2924 J. Acoust. Soc. Am. 110 (6), December /2001/110(6)/2924/22/$ Acoustical Society of America

Transcription

1 Extncton theorem for object scatterng n a stratfed medum Purnma Ratlal and Ncholas C. Makrs Massachusetts Insttute of Technology, Cambrdge, Massachusetts 0239 Receved 20 December 2000; revsed 3 June 200; accepted 0 July 200 A smple relaton for the rate at whch energy s extngushed from the ncdent wave of a far feld pont source by an obstacle of arbtrary sze and shape n a stratfed medum s derved from wave theory. Ths relaton generalzes the classcal extncton theorem, or optcal theorem, that was orgnally derved for plane wave scatterng n free space and greatly facltates extncton calculatons by elmnatng the need to ntegrate energy flux about the obstacle. The total extncton s shown to be a lnear sum of the extncton of each wave gude mode. Each modal extncton nvolves a sum over all ncdent modes that are scattered nto the extngushed mode and s expressed n terms of the object s plane wave scatter functon n the forward azmuth and equvalent plane wave ampltudes of the modes. The only assumptons are that multple scatterng between the object and wave gude boundares s neglgble, and the object les wthn a constant sound speed layer. Modal extncton cross sectons of an object for the extncton of the ndvdual modes of a wave gude are then defned. Calculatons for a shallow water wave gude show that, after correctng for absorpton loss n the medum, the modal cross secton of an object for mode n a typcal ocean wave gude s very nearly equal to ts free space cross secton. Ths new extncton theorem may be appled to estmate the cross secton of an object submerged n a wave gude from a measurement of ts forward scattered feld. 200 Acoustcal Socety of Amerca. DOI: 0.2/ PACS numbers: Ft, Bp, Gr DLB I. INTRODUCTION If an object s placed n the path of an ncdent wave, some of the ntercepted power s scattered n all drecton and the remander s absorbed. The total power removed from the ncdent feld as a result of scatterng and absorpton by the object s called extncton. Van de Hulst has shown, n what has become known alternatvely as the extncton theorem, optcal theorem, and forward scatter theorem, that the extncton can be derved from the scattered far feld n the forward drecton. Specfcally, the total extncton of a plane wave ncdent on an object n free space equals the magnary part of the forward scatter ampltude multpled by the ncdent ntensty and 4/k 2 where k s the wave number. 2 4 Ths remarkably smple relatonshp reflects the fact that the extncton caused by the obstacle leads to shadow formaton va destructve nterference between the ncdent and forward scattered felds. The permanence of the extncton s mantaned by the formaton of a regon of destructve nterference that survves as an actve shadow remnant 3 n perpetuty beyond the deep shadow. The total power scattered by an object can be found by ntegratng the scattered ntensty over a large control surface enclosng the object n the far feld. Ths ntegraton s usually dffcult to perform and makes an alternatve approach attractve. For nonabsorbng objects, the total power scattered by the object s the extncton. 5,6 One great advantage of the extncton theorem s that t elmnates the need to ntegrate the scattered energy flux around the object. The extncton theorem s typcally appled n acoustcs to measure the extncton cross secton of objects. 7 Ths equals twce the object s projected area n the hgh frequency lmt, and so provdes a useful method for estmatng an object s sze. The extncton theorem has many dverse applcatons n acoustcs, such as those gven n Refs. 8 and 9. It can also be used as a burglar alarm to detect and classfy ntrudng objects that pass between a source and an acoustc recever array. In 985 Guo 0 extended the extncton theorem to scatterng by an object located next to an nterface between two dstnct acoustc half spaces. He found an expresson for the extncton of an ncdent plane wave n terms of the far-feld scattered pressures n the specular reflecton and transmsson drectons, determned by Snell s law. In a wave gude, the effect of multmodal propagaton ensures that the feld ncdent on the object wll arrve from many dstnct drectons. Ths, combned wth the effect of absorpton loss n the wavegude, wll modfy the extncton and scatterng cross sectons from ther free space values. The free space extncton theorem and the half-space extenson of Guo are therefore not applcable n a wave gude. Here we use wave theory to derve a generalzed extncton theorem by developng a relaton for the rate at whch energy s extngushed from the ncdent wave of a far feld pont source by an object of arbtrary sze and shape n a stratfed medum. Lke ts free space analogue, the relaton s agan remarkably smple. The total extncton s shown to be a lnear sum of the extncton of each wave gude mode. Each modal extncton nvolves a sum over all ncdent modes that are scattered nto the gven mode and s expressed n terms of the object s plane wave scatter functon n the forward azmuth and equvalent modal plane wave ampltudes. For the multple ncdent plane waves n a wave gude, extncton s a functon of not only the forward scatter ampltude for each of the ncdent plane waves but also depends on the scatter functon ampltudes couplng each ncdent plane wave to all other plane waves wth dstnct drec J. Acoust. Soc. Am. 0 (6), December /200/0(6)/2924/22/$ Acoustcal Socety of Amerca

