Research Article Effects of Medium Characteristics on ... · 2 = * d r01 * d r02 * d r1 F* d r 2 % r r01 | r 1 % r02 | r 2 Fexp K L ! 1 7 M 2 N exp K L ! 2 7 M 2 N F & r | r01 & r

Research ArticleEffects of Medium Characteristics on Laser RCS ofAirplane with E-Wave Polarization

Hosam El-Ocla

Lakehead University 955 Oliver Road Thunder Bay ON Canada P7B 5E1

Correspondence should be addressed to Hosam El-Ocla hosamlakeheaduca

Received 27 May 2015 Accepted 30 August 2015

Academic Editor Jun Hu

Copyright copy 2015 Hosam El-Ocla This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Plane wave incidence should be postulated to have an authentic target detection Practically the plane wave is incapable usually ofkeeping its power in the far field especially when propagating through an inhomogeneous medium Consequently we assume anincident beam wave with a finite width around the target In this work we calculate numerically a laser radar cross section (LRCS)of conducting targets having smooth cross sections with inflection points such as airplane in randommedia Effects of fluctuationsintensity of randommedia on the LRCSperformance are studied in this paper E-wave polarization (E-wave incidence) is consideredwhile the mean target size is approximately twice the wavelength

1 Introduction

Methods to formulate scattered waves were conducted overdecades [1ndash3] Yet these methods require long processingtime and severalmemory resourcesThese restrictionswouldtherefore lessen the usefulness of such methods particularlywhen we consider objects detection applications in real timeA method has been presented in the previous years to solvethe scattering problem numerically [4ndash6] In this methodthe problem is solved as a boundary value problem wherethe current is calculated on the outer surface of the targetAs a result waves intensity and accordingly target detectioncoefficients are accurate in a forward scattering environmentsuch as turbulence Effects of targets configuration randommedia strength and the polarization on the LRCS and thebackscattering enhancement for a beam wave incidence werestudied in several publications (eg [7ndash9]) Our resultsproved to be in excellent agreement with those assuminga cylinder with circular cross section in free space in [10]Lately thismethodwas verified using FDFDmethod showinga fair agreement with an accuracy below 5 error rate forobjects in randommedia and even less error in the free space[11]

Approximate solutions for beam wave functions wereobtained and research on laser radar was pursued as in [12

13] Most of the proposed techniques have limitations ontargets configuration as being restricted to be point targetsor simple canonical cross sections Here LRCS is calculatedfor targets of analytical cross sections with inflection pointsin random medium such as turbulence Targets applicationsinclude civil and military airplane

Propagation of wide-pulse in sparse medium such asatmospheric and other aerosols was studied [14 15] Mediumdispersion likely causes an unwanted signal distortion andretrogradation [16]Thework in this paper is to probe the fre-quency range where the proper selection of beam width mayresult in a minimal effect of random medium parameters onthe LRCS Consequently we will be able to find a mechanismthat maximizes the accuracy of radar detection in a randommedium particularly in the far field Correspondingly wepresent a study on random media parameters and analyzetheir performance to examine their effects on the scatteredwaves These parameters include both medium fluctuationsintensity implemented by the medium correlation functionand the spatial coherence length (SCL) of waves around thetarget

Having the radius of the scatterer much smaller than thesize of the wavelength assumed in [17] is not the general caseparticularly in the high frequency band As a result a targetof a large size compared to the wavelength is a crucial issue

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 515170 7 pageshttpdxdoiorg1011552015515170

2 International Journal of Antennas and Propagation

L

Target

Randommedium

x

z

z

Incidentwave

Scatteredwave

S

fluctuation intensityof random medium

a

y

L ≫ a

0

B0

120579

120601

B(rr) normalized

r = (x z)

Figure 1 Geometry of the problem of wave scattering from aconducting cylinder in a random medium

for the frequency spectrum of waves propagating in highlyfluctuating media that in turn affects the electromagneticwaves detection capabilities Putting this in mind our workassumes a target size approximately twice the wavelength tosuit the remote sensing using high frequency range On theother hand the case where the beam width is smaller thanthe target dimensions is considered tomeet the needs of radarapplications LRCS is calculated using our prescribedmethodfor convex illumination region of partially convex targets

