Measurement of Characteristics of Sound Reflection and Scattering by Elastic Cylindrical Shells in C...

A. Kleshchev

  Open Access OPEN ACCESS  Peer Reviewed PEER-REVIEWED

Measurement of Characteristics of Sound Reflection and Scattering by Elastic Cylindrical Shells in Conditions of Hydroacoustic Basin

A. Kleshchev

St. Petersburg State Marine Technical University, St. Petersburg, Russia

Abstract

A method and experimental setup intended for measurement the amplitude and phase of acoustic field in the near zone a scatterer are described. The results of measurement the scattering characteristics of low-frequency sound signals scattered by elastic cylindrical shells are analyzed.

At a glance: Figures

Cite this article:

  • Kleshchev, A.. "Measurement of Characteristics of Sound Reflection and Scattering by Elastic Cylindrical Shells in Conditions of Hydroacoustic Basin." International Journal of Physics 1.1 (2013): 5-14.
  • Kleshchev, A. (2013). Measurement of Characteristics of Sound Reflection and Scattering by Elastic Cylindrical Shells in Conditions of Hydroacoustic Basin. International Journal of Physics, 1(1), 5-14.
  • Kleshchev, A.. "Measurement of Characteristics of Sound Reflection and Scattering by Elastic Cylindrical Shells in Conditions of Hydroacoustic Basin." International Journal of Physics 1, no. 1 (2013): 5-14.

Import into BibTeX Import into EndNote Import into RefMan Import into RefWorks

1. Introduction

The following two problems are solved on the basis of experimental data on the amplitude - phase characteristics of the scattered sound field in the near zone of a scatterer made in the form of an elastic finite cylindrical shell [1, 2]:

1). determination of the nature of sound scattering by the shell at low sounding frequencies;

2). calculation of the angular scattering in the Fraunhofer zone from the results of measurement the amplitude and phase of the scattered field in the Fresnel zone

2. The Criterions of the Acoustical Diffracted Measurements

The measurement of the characteristics of the sound reflection and scattering very offen conduct at the models with an use for these intentions of the hydroacoustic basins and the shallow ponds and the water areas [3, 4]. A principal difference of the diffracted experiment (DE) in the conditions of the hydroacoustic basin (HB) and the shallow water area (SWA) along a comparison with the natural conditions of a deep sea consistes in a presence of the reflective boundaries disposed near to the source and scatterer (a free surface, a bottom, the walls of the basin). As a rule, the experimenters are interested by the characteristics of the reflection of a Fraunhofer zone (of a distant field) correspond-ding an infinite medium in that from the infinitely removed source on the scatterer falls a sound plane wave.

Therefore by the DE in the conditions of the HB or of the SWA by the principal problem is an except of an influence of the reflective boundaries of the division of the mediums on the characteristics of the scatterer. The solution of the this problem is begun witch a choise of the form of the sound signal. The optimum properties with this point of view has a pulse signal with a harmonic or frequency – modulated filling. An use of the signal of the such type permits distinguish at a time the useful reflected signal by the body on a background of the preventing reflections from the boundaries. Therefore we are sparing the principal attention in the future the signals of the pulse type.

A distance between the source of pulse signal and the scatterer we will choose so as the wave, falling on the body, at least in the limits of the scatterer, could be received for the plane wave (littlely from it differed). A receiver (a hydrophone) of the reflected signal we will place in the Fraunhofer zone. The criterions of the minimum distance between the combined antenna and the scatterer answering the two brought demands and founded on a known formula (D is the maximum dimension of the scatterer, λ is the length of the sound wave in the medium) were considered in [3, 5]. However in the diffracted measurements the criterion of the minimum distance can be found with the help of the radial wave functions being by the fundamental solutions of the of the scalar Helmholtz equation [6, 7]. We are showing this on the example of the bodies of the simplest form (a sphere, an infinite cylinder, prolate and oblate spheroids) or the bodies round that can circumscribe the spherical, cylindrical or spheroidal surfaces. At first we are determining a difference in the angular characteristics of the scattering by these bodies of the plane and spherical sound waves (for the cylinder we will conduct the comparison between the plane and cylindrical waves). For this we are write the decomposition of a potential of the spherical (or cylindrical) wave in series at the own functions of the Helmholtz equation [8]:

1). in the spherical coordinate system

(1)

where R - the distance between the source and the point of an observation; r - the radial coordinate of the point of an observation; r1 - the radial coordinate of the source; θ - the angular coordinate of the point of the observation relatively of a polar axis;

2). in the circular cylindrical coordinate system

(2)

where r1, r and R form a tringle by that the angle φ is concluded between sides r and r1;

3). in the prolate spheroidal coordinate system

(3)

where ξ1, η1(η1=cos θ1), φ1=0 - the spheroidal coordinates of the source; , (), φ - the spheroidal coordinates of the point of the observation.

Figure 1. The moduluses of the angular characteristics of the soft spheroid

Analogously (3) will look the decomposition of the potential of the spherical wave and in the oblate spheroidal coordinate system, only С must substitute on iC, ξ1 on 1 and on . So in future we are limited by the prolate spheroidal coordinate system, remembered what we can cross to the oblate coordinates by a simple substitution of the wave dimension and the radial coordinate.

The formulas for the potential of the plane wave are got from (1) - (3), if r1 and ξ1 direct to the infinite and instead of the radial functions of third type and the Hankel functions use by their asymptotical meanings [2, 6]. We are writing these asimptotics for the radial functions in the chosen us coordinate systems [2, 6]:

1). in the spherical coordinate system

(4)

2). in the cylindrical coordinate system

(5)

3). in the prolate spheroidal coordinate system

(6)
Figure 2. The moduluses of the characteristics of the sound scattering of the hard spheroid

In the real situations we have an affair with the terminal meanings r1 and ξ1, however and in this case we can accomplish correctly a passage from the spherical (or cylindrical) wave to the plane wave (the field of the falling wave) or from the Fresnel zone in the Fraunhofer zone (the field of scattered wave). We apply to the decompositions (1) - (3). The convergence of the serieses standing in the right parts of these formulas depends from this as quickly are diminiched (over the modulus) with a growth of the indexes of the summing up the productes of the radial functions of the first and third types. The number of the member of the series (nlim, mlim), on that must finish the summing up can be determined from the tables of the radial functions, it will by the function of the wave dimension kr0 (the sphere, the cylinder) or С (the spheroid), of the coordinate r or ξ [in the given case it corresponds the coordinate of the external surface of the scatterer (r0 or ξ0) or circumscribed round it the the coordinate surface] and of the requisite precision. For a practical intention we can be limited by the members by that the modulus of the product of the radial functions of the first and third types at 3 - 4 order less of the biggest modu-lus of the product of these functions by the less meanings of the indexes. Now at the tables of the radial functions we are finding the limital meanings of the indexes of the summing up to that the and the sphere nlim has the order kr0, by the spheroid nlim0 (m=0) of the order 0, by m>0 the meaning nlim m is diminiched with the growth m, in other words nlim mnlim m+1 has one and the same order what and 0/2. We are showing an use of the given method of the choise of the position of the point source relatively of the scatterer on example of the prolate spheroid with the coordinate of the external surface ξ0=1,005, the wave dimension of the body С we are placing equal 10,0. The angular coordinates of the source will invariable (θ0=90˚, φ0=0˚), but the it radial coordinate ξ1 are giving two meanings ; . From the tables of the radial functions we are putting, what the functions of the third type must answer to the asymptotic (6) at least by the following meanings of the indexes: m=0; n=0; 1; 2; 3;…; 13; m=1; n=1; 2;…; 13; m=2; n=2; 3;…; 12; m=3; n=3; 4;…; 11; m=4; n=4; 5;…; 10; m=5; n=5; 6; 7; 8. In the tables 1 and 2 are presented the tabulated and asymptotical [from (6)] meanings of the radial functions of the third type with the indexes m=0; n=0-13 and the radial coordinate ξ1= 2,4; 10,0. The difference between the tabulated and asymptotical meanings of the functions of the third type by ξ1= 2,4 leads to the difference of the angular characteristics of the backscattering of the soundsoft spheroid of the wave from the point source with these radial coordinates, over the comparison with the angular characteristics of the scattering of the plane falling wave (Figure 1). As well can be placed the boundary of the Fraunhofer zone for the ideal scatterers in the form of the sphere, the cylinder or the spheroid (or near to them forms). Howerer in this case the numbers of the indexes nlim (the sphere, the cylinder) or nlim and mlim (the spheroid) of the radial functions of the third type for that must be fulfilled the asimtotical formulas (4) - (6) are determined more complicated combination of the radial functions of the first and third types (by the hard scatterer appear and the derivatives of these functions), entering in the coefficients of the decomposition of the pressure in the scattered wave over the own functions of the Helmholtz equation. The concrete formula for these coefficients is settled proceeded from the form of the boundary conditions and the type of the source of the falling on the body wave. For the ideal scatterer in the form of the prolate spheroid the combination of the radial functions has the following form:

the point source

а) the soft spheroid

(7)

б) the hard spheroid

(8)

the plane wave

а) the soft spheroid

(9)

б) the hard spheroid

(10)

Table 1. The radial spheroidal functions

Table 2. The radial spheroidal functions

If used by the formulas (9) and (10), we are finding the boundary ξ of the Fresnel zone and Fraunhofer zone of the scattered pressure for the plane falling wave and place on this boundary the point source (ξ1=ξ), that as is seen from the comparison (7) and (8) with (9) and (10), this boundary will general for both types indignation. In a confirmation of this in Figure 2 come up the angular characteristics of the sound scattering by the hard spheroid of the pressure in the Fraunhofer zone for the plane and spherical waves. In Figure 2 the curve 1 corresponds the modulus of the angular characteristic of the sound scattering by the hard spheroid for the plane falling wave; the curve 2 corresponds the point source with the coordinate ξ1=10,0 and the distribution of the modulus of the scattered pressure for the radial coordinate ξ1=10,0 by the falling plane wave*; the curve 3 characterises the distribution of the modulus of the scattered pressure for the radial coordinate ξ1=10,0 (it is by the coordinate of the source).

The fulfiled numerical estimations shown, what the minimal distance Rmin from the scatterer to the boundaries of the Fresnel and Fraunhofer zones submits to an inequality Rmin≥2(D2/λ).

We see, what for the scatterers of the spheroidal form the linear distance from the surface of the scatterer ξ0 until a line of the boundary ξ2 depends from the angle of the observation θ: by biggest it will for the angle θ=0˚ and by least for the angle θ=90º (the oblate spheroid, for the prolate spheroid– backwards: maximum by θ=90º, minimum by θ=0˚).

We are noticing as well, what by the angle θ=0˚ (the axially symmetric problem), the index m in the all higher written formulas has one meaning m =0.

By the diffracted measurements in the natural ponds must remember about the following principal demands that they must answer [3]: 1) have enough the big dimensions so as an interference evoked by the reflections from the way of an use of the pulse regime; 2) posses by a low level of a surrounded noise; 3) the liquid medium must be free from all, what can evoke the refraction or the scattering of the sound (of the flows, of the gradients of the temperarature, of the sea organisms, of the bubles and of the soilings).

For a securing of the low level of the surrounded noise, of a stability of a platform and of the conveniences of a production of the measurements must be fulfilled a protection from rainy weather.

By the source of the noise are the ships, disposed near from the basin, the industrial plants (especially the connected with a water pumps), a freight transport, the railroads, a rain, and the waves.