2 FIG.. The geometry of the problem showng an object n a stratfed medum composed of a water column of thckness H overlyng a bottom. The orgn of the coordnate system s at the center of the object and the source s located at (x 0,0,z 0 ). The screen s normal to the x axs wth wdth L and s sem-nfnte n the z-drecton penetratng nto the bottom wth an edge at the top of the water column. tons that make up the ncdent feld. The fnal relaton greatly facltates extncton calculatons by elmnatng the need to ntegrate energy flux about the object. Our dervaton begns wth the tme-harmonc scattered feld from an object n a wave gude that s derved drectly from Green s theorem.,2 The only smplfyng assumptons are that multple scatterng between the object and wave gude boundares s neglgble and that the object les wthn a constant sound speed layer. The smplcty of the resultng extncton relaton n the wave gude follows from the fact that the full extncton s mantaned n the regon of actve nterference and that ths regon extends nto the far feld where separaton of varables can be nvoked. Energy fluxes necessary for the dervaton can then be calculated n the far feld n terms of wave gude modes and the object s plane wave scatterng functon. 2,3 The extncton cross secton of an object s defned as the rato of ts extncton to the rate at whch energy s ncdent on unt cross sectonal area of the object. The extncton cross secton reduces to the scatterng cross secton for nonabsorbng objects, and s useful n actvely classfyng targets snce, as the rato of the total extncton to the ncdent ntensty, t depends only upon scatterng propertes of the target. Ths defnton, however, s ambguous n a wave gude because both the ncdent and scattered felds are comprsed by superpostons of plane waves. Here scatterng and propagaton effects are not generally separable snce they are convolved together n the extncton. Addtonally, the ncdent ntensty s not spatally constant. In spte of these dffcultes, we fnd t convenent to nterpret the extncton cross secton for an object n a wave gude as the rato of the extncton to the ncdent energy flux per unt area n the radal drecton at the object s centrod. Ths defnton s sensble when the object s n a constant sound speed layer and n the far feld of the source. Calculatons for a shallow water wave gude, whch have great relevance to actve detecton problems n ocean acoustcs, show that an object s cross secton for the combned extncton of all the modes of the wave gude s hghly dependent on measurement geometry, medum stratfcaton, as well as ts scatterng propertes. In addton, the combned cross secton fluctuates rapdly wth range due to coherent nterference between the modes. The presence of absorpton n the medum can also sgnfcantly modfy a measurement of the total scatterng cross secton. The practcal mplcaton of these fndngs s that expermental measurements of the total scatterng cross secton of an obstacle n a wave gude may dffer greatly from those obtaned for the same obstacle n free space and may lead to errors n target classfcaton f the wave gude effects are not properly taken nto account. For an object submerged n a wave gude, we also defne modal cross sectons of the object for the extncton of the ndvdual modes of the wave gude. The modal cross secton of an object for the extncton of mode n a typcal ocean wave gude was found to be nearly equal to the free space cross secton of the object. A potental applcaton of the extncton theorem n a wave gude s then the estmaton of the sze of an object submerged n the wave gude from a measurement of the extncton t causes to mode. The generalzed extncton theorem can also be used to determne the attenuaton due to volume and surface scatterng of guded waves propagatng through stratfed meda such as the ocean or the earth s crust. II. THE GENERALIZED EXTINCTION THEOREM In ths secton, we derve the extncton n the ncdent feld of a far feld pont source due to an obstacle of arbtrary sze and shape n a stratfed medum. The general approaches for calculatng extncton are dscussed n Appendx A. Here, we adopt the ntutve approach of Van de Hulst,2,4 whch nvolves ntegratng the energy flux, or ntensty, over a screen placed a dstance away from the object suffcently large to regster Fraunhofer dffracton, Eq. A. Inthe absence of the object, the total energy flux across the screen s maxmal. In the presence of the object, the total energy flux across the screen s dmnshed by the shadow remnant. For a suffcently large screen, the dfference between these fluxes s the total extncton. We focus on the Van de Hulst screen method for calculatng extncton because t s of more practcal use snce t represents the type of measurement that can be made wth a standard 2D planar or bllboard array. Ths s dscussed further n Sec. V. The other approach for calculatng extncton usng a control surface that encloses the object n a stratfed wave gude s dscussed n Appendx D. A control volume measurement would be very dffcult to mplement snce t would requre an array that completely encloses the object. The orgn of the coordnate system s placed at the object centrod wth z axs vertcally downward, and x axs parallel to the boundares as shown n Fg.. The coordnates of the source are defned by r 0 (x 0,0,z 0 ). The screen s postoned n forward azmuth on the y z plane at a horzontal range x from the object center. The wdth of the screen s L along the y drecton and s sem-nfnte n the z drecton wth an edge at the surface of the wave gude. Let r (x,y,z) be the coordnates of a pont on the screen. Spatal cylndrcal (,,z) and sphercal systems (r,,) are de- J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 2925