Applications for our study consider conducting targetssuch as airplane where models have different contour com-plexity and shapes We deal with the scattering problem two-dimensionally considering the horizontal polarization (E-wave incidence) The time factor exp(minus119894119908119905) is assumed andsuppressed in the following section

2 Formulation

Geometry of the problem is shown in Figure 1 A randommedium is assumed as a sphere of radius 119871 around a target ofthemean size 119886 ≪ 119871 and also to be described by the dielectricconstant 120576(r) the magnetic permeability 120583 and the electricconductivity ] For simplicity 120576(r) is expressed as

120576 (r) = 1205760 [1 + 120575120576 (r)] (1)

where 1205760is assumed to be constant and equal to free space

permittivity and 120575120576(r) is a random function with

⟨120575120576 (r)⟩ = 0

⟨120575120576 (r) 120575120576 (r1015840)⟩ = 119861 (r r1015840)

(2)

119861 (r r) ≪ 1

119896119897 (r) ≫ 1

(3)

Here the angular brackets denote the ensemble average and119861(r r) 119897(r) are the local intensity and local scale-size of therandom medium fluctuation [18 19] respectively and 119896 =

120596radic12057601205830is the wavenumber in free space Also 120583 and ] are

assumed to be constants 120583 = 1205830 ] = 0 For practical

turbulent media condition (3) may be satisfied Thereforewe can assume the forward scattering approximation andthe scalar approximation [20] Consider the case where adirectly incident beamwave is produced by a line source119891(r1015840)along the 119910 axis Here let us designate the incident waveby 119906in(r) the scattered wave by 119906

119904(r) and the total wave by

119906(r) = 119906in(r)+119906119904(r)The target is assumed to be a conducting

cylinder of which cross section is expressed by

119903 = 119886 [1 minus 120575 cos 3 (120579 minus 120601)] (4)

where 120601 is the rotation index and 120575 is the concavity indexWe can deal with this scattering problem two-dimensionallyunder condition (3) therefore we represent r as r = (119909 119911)Assuming a horizontal polarization of incident waves (E-wave incidence) we can impose the Dirichlet boundarycondition for wave field 119906(r) on the cylinder surface 119878 Thatis 119906(r) = 0 where 119906(r) represents 119864

119910

Using the current generator 119884119864and Greenrsquos function in

randommedium119866(r | r1015840) we can express the surface currentwave as

119869 (r2) = int

119878

119884119864(r2| r1) 119906in (r

1| r119905) dr1 (5)

where r119905represents the source point location and it is

assumed as r119905= (0 119911) in Section 3 The first order of ⟨119866(r |

r1015840)⟩ is 11987210and expressed as

11987210 (119911) = ⟨119866 (r | 119911)⟩ = 119866

0 (r | 119911) 119890minus120572(119911)119911

(6)

where 1198660(r | 119911) is Greenrsquos function in free space [20] and

120572 is the coherence attenuation index [21] at the far field inthe 119911 direction We consider 119906in(r1 | r

119905) whose dimension

coefficient is understood to be represented as [20]

119906in (r1| r119905) = 119866 (r

1| r119905) exp[minus(

1198961199091

119896119882)

2

] (7)

where 119882 is the beam width The beam expression is approx-imately useful only around the cylinder Then the scatteredwave is given by

119906119904 (r) = int

119878

119869119864(r2) 119866 (r | r

2) dr2 (8)

This can be represented by

119906119904(r)

= int

119878

dr1int

119878

dr2[119866 (r | r

2) 119884119864(r2| r1) 119906in (r

1| r119905)]

(9)

Here 119884119864is the operator that transforms incident waves

into surface currents on 119878 and depends only on the scatteringbody [4 6] The current generator can be expressed in termsof wave functions that satisfy Helmholtz equation and theradiation condition

That is the surface current is obtained as

int

119878

119884119864(r2| r1) 119906in (r

1| r119905) dr1

≃ Φlowast

119872(r2) 119860minus1

119864int

119878

(Φ119879119872

(r1) 119906in (r

1| r119905)) dr

1

(10)

International Journal of Antennas and Propagation 3

where

int

119878

(Φ119879119872

(r1) 119906in (r

1| r119905)) dr

1

equiv int

119878

[120601119898

(r1)120597119906in (r

1| r119905)

120597119899

minus120597120601119898

(r1)