The auxiliary platforms, using by the measurements, are different: the piers, the bridges, the barges, but as well the ships. The piers and the bridges secure the best conveniences and the stability of the conditions of a work. If the basin is big and deep must use a sailing construction, but the stability and the conveniences must secure by a vertical dimension of the construction and by the service lines going under the water or coming to a shore.

Of the simple criterion for a definition of the minimal acceptable depth by the production of the measurements in the basin is not, but the requisite depth in a first turn is determined by the frequency range of the reflective body.

A quality of the hydroacoustic basin depends from that in which degree can remove an influence of the reflections or of the hindrance from the reflections. If the reflections are abolished or absorbed by the absorbes an the boundaries of the basin then last is named by drowned.

The big part of the such hydroacoustic basins are not above and they are similar the natural basins, but less them at the dimensions. By an exeption is the reserved drowned basin, in that can imitate the conditions of the measurements of the deep ocean given the high static pressures and the low temperature of the water.

By the pulse regime of the work the form of the basin is not important. By a limiting factor is the distance until reflective boundaries surfaces nearest to the way of the propagation of the straight acoustic signal from a source to ta scatterer and to a receiver.

3. The Structure Schema of the Experiment and it Methodology

The block diagram of the plant for the amplitude - phase measurements of the reflected (scattered) signal is presented in Figure 3. The block diagram includes in itself the radiating section, the scatterer (model) and the liquid medium. We explain the principle its work, used the time diagrams of voltages in the radiating section (Figure 4). The generator of the harmonic oscillations G creates the harmonic signal of the frequency ω (see the diagram I on the Figure 4), the frequency that is controled by the crystal - controlled digital frequency meter Fr. Entered on the modulator Mod, creating the rectangular video pulse by the durationn τ (see the diagram II in Figure 4), the harmonic signal is turned in the pulse signal with the rectangular envelope and the harmonic filling of the frequency ω (see the diagram III in the Figure 4). This pulse signal with the help of the source S exposes to the sound the liquid medium of the the basin or of the water area and the finding in it model M. The hydrophone H (the receiver in Figure 3) can receive the different signals (in the dependence from the situation). If turns out well divide the useful signal ps (reflected from the model) and the hindrance pp (the reflection from the boundaries of the division of the mediums), then the receiver fixes the signal ps, scattered of the model. If the model is absent, then the hydrophone receives only the straight signal pi. If the receiver is disposed near of the model, then the division of the straight and scattered signals can not and the hydrophone fixes the diffracted pulse signal.

Figure 3. The block diagram of the plant for the measurement of the sound scattering
Figure 4. The time diagrams of the voltages in the radiating section of the plant

At the low frequencies, interfering reflections can be avoided only near the model. In this case, it is almost impossible to separate the direct and reflected signals and the measurement receiver detects the total (diffracted) signal. If the undistorted signal (obtained without the model) is known at the same points, the scattered signal can be determined as the difference between the diffracted and straight signals (under the condition that the amplitudes and phases of these signals are measured simultaneously). From the distributions of the scattered or diffracted signals, measured near the model, it is possible to calculate the angular charactristics (the characteristics of the far field of the scatterer) by using the Kirchhof integral [7, 9, 10]. In acoustics, the method of extrapolating the near - field data to the far field was first applied to hydroacoustics arrays (in the literature, this method has been called the DRL method) [3, 5, 8, 11]).

The sound pulse (ps, pi or pΣ), turned by the hydrophone in electric signal, dos at the preliminary amplifier >, but after at the spectrometer Sp of the ultrasonics frequencies, tuned in the frequency of the filling of the pulse ω. Further the signal follows over the two channels: over the first channel the signal filtered out by the spectrometer Sp is given at the time selector TS that, cuts out 1 - 2 the periods of the placing part of the pulse. The pulse signal with the time selector and the harmonic signal with the generator G of the harmonic oscillations do at the two entrances of the digital phase - meter DPh, that measures the phase of the signal, receiving by the hydrophone, relatively of the supportal signal with the generator G.поступают Simultaneously the signal with the spectrometer Sp dos at the first entrance of the commutator C, at it the second entrance dos the signal with the time selector TS and in the dependence from the position of the relay of the commutator on the screen of the cathode - ray oscilloscope EOs we observe either the all time picture of the pulses in the liquid medium (in the time of the time - base) or the placing part of the useful signal cut out by the time selector TS.

