the antennae in each of said rows being aligned with the antennae in each of every other of said rows so that a plurality of columns of antennae are formed;

means for shading each of said antennae so that its sensitivity satisfies the equation S=S1S2 where S=the sensitivity of the antenna in the plane array; S1=tl1e sensitivity of the antenna in a first line array which occupies the sarne position with respect to the other antennae in said first line array as said antenna of said plane array occupies with respect to the other antennae of the row in which it is included, said rst line array containing the same number of antennae as contained within said row being spaced identically to said antennae in said row, the sensitivities of the antenna in said rst line array being shaded so as to increase from the extremities toward the center thereof according to the coefficients of a binomial probability distribution function; and S2=the sensitivity of the antenna in a second line array which occupies the same position with respect to the other antennae in said second line array as said antenna of said plane array occupies with respect to the other antennae of the column in which it is included, said second line array containing the same number of antennae as contained within said column being spaced identically to the antennae in said column, the sensitivities of the antennae in said second line array being shaded so as t increase from the extremities toward the center thereof according to the coefficients of a binomial probability distribution function.

24. The plane array of claim 23 wherein:

said plane array has the same number of rows as columns; and

the antennae in said first and second line arrays are shaded so that their sensitivities increase from the extremities toward the centers of the respective line arrays according to the coefcients of the same binomial probability distribution function.

25. An antenna plane array having approximate circular symmetry, a constant pressure, -plane wave near-field, and an operating frequency comprising:

a plurality of equidistantly spaced rows of antennae lying in a common plane;

said rows being aligned so that each antenna in a row is aligned in a column with an antenna from each of every other row;

means for shading each of said antennae so that its sensitivity satisfies the equation S=S1S2 where S=the sensitivity of the antenna in the plane array; S1=the sensitivity of the antenna in a first line array which occupies the same lposition with respect to the other antennae in said first line array as said antenna of .said plane array occupies with respect to the other antennae of the row in which it is included, said first line array having the same operating frequency as said plane array and containing the same number of antennae as contained within said row spaced the same distance apart as the antennae in said row, the .sensitivities of the antennae in said first line array being shaded so as to increase from the extremities toward the center thereof according to the coetiicients of a binomial probability distribution functio having the general formula n=the number of independent binomial trails, and r=l, 2, 3 n; and S2=the sensitivity of the antenna in a second line array which occupies the same position with respect to the other antennae in said second line array as said antenna of said plane array occupies with respect to the other antennae of the column in which it is included, said second line array having the sarne operating frequency as said plane array and containing the same number of antennae as contained within said column of said plane array spaced the same distance apart as the antennae in said column, the sensitivities of the antennae in said second line array being shaded so as to increase from the extremities toward the center thereof according to the coeicients of a binomial probability distribution function having said general formula; said operating frequency being such that where d=the distance between adjacent antennae in one of said line arrays, 7\=the wavelength of said operating frequency, and R=the distance between the points each side of center of said one of said line arrays around which a symmetry exists with respect to the sensitivities of the antennae on the respective sides of said center whose sensitivities are affected by said shading.

26. An antenna plane array comprising:

a plurality of rectangular groups of antennae lying in a common plane;

the antennae in each of said groups being so arranged that each of said antennae lies in a row and a column of antennae;

said groups being so arranged that the perpendicular bisectors of the sides of the rectangles formed thereby meet at a common point;

means for shading each of said groups so that the sen- .sitivity of the antennae in the jD group is expressible as a function of the number and sensitivities of the antennae in a line array where said number of antennae in said line array equals twice the number of said groups in .said plane array and the sensitivities of the antennae in said line array are shaded so as to increase from the extremities toward the center thereof according to the coeiiicients of a binomial probability distribution function, the sensitivities of the antennae in said jp group being expressible as a function of the number and sensitivities of the antennae in said line array according to the equation where jp identifies any one of said groups by the number of groups said one of said groups is removed from said common point and =1, 2, 3 the total number of groups, Sjp=the sensitivity of the antenna in the jp group, iL identities any one of the antennae in said line array by the number of antennae said one of said antennae is removed from the center of said line array and =jp, nJ-L=the number of antennae the jL antenna is removed from the center of said line array, SjL=the sensitivity of the iL antenna, tL=onehalf the total number of antennae in said line array, x identities any one of the antennae in said line array, nX--the number of antennae the x antenna is removed from the center of said line array, and Sx=the sensitivity of the x antenna.

. tp, rp:

a common plane;

the antennae in each of said groups being so arranged that each of said antennae lies in a row and a co1- umn of antennae;

means for shading each of said groups so that the sensitivity of the antennae in the jp group is expressible as a function of the number and sensitivities of the antennae in a line array having the same operating frequency as said plane array and having the same distance between adjacent antennae as the distance between adjacent groups of antennae in said plane array, said number of antennae in said line array being equal to twice the number of said groups and the sensitivities of the antennae in said line array being shaded so as to increase from the extremities to the center thereof according to the coefcients of a binomial probability distribution function having the general formula r=1, 2, 3 n; the sensitivity of the antennae in 2O said jp group being expressible as a function of the number and sensitivities of the antennae in said line array according to the equation 1 tL 1 'SW- m1, SIL z=j+1 m01:- 1) Sx where jp identifies any one of said groups by the number of groups said one of said groups is removed from said common point and =1, 2, 3 the total number of groups, Sjp=the sensitivity of the antennae in the jp groups, jL identies any one of the antennae in said line array by the number of antennae said one of said antennae is removed from the center of said line array and :iw njL=the number of antennae the jL antenna is removed from the center of said line array, SjL=the sensitivity of the jL antenna, tL=onehalf the total number of antennae in said line array, x identifies any one of the antennae in said line array, nx=the number of antennae the x antenna is removed from the center of said line array, and Sx: the sensitivity of the x transducer;

. fp, tp:

said operating frequency being such that where d=the distance between adjacent antennae in said line array, t=the wavelength of said operating frequency, and R=the distance between the points each said of the center of said line array around which a symmetry exists with respect to the sensitivities of the transducers on the respective sides of said center whose sensitivities are alected by said shading.

References Cited UNITED STATES PATENTS 1,643,323 9/1927 Stone 343-844 2,407,329 9/1946 Turner 340-8 2,407,643 9/ 1946 Batchelder 340-16 X 2,419,562 4/ 1947 Kandoian. 2,698,927 1/1955 Parr 340-15 OTHER REFERENCES Dolph, Proc. of the I.R.E., June 1946, pp. 335-339.

RICHARD A. FARLEY, Primary Examiner.

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