120597119899119906in (r1| r119905)] dr1

(11)

The above equation is sometimes called ldquoreactionrdquo as namedby Rumsey [22] In (10) the basis functionsΦ

119872are called the

modal functions and constitute the complete set ofwave func-tions satisfying the Helmholtz equation in free space and theradiation conditionΦ

119872= [120601minus119873

120601minus119873+1

120601119898 120601

119873]Φlowast119872

and Φ119879

119872denote the complex conjugate and the transposed

vectors of Φ119872 respectively 119872 = 2119873 + 1 is the total mode

number 120601119898(r) = 119867

(1)

119898(119896119903) exp(119894119898120579) and 119860

119864is a positive

definite Hermitian matrix given by

119860119864

= (

(120601minus119873

120601minus119873

) sdot sdot sdot (120601minus119873

120601119873)

d

(120601119873 120601minus119873

) sdot sdot sdot (120601119873 120601119873)

) (12)

in which its 119898 119899 element is the inner product of 120601119898and 120601

119899

(120601119898 120601119899) equiv int

119878

120601119898

(r) 120601lowast119899(r) dr (13)

119884119864is proved to converge in the sense of mean on the true

operator when 119872 rarr infinTherefore the average intensity of backscatteringwave for

E-wave incidence is given by

⟨1003816100381610038161003816119906119904 (r)

1003816100381610038161003816

2⟩ = int

119878

dr01

int

119878

dr02

int

119878

dr10158401

sdot int

119878

dr10158402119884119864(r01

| r10158401) 119884lowast

119864(r02

| r10158402)

sdot exp[minus(1198961199091015840

1

119896119882)

2

] exp[minus(1198961199091015840

2

119896119882)

2

]

sdot ⟨119866 (r | r01) 119866 (r | r

02) 119866lowast(r | r10158401)119866lowast(r | r10158402)⟩

(14)

Here we calculate the current generated at r01

as a result ofthe waves incidence at r1015840

1on the object surface and similarly

at the point r02

as a result of the waves incidence at r10158402

In our representation of ⟨|119906119904(r)|2⟩ we use an approximate

solution for the fourthmoment ofGreenrsquos function in randommedium 119872

22[5]

11987222

= ⟨119866 (r | r10158401)119866 (r | r

01) 119866lowast(r | r10158402)119866

lowast(r | r02)⟩

≃ ⟨119866 (r | r10158401)119866lowast(r | r10158402)⟩ ⟨119866 (r | r

01) 119866lowast(r | r02)⟩

+ ⟨119866 (r | r10158401)119866lowast(r | r02)⟩

sdot ⟨119866 (r | r01) 119866lowast(r | r10158402)⟩

(15)

0

02

04

06

08

1

0 2 4 6 8

SCL = 52SCL = 3

kl = 58120587kl = 20120587

k120588

Γ(k120588)

eminus1

Figure 2The degree of spatial coherence of an incident wave aboutthe cylinder

We can obtain LRCS 120590119887using (14)

120590119887= ⟨

1003816100381610038161003816119906119904 (r)1003816100381610038161003816

2⟩ sdot 119896 (4120587119911)

2 (16)

3 Numerical Results

Although the incident wave becomes sufficiently incoherentwe should pay attention to the spatial coherence length (SCL)of incident waves around the target In [6] the degree ofspatial coherence is defined as

Γ (120588 119911) =⟨119866 (r1| r119905) 119866lowast(r2| r119905)⟩

⟨1003816100381610038161003816119866 (r0| r119905)1003816100381610038161003816

2⟩

(17)

where r1= (120588 0) r

2= (minus120588 0) r

0= (0 0) and r

119905= (0 119911) In

this study we consider different random medium intensity119861(r r) defined in (2) and it is assumed to be constant 119861

0as

shown in Figure 1The condition of the coherence attenuationindex 120572 defined in [4] holds with all values of 119861

0 119871 and 119896119897

considered in this paper The SCL is defined as the 2119896120588 atwhich |Γ| = 119890

minus1≃ 037 Figure 2 shows a relation between

SCL and 119896119897 in this case and SCL accordingly is equal to 3and 52 We use both 119861

0and SCL to represent the random

medium effects on the LRCS We consider a variety of valuesfor these parameters to investigate the impact of media withdifferent inhomogeneous fluctuations strength