We are telling about the method DRL on the example of the diffracted problem [7, 9]. If we know the distribution of the scattered ps or diffracted pressures on the surface S, surrounding the scatterer, then with the help of the Kirchhoff integral we can determine or in the any arbitrary point of the observation Р [1, 2, 3, 5, 7, 9, 10]:

(11)

where: the index d is s or Σ, Q - the point of the surface S; G(P; Q) - the Green function, submitting to the non - homogeneous Helmholtz equation [2].

(12)
(13)

For the surfaces S of the spheroidal or spherical forms the formulas for G1 and G2 have according the following form [2]:

(14)
(15)
(16)
(17)

where ξ, η, φ; r, θ, φ – the spheroidal and spherical coordinates of the point of the observation Р; ; - the spheroidal and spherical coordinates of the point Q of the surface S; in the given case ξ=ξ0 and .

An advantage of the termal integrals (12) and (13) at the comparison with the binominal (11) obviously: for the definition of the pressure in the distant field enough know the distribution only of the pressure or only its normal derivative along the surface S. The formulas (14)–(17) get simplified [13] for the account of the substitution of the radial functions and their asymptotical meanings, what is fairly for the distant field (for the Fraunhofer zone).

By the use of the binominal integral (11) as the Green function G(P;Q) we choose

(18)

where r - the distance between the points Р и Q. Besides that, from the absence the enough miniature receivers of the oscillatory velocity use one from two approximation:

1) the scattered wave near of the surface S we account by plane and spreading normally to it, in the result the measurement of the normal derivative is become unnecessary as ;

2) the measurement of the derivative we substitute by the relation of the terminal differences of the pressures and the distances Δq between the surfaces of the measurements S1 and S2:

In the Fraunhofer zone (r is enough big) are fair the usual approximations [6]:

(19)

where R - the distance from the beginning of the coordinates of the system О to the point of the observation; D(θ,φ) - the angular characteristic of the scattering in the spherical coordinates θ, φ;

(20)

In the distant zone the single vector from Q in Р can substitute by the single vector from the point О in the point Р, so what

On the practice the continuous distributions of the pressure and of the normal derivative are substituted by discrete and from the integrals we come to the numeri- cal quadraturs for the both approximations [1, 2, 7, 9]

(21)
(22)

where and - the meanings of the corresponding integrand functions in the nodes of the numerical quadratures; L and М – the number of the nodes after the coordinates and according; Al and Dm – the main coefficients; the formulas for and В are given in [8]. By the use (22) for the calculation of the relation we measure the pressures on the two enough near disposed confocal surfaces (the distance between them much less of the length of the sound wave λ in the liquid). For the exception of the influence of the nonhomo-geneous flexural waves (for the scatterers in the form of the elastic shells) the distance from the surface of the scatterer to the nearest (main) point must not be λ/2, the distance between the main points so as we could evoid of the false additional maximums must be less λ/2.

Figure 5. The angular gear for the measurements of the distributions in the near zone

The measurement of the distributions of the diffracted pressure near of the model can be fulfield with the help of the construction, presented in Figure 5. The model 1 is suspended to the construction 2 with the help of the metallic strings. The miniature hydrophones 3 are fixed to the strings of angular gear 4 made from hollow (water - filled) metal tubes. For correct phase meas-urements, the hydrophone size should be small compared to the wavelength of sound. The low-speed electric motor 5 served to rotate the angular gear around the model at the required step in angle. At each fixed position of the system, the amplitude - phase distribution of the diffracted pressure was measured with the use of switch 6 and the receiving channel of the experimental setup (see Figure 5). Through the A/D converter, the distribution was sent to the PC, where the angular characteristic was calculated.

Figure 6. The measuring directions AB, CD, EF and OM; the measuring points; the contour of the cylindrical shell on the measuring template

The field, scattered by the elastic shell, was measured in the Fresnel zone by using the setup shown in Figure 5 and the structure shown in Figure 6. The latter consisted of the rigid template fixed above the water surface. The paper sheet was glued to the upper surface of the template. On the paper, the contour of a cylindrical shell with flat ends, the approximating contour of an elliptic shape, the measuring elliptic contour and the measurement lines AB, CD, EF and OM were drawn. Along the shell contour, the measuring elliptic contour and the AB, CD, EF and OM directions, holes served to suspend miniature spherical hydrophones with the diameter d = 5mm. The strings were strained with the use of a load, which was positioned near the bottom of the tank to eliminate its effect on the scattered sound field measurements [10].

In both types of the experimental setup, the angle at which the model was insonified could be varied by varying the position of the source with respect to the scatterer. The insonifying pulse length was always greater than the maximal size of the model.

As in the setup shown in Figure 1, the measurements were performed twice: first, in the pre-sence of the shell, the magnitude of the diffracted pressure and its phase were measured, then the shell was lifted to the surface and the pressure magnitude and the phase of the incident wave were measured at the same points as before.

4. The Analysis of the Results of the Measurements

The measured amplitude – phase distributions of the diffracted or scattered field near the scattered allow determination of the factors that govern the scatterer field formation. Figure 7 shows the meanings of the magnitude and the phase of the scattered pressure for the finite cylinnddrical shell insonified along its rotation axis. At this axis (the OM direction in Figure 6), the distributions of the magnitude and the phase were measured at different wave distances kz from the end of the shell with the radius a. Curves 1 and 2 represent the experimental values of thmag-nitude and the phase of the scattered pressure respectively. Curves 3 and 4 represent the corresponding distributions for the elastic hollow oblate spheroidal shell with the major semiaxis a in the case of its insonification along the OM direction [14]. Distributions 5 and 6 characterize the magnitude and the phase of the scattered pressure soft oblate spheroid, the wave size of the body С = ka = 3,1.

Figure 8 shows similar distributions of the magnitude and the phase of the scattered pres-sure for aforementioned spheroidal shell along the OM direction with another wave size C = 1,0.

Figure 9 and Figure 10 compare the experimental and calculated (for the acoustically soft in-finite cylinder) distnticributions of the magnitude and the phase of the diffracted pressure, the distributions were measured along the EF direction, the data were obtained for bounded cylinders with the wave size C = ka = 1,0 (Figure 9) and C = ka = 3,1 (Figure 10). In addition, Figure 9 compares the distributions of the magnitude and phase of the diffracted pressure obtained for two shells (model 1 and model 2) with identical lengths and diameters but with different thicknesses of their ends. In both cases, the shells were insonified along the EF direction.

The structure represented in Figure 6 makes it possible (by recalculating the near field to the far field) to obtain the angular scattering characteristic D(θ) for axisymmetric models insoni- fied along their rotation axis, because, in this case, it is not necessary to measure and ψs for different values of the angle φ. As the illustration, Figure 11 displays the results of the measure- ments the magnitude of the angular chararacteristic for the aforementioned cylindrical shell with the use of the setup and method of measurement described above.

Figure 7. The magnitude – phase distributions of the scattered field
Figure 8. The calculated and experimental distributions of the magnitude and the phase of the scattered field
Figure 9. The calculated and experimental distributions of the magnitude and the phase of the diffracted field
Figure 10. The calculated and experimental distributions of the magnitude and the phase of the diffracted field
Figure 11. The modulus of the angular scattering characteristic of the hollow cylindrical shell

5. Conclusions

If the pulsed signal from the scatterer can be singled out in the Fraunhofer zone (the deep- water region), the angular scattering characteristic can be measured directly. Howerer, this procedure is rather complicated, as compared to the method described above, be-cause it requires scanning of the great number of points at the uniform step on the spherical surface in the far zone. In addition, it is necessary to select the useful scattered signal against interfering reflegtions at all of the measurement points. Therefore, in experiments, as the rule, only two - dimensional (rather than three - dimensional) scattering characteristics are measured (or only parts of these characteristics).