In the far field as a result of waves propagation throughrandom medium beam waves may lose some of its concen-tration by diffraction so we assume here a limited 119896119882 in ournumerical results

31 Radar Cross Section RCS Wings of airplane have variousslopes with fuselages according to their shapes and models


1

2

3

4

5120590b(2a

)

ka

B0 = 2 lowast 10minus4


0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)B0 = 2 lowast 10minus4


ka

0 2 4 6 8 10 12

(b)

Figure 3 LRCS versus target size for 120575 = 0 and 119896119882 = 15 where the target is located in random medium with (a) SCL = 3 (b) SCL = 52

As a result the intensity of scattered waves is different withthe angle of waves incidence on suchmodels and alsowith therandom medium coherence function SCL and the mediumfluctuations intensity 119861

0 We conduct results in Figures 3

to 6 presenting effects of these parameters on the LRCS Inother words when waves are scattered from fuselage onlywe postulate 120575 = 0 where the cross section has a circularcontour On the other hand waves scattering from spots inthe neighborhood of the fuselage attached to the wings wouldassume other values for 120575 such as Q-170 and X-47A airplane(120575 ≃ 009)

At relatively small 119896119886 LRCS 120590119887suffers from oscillating

behavior due to the contributions from the illuminationregion particularly in addition to the stationary and inflectionpoints The scattered waves are sometimes in-phase so theyadd up and sometimes out-of-phase so they cancel outdepending on the scattered rays directions that result in suchsubstantial oscillating behavior [8] As 119896119886 increases LRCSgradually lessens as a result of the lack in the surface currentgenerated when 119896119886 gets greater than 119896119882

When 119896119886 is comparable with 119896119882 LRCS behaves almostindifferently with 119861

0 This behavior is more obvious with

wider SCL since waves are correlated enough around thetarget that 119861

0does not alter the LRCS In this range of 119896119886 the

target illumination region is almost covered by incidentwavesand therefore the scattered waves are relatively correlatedand that is why 119861

0has a less effect on the LRCS As 119896119886 gets

greater than 119896119882 LRCS changes with 1198610particularly with

shorter SCL but yet slightly with longer SCL In this rangethe target is not fully covered by incident waves so scatteredwaves particularly scattering from unilluminated portion

of the target surface are randomly back to the observationpoint propagating through the medium and then 119861

0has a

larger influence on the LRCS It should be noted that thescattering intensity and therefore 120590

119887 is decreased for strong

correlation function 1198610through the propagation due to the

strong absorbance of the medium and this is in a completeagreement with [23]

The similarity to some extent of the LRCS behavior with120575 is obvious regardless of the range of both 119896119886 and 119861

0in

Figures 3 and 4As shown in Figure 2 the scale size 119896119897 ofmediumparticles

relates to the SCL With a small SCL waves transversesuffers from relatively sharp transitions on the particles edgesConsequently waves scatter in a wide spectrum of mediumparticles leading to having a higher effect of 119861

0on (14) as

shown in Figures 3(a) and 4(a) while with having a widerSCL or alternatively big 119896119897 in Figures 3(b) and 4(b) particleswould have smoother surface and therefore waves transversestraightly through the particles in the forward propagationdirection reducing the scattering in other directions As aresult waves do not penetrate that much through mediumparticles rather than the forward direction and accordinglythe correlation intensity 119861

0would have a little effect on the

scattering intensity defined in (14) To confirm such behaviorwe reduce the SCL width in Figure 5 and compare it withFigure 3 We can notice that the effect of 119861

0is increased on

the LRCS in accordanceEffect of the 119896119882 on the LRCS is considered in Figure 6

Greater 119896119882 would result in major effects of 1198610on the LRCS

and vice versa This is owing to having a more uncorrelatedamount of waves in a wider range of the medium within 119896119882


1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)


1

2

3

4

5

120590b(2a

)

ka



00

2 4 6 8 10 12

Figure 5 LRCS versus target size for 120575 = 0 and 119896119882 = 15 where the target is located in random medium with SCL = 22

Propagation of such uncorrelated waves would return scat-tering waves intensity that is in turn obviously different fromthemedium fluctuations density 119861

0This affirms that to have

an authentic radar detection beam width should be narrowenough so the severity of the medium heterogeneity wouldhave a minimal impact

4 Conclusion

In this work the randommedium parameters object config-uration parameters and beam wave size that influences thebehavior of waves scattering from conducting targets withpartially convex contour such as airplane were discussed


1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)

Figure 6 LRCS versus target size for 120575 = 0 and SCL = 22 in random medium where (a) 119896119882 = 2 (b) 119896119882 = 25

In doing this laser RCS (LRCS) of partially convex targetsis calculated and analyzed numerically Medium parametersinclude the spatial coherence length (SCL) around the targetand the fluctuations intensity119861

0 LRCSdoes not changemuch

with 1198610when target size is comparable with the beam width

however 1198610has an obvious effect on the LRCS when airplane

mean size is greater than the beam width particularly withshorter length of SCL Also it should be noted that beamsize around the object should be limited enough to reducethe effect of the intensity of the medium fluctuations On theother hand and with a wider SCL or alternatively greaterscale size 119896119897 intensity of waves scattering by medium parti-cles lessens and the amount of waves correlation increasesAccordingly this lets 119861

0have a limited impact on the LRCS

As a result to have an accurate radar detection it isrecommended to have a beamwidth of incidentwaves narrowin a way to avoid themedium randomness effectsThis wouldbe quite essential mainly in the high frequency band neededin the far field propagation Complexity of the airplane shapehas a slight effect on the scattering waves propagating inmedia having different 119861

0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J B Keller andW Streifer ldquoComplex rays with an application toGaussian beamsrdquo Journal of the Optical Society of America vol61 no 1 pp 40ndash43 1971

[2] C L Bennett and H Mieras ldquoTime domain scattering fromopen thin conducting surfacesrdquo Radio Science vol 16 no 6 pp1231ndash1239 1981

[3] B Gallinet A M Kern and O J F Martin ldquoAccurate andversatile modeling of electromagnetic scattering on periodicnanostructures with a surface integral approachrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2261ndash2271 2010

[4] M Tateiba and Z Q Meng ldquoWave scattering from conductingbodies in random mediamdashtheory and numerical resultsrdquo inElectromagnetic Scattering by Rough Surfaces and RandomMedia M Tateiba and L Tsang Eds PIER 14 pp 317ndash361PMW Publications Cambridge Mass USA 1996

[5] H El-Ocla and M Tateiba ldquoThe effect of H-polarization onbackscattering enhancement for partially convex targets of largesizes in continuous random mediardquo Waves in Random Mediavol 13 no 2 pp 125ndash136 2003

[6] H El-Ocla ldquoTarget configuration effect on wave scattering inrandommedia with horizontal polarizationrdquoWaves in Randomand Complex Media vol 19 no 2 pp 305ndash320 2009

[7] H El-Ocla and M Tateiba ldquoAn indirect estimate of RCS ofconducting target in random mediumrdquo IEEE Antennas andWireless Propagation Letters vol 2 pp 173ndash176 2003

[8] H El-Ocla ldquoLaser backscattered from partially convex targetsof large sizes in randommedia for E-wave polarizationrdquo Journalof the Optical Society of America A vol 23 no 8 pp 1908ndash19132006

[9] H El-Ocla ldquoOn laser radar cross section of targets with largesizes for Epolarizationrdquo Progress in Electromagnetics Researchvol 56 pp 323ndash333 2006

[10] M Kerker The Scattering of Light and Other ElectromagneticRadiation Academic Press New York NY USA 1969

[11] M Al Sharkawy and H El-Ocla ldquoElectromagnetic scatteringfrom 3-D targets in a random medium using finite difference


frequency domainrdquo IEEE Transactions on Antennas and Propa-gation vol 61 no 11 pp 5621ndash5626 2013

[12] A V Jelalian Laser Radar Systems ArtechHouse BostonMassUSA 1992

[13] V Cerveny ldquoExpansion of a plane wave into Gaussian beamsrdquoStudia Geophysica et Geodaetica vol 46 no 1 pp 43ndash54 2002

[14] R Marty and K Solna ldquoAcoustic waves in long range randommediardquo SIAM Journal on AppliedMathematics vol 69 no 4 pp1065ndash1083 2009