Acknowledgments

The work was supported as of research under State Contract no P242 of April 21, 2010, within the Federal Target Program “Scientific and scientific - pedagogical personnel of innova-tive Russia for the years 2009 - 2013”.

Notes




1*By writing down (11) – (13)took into account, what the Kirchhoff integral for the pressurepiin the falling wave equally zero [12].

References

[1]  Kleshchev A. A., “Scattering of Low - Frequency Pulsed Sound Signals from Elastic Cylindrical Shells”, Acoust. Physics. 57(3): 375-380, 2011.
In article      CrossRef
 
[2]  Kleshchev A. A., Hydroacoustic Scatterers, Prima, St. Petersburg, 2012 [in Russian].
In article      
 
[3]  Bobber R., Underwater Electroacoustic Measurements, Peace, Moscow, 1974 [in Russian].
In article      
 
[4]  Klyukin I. I., Kolesnikov A. E., Acoustic Measurements in Shipbuilding, Shipbuil- ding, Leningrad, 1982 [in Russian].
In article      
 
[5]  Horton C. W., “Acoustic Impedance of an Outgoing Cylindrical Wave”, J. Acoust. Soc. Am. 34 (10). 1663. Oct. 1962.
In article      
 
[6]  KleshchevA. A., Klyukin I. I., Principles of Hydroacoustics, Shipbuilding, Lenin- grad, 1987 [in Russian].
In article      
 
[7]  Kleshchev A. A., Klyukin I. I., “The experimental method of the production of the characteristics of the far - field of the scatterers”, Tr. LKI. 77: 37-40, 1972.
In article      
 
[8]  Horton C. W., Innis G. S., “The Comparison of Far - Field Radiation Patterns from Measurements Made Near the Source”, J. Acoust. Soc. Am. 33(7): 877-880, 1961.
In article      CrossRef
 
[9]  Kleshchev A. A., Klyukin I. I., “The use of the diffracted Kirchhoff integral for the calculation of the angular characteristics of scattering”, in 7th All - Union Acoustic Conference, Nauka, Leningrad, 123-125, 1973.
In article      
 
[10]  Kleshchev A. A., “The experimental characteristics of the scattering of the non - sta- tionary sound signal by the cylindrical shells at the low frequencies in the Fresnel zone”, in 21st Session of Russ. Acoust. Soc. Geos, Moscow, 163-166, 2009.
In article      
 
[11]  Baker D. D., “Determination of Far - Field Characteristics of Large Underwater Sound Transduccers from Near - Field Measurements”, J. Acoust. Soc. Am. 34 (11): 1737-1744. Nov. 1962.
In article      CrossRef
 
[12]  Shenderov E. L., Wave Problems of Hydroacoustic, Shipbuilding, Leningrad, 1982 [in Russian].
In article      
 
[13]  Kleshchev A. A., “The some criterions of the acoustical diffracted measurements”, in 7th All - Union Acoustic Conference. Nauka, Leningrad, 143-146, 1973.
In article      
 
[14]  Kleshchev A. A., “The scattering of the sound by the elastic oblate spheroidal shell”, Sov. Phys. Acoust. 21 (6): 571-573, 1975.
In article      
 
  • CiteULikeCiteULike
  • Digg ThisDigg
  • MendeleyMendeley
  • RedditReddit
  • Google+Google+
  • StumbleUponStumbleUpon
  • Add to DeliciousDelicious
  • FacebookFacebook
  • TwitterTwitter
  • LinkedInLinkedIn