[15] E H Bleszynski M C Bleszynski T Jaroszewicz and RAlbanese ldquoAnalysis of dispersive effects and enhanced mediumpenetrability in wide-band pulse propagation through sparsediscrete mediardquo IEEE Transactions on Antennas and Propaga-tion vol 60 no 7 pp 3265ndash3277 2012

[16] K E Oughstun ldquoPulse propagation in a linear causally disper-sive mediumrdquo Proceedings of the IEEE vol 79 no 10 pp 1379ndash1390 1991

[17] N G Alexopoulos and P K Park ldquoScattering of waves with nor-mal amplitude distribution from cylindersrdquo IEEE Transactionson Antennas and Propagation vol 20 no 2 pp 216ndash217 1972

[18] V G Gavrilenko and S S Petrov ldquoCorrelation characteristics ofwaves in lossy media with smooth fluctuations of the refractiveindex with oblique incidence on boundaryrdquo Waves in RandomMedia vol 2 no 4 pp 273ndash287 1992

[19] A Ishimaru Wave Propagation and Scattering in RandomMedia IEEE Press 1997

[20] A Ishimaru Electromagnetic Wave Propagation Radiation andScattering Prentice-Hall 1991

[21] M Tateiba ldquoMultiple scattering analysis of optical wave propa-gation through inhomogeneous randommediardquo Radio Sciencevol 17 no 1 pp 205ndash210 1982

[22] V H Rumsey ldquoReaction concept in electromagnetic theoryrdquoPhysical Review vol 94 pp 1483ndash1491 1954

[23] Y H Rim ldquoForward scattering of light in inhomogeneousbinary dielectric mediardquo Physical Review E vol 52 no 2 pp2110ndash2113 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



L

Target

Randommedium

x

z

z

Incidentwave

Scatteredwave

S

fluctuation intensityof random medium

a

y

L ≫ a

0

B0

120579

120601

B(rr) normalized

r = (x z)

Figure 1 Geometry of the problem of wave scattering from aconducting cylinder in a random medium

for the frequency spectrum of waves propagating in highlyfluctuating media that in turn affects the electromagneticwaves detection capabilities Putting this in mind our workassumes a target size approximately twice the wavelength tosuit the remote sensing using high frequency range On theother hand the case where the beam width is smaller thanthe target dimensions is considered tomeet the needs of radarapplications LRCS is calculated using our prescribedmethodfor convex illumination region of partially convex targets

Applications for our study consider conducting targetssuch as airplane where models have different contour com-plexity and shapes We deal with the scattering problem two-dimensionally considering the horizontal polarization (E-wave incidence) The time factor exp(minus119894119908119905) is assumed andsuppressed in the following section

2 Formulation

Geometry of the problem is shown in Figure 1 A randommedium is assumed as a sphere of radius 119871 around a target ofthemean size 119886 ≪ 119871 and also to be described by the dielectricconstant 120576(r) the magnetic permeability 120583 and the electricconductivity ] For simplicity 120576(r) is expressed as

120576 (r) = 1205760 [1 + 120575120576 (r)] (1)

where 1205760is assumed to be constant and equal to free space

permittivity and 120575120576(r) is a random function with

⟨120575120576 (r)⟩ = 0

⟨120575120576 (r) 120575120576 (r1015840)⟩ = 119861 (r r1015840)

(2)

119861 (r r) ≪ 1

119896119897 (r) ≫ 1

(3)

Here the angular brackets denote the ensemble average and119861(r r) 119897(r) are the local intensity and local scale-size of therandom medium fluctuation [18 19] respectively and 119896 =

120596radic12057601205830is the wavenumber in free space Also 120583 and ] are

assumed to be constants 120583 = 1205830 ] = 0 For practical

turbulent media condition (3) may be satisfied Thereforewe can assume the forward scattering approximation andthe scalar approximation [20] Consider the case where adirectly incident beamwave is produced by a line source119891(r1015840)along the 119910 axis Here let us designate the incident waveby 119906in(r) the scattered wave by 119906

119904(r) and the total wave by

119906(r) = 119906in(r)+119906119904(r)The target is assumed to be a conducting

cylinder of which cross section is expressed by

119903 = 119886 [1 minus 120575 cos 3 (120579 minus 120601)] (4)

where 120601 is the rotation index and 120575 is the concavity indexWe can deal with this scattering problem two-dimensionallyunder condition (3) therefore we represent r as r = (119909 119911)Assuming a horizontal polarization of incident waves (E-wave incidence) we can impose the Dirichlet boundarycondition for wave field 119906(r) on the cylinder surface 119878 Thatis 119906(r) = 0 where 119906(r) represents 119864

119910

Using the current generator 119884119864and Greenrsquos function in

randommedium119866(r | r1015840) we can express the surface currentwave as

119869 (r2) = int

119878

119884119864(r2| r1) 119906in (r

1| r119905) dr1 (5)

where r119905represents the source point location and it is

assumed as r119905= (0 119911) in Section 3 The first order of ⟨119866(r |

r1015840)⟩ is 11987210and expressed as

11987210 (119911) = ⟨119866 (r | 119911)⟩ = 119866

0 (r | 119911) 119890minus120572(119911)119911

(6)

where 1198660(r | 119911) is Greenrsquos function in free space [20] and

120572 is the coherence attenuation index [21] at the far field inthe 119911 direction We consider 119906in(r1 | r

119905) whose dimension

coefficient is understood to be represented as [20]

119906in (r1| r119905) = 119866 (r

1| r119905) exp[minus(

1198961199091

119896119882)

2

] (7)

where 119882 is the beam width The beam expression is approx-imately useful only around the cylinder Then the scatteredwave is given by

119906119904 (r) = int

119878

119869119864(r2) 119866 (r | r

2) dr2 (8)

This can be represented by

119906119904(r)

= int

119878

dr1int

119878

dr2[119866 (r | r

2) 119884119864(r2| r1) 119906in (r

1| r119905)]

(9)

Here 119884119864is the operator that transforms incident waves

into surface currents on 119878 and depends only on the scatteringbody [4 6] The current generator can be expressed in termsof wave functions that satisfy Helmholtz equation and theradiation condition

That is the surface current is obtained as

int

119878

119884119864(r2| r1) 119906in (r

1| r119905) dr1

≃ Φlowast

119872(r2) 119860minus1

119864int

119878

(Φ119879119872

(r1) 119906in (r

1| r119905)) dr

1

(10)


where

int

119878

(Φ119879119872

(r1) 119906in (r

1| r119905)) dr

1

equiv int

119878

[120601119898

(r1)120597119906in (r

1| r119905)

120597119899

minus120597120601119898

(r1)

120597119899119906in (r1| r119905)] dr1

(11)




119872= [120601minus119873

120601minus119873+1

120601119898 120601

119873]Φlowast119872

and Φ119879



number 120601119898(r) = 119867

(1)

119898(119896119903) exp(119894119898120579) and 119860

119864is a positive


119860119864

= (

(120601minus119873

120601minus119873


120601119873)

d

(120601119873 120601minus119873

) sdot sdot sdot (120601119873 120601119873)

) (12)


119899

(120601119898 120601119899) equiv int

119878

120601119898

(r) 120601lowast119899(r) dr (13)




⟨1003816100381610038161003816119906119904 (r)

1003816100381610038161003816

2⟩ = int

119878

dr01

int

119878

dr02

int

119878

dr10158401

sdot int

119878

dr10158402119884119864(r01

| r10158401) 119884lowast

119864(r02

| r10158402)

sdot exp[minus(1198961199091015840

1

119896119882)

2

] exp[minus(1198961199091015840

2

119896119882)

2

]

sdot ⟨119866 (r | r01) 119866 (r | r


(14)




at the point r02




22[5]

11987222

= ⟨119866 (r | r10158401)119866 (r | r

01) 119866lowast(r | r10158402)119866

lowast(r | r02)⟩

≃ ⟨119866 (r | r10158401)119866lowast(r | r10158402)⟩ ⟨119866 (r | r

01) 119866lowast(r | r02)⟩

+ ⟨119866 (r | r10158401)119866lowast(r | r02)⟩

sdot ⟨119866 (r | r01) 119866lowast(r | r10158402)⟩

(15)

0

02

04

06

08

1

0 2 4 6 8

SCL = 52SCL = 3

kl = 58120587kl = 20120587

k120588

Γ(k120588)

eminus1



120590119887= ⟨

1003816100381610038161003816119906119904 (r)1003816100381610038161003816

2⟩ sdot 119896 (4120587119911)

2 (16)

3 Numerical Results


Γ (120588 119911) =⟨119866 (r1| r119905) 119866lowast(r2| r119905)⟩

⟨1003816100381610038161003816119866 (r0| r119905)1003816100381610038161003816

2⟩

(17)

where r1= (120588 0) r

2= (minus120588 0) r

0= (0 0) and r

119905= (0 119911) In


0as


0 119871 and 119896119897









1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a



ka

0 2 4 6 8 10 12

(b)
















0has a






0in



0on (14) as




0is increased on





1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)


1

2

3

4

5

120590b(2a

)

ka



00

2 4 6 8 10 12





4 Conclusion



1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)









0



References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

int

119878

(Φ119879119872

(r1) 119906in (r

1| r119905)) dr

1

equiv int

119878

[120601119898

(r1)120597119906in (r

1| r119905)

120597119899

minus120597120601119898

(r1)

120597119899119906in (r1| r119905)] dr1

(11)




119872= [120601minus119873

120601minus119873+1

120601119898 120601

119873]Φlowast119872

and Φ119879



number 120601119898(r) = 119867

(1)

119898(119896119903) exp(119894119898120579) and 119860

119864is a positive


119860119864

= (

(120601minus119873

120601minus119873


120601119873)

d

(120601119873 120601minus119873

) sdot sdot sdot (120601119873 120601119873)

) (12)


119899

(120601119898 120601119899) equiv int

119878

120601119898

(r) 120601lowast119899(r) dr (13)




⟨1003816100381610038161003816119906119904 (r)

1003816100381610038161003816

2⟩ = int

119878

dr01

int

119878

dr02

int

119878

dr10158401

sdot int

119878

dr10158402119884119864(r01

| r10158401) 119884lowast

119864(r02

| r10158402)

sdot exp[minus(1198961199091015840

1

119896119882)

2

] exp[minus(1198961199091015840

2

119896119882)

2

]

sdot ⟨119866 (r | r01) 119866 (r | r


(14)




at the point r02




22[5]

11987222

= ⟨119866 (r | r10158401)119866 (r | r

01) 119866lowast(r | r10158402)119866

lowast(r | r02)⟩

≃ ⟨119866 (r | r10158401)119866lowast(r | r10158402)⟩ ⟨119866 (r | r

01) 119866lowast(r | r02)⟩

+ ⟨119866 (r | r10158401)119866lowast(r | r02)⟩

sdot ⟨119866 (r | r01) 119866lowast(r | r10158402)⟩

(15)

0

02

04

06

08

1

0 2 4 6 8

SCL = 52SCL = 3

kl = 58120587kl = 20120587

k120588

Γ(k120588)

eminus1



120590119887= ⟨

1003816100381610038161003816119906119904 (r)1003816100381610038161003816

2⟩ sdot 119896 (4120587119911)

2 (16)

3 Numerical Results


Γ (120588 119911) =⟨119866 (r1| r119905) 119866lowast(r2| r119905)⟩

⟨1003816100381610038161003816119866 (r0| r119905)1003816100381610038161003816

2⟩

(17)

where r1= (120588 0) r

2= (minus120588 0) r

0= (0 0) and r

119905= (0 119911) In


0as


0 119871 and 119896119897









1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a



ka

0 2 4 6 8 10 12

(b)
















0has a






0in



0on (14) as




0is increased on





1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)


1

2

3

4

5

120590b(2a

)

ka



00

2 4 6 8 10 12





4 Conclusion



1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)









0



References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a



ka

0 2 4 6 8 10 12

(b)
















0has a






0in



0on (14) as




0is increased on





1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)


1

2

3

4

5

120590b(2a

)

ka



00

2 4 6 8 10 12





4 Conclusion



1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)









0



References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)


1

2

3

4

5

120590b(2a

)

ka



00

2 4 6 8 10 12





4 Conclusion



1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)









0



References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

3

4

5120590b(2a

)

ka



0 2 4 6 8 10 12

(a)

1

2

3

4

5

120590b(2a

)

ka



0 2 4 6 8 10 12

(b)









0



References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Effects of Medium Characteristics on ... · 2 = * d r01 * d r02 * d r1 F* d r 2 % r r01 | r 1 % r02 | r 2 Fexp K L ! 1 7 M 2 N exp K L ! 2 7 M 2 N F & r | r01 & r