3 fned by xr sn cos, yr sn sn, zr cos and x 2 y 2. The horzontal and vertcal wave number components for the nth mode are, respectvely, n k sn n and n k cos n where k 2 n 2 n 2 and the wave number magntude k equals the angular frequency dvded by the sound speed c n the object layer. As dscussed n Appendx A to measure the full extncton n the wave gude, we requre L x, where x s the horzontal range of the screen from the object. Assumng that the object s far from the source and the screen so that the range from the screen to the source s large, the ncdent feld at locaton r on the screen for a source at r 0, can be expressed as a sum of normal modes, rr 0 dz 0 8 e/4 l u l zu l z 0 e l 0 l 0, where u l (z) and l are the lth modal ampltude and horzontal wave number, respectvely, and d(z) s the densty at depth z. Usng the formulaton of Refs. and 2 based on Green s theorem, the scattered feld from the object at recever r for a source at r 0 s 4 s rr 0 m n k A m ra n r 0 S m,; n, 0 B m ra n r 0 S m,; n, 0 A m rb n r 0 S m,; n, 0 B m rb n r 0 S m,; n, 0, 2 where A m r d0 8 m /2 u m zn m e m m D/4, B m r d0 8 m /2 u m zn m e m m D/4, 3 A n r 0 dz 0 8 n 0 /2 u n z 0 N n e n 0 n D/4, B n r 0 dz 0 8 n 0 /2 u n z 0 N n e n 0 n D/4 are the down and up gong plane waves n the layer of the object, D s the depth of the object center from the sea surface and S(,;, ) s the object s plane wave scatter functon. The defnton of the plane wave scatter functon here follows that defned n Ref. 2 where the ncdent plane wave on the object s descrbed n terms of the drecton t goes to, so that for forward scatter n free space,,. The mode functons are normalzed accordng to nm D u m zu n *z dz, dz and are decomposable nto up- and down-gong plane waves n the layer of the object va u m zn n e n zd N n e n zd. N n and N n are the ampltudes of the down- and up-gong plane waves n ths layer. A number of assumptons have to be satsfed for the above formulaton for the scattered feld to be vald as noted n Ref. 2. In partcular, multple scatterng between the object and wave gude boundares s neglgble, the object les wthn a layer of constant sound speed, and the range from the object to source or recever must be large enough that the scattered feld can be approxmated as a lnear functon of the object s plane wave scatter functon. The last condton does not lmt the generalty of the fnal extncton expresson snce the full extncton can be regstered on suffcently large screens n the object s far feld, but nstead smplfes ts dervaton. To calculate the extncton usng the general formula of Eq. A, we frst evaluate the ntegrand for the pont r on the screen. The frst term n the ntegrand of Eq. A usng Eqs. A2,, 2, and 3 s 4 5 V * s dzdz 0 d02k l m n u l *z 0 u m z 2 z u l *z z l *u l *z x y 2 erlx0x l *x 0 x e R m x 2 y 2 m x N m e R md A n r 0 S m,; n,0n m e R md A n r 0 S m,; n,0 N m e R md B n r 0 S m,; n,0n m e R md B n r 0 S m,; n,0 e I lx 0 x e I mx e I md. 6 In the above expresson, the terms representng absorpton by the wave gude have been factored out explctly to avod confuson when conjugatng the felds and also to keep track of absorpton losses due to the wave gude. The exact expressons for 0 (xx 0 ) 2 y 2 and x 2 y 2 were kept n the terms that determne the phase of the ntegrand whle the approxmatons 0 (xx 0 ) and x were used n the spreadng and absorpton loss factors, snce x, x 0 y can be satsfed for a screen that measures the full extncton. Next we ntegrate Eq. 6 over the area of the screen. Wth the screen lyng parallel to the y z plane, an area ele J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng

4 ment of the screen s ds x dy dz. We use the orthorgonalty relaton n Eq. 4 between the modes u l *(z) and u m (z) to ntegrate Eq. 6 over the sem-nfnte depth of the screen n the wave gude. Ths reduces the trple sum over the modes to a double sum: L/2 V * s Sc ds L/2 V * s x dz dy D dz 0 d02k m n m * m xx 0 x u m *z 0 L/2 N m e RmD A n r 0 S m,; n,0n m e RmD A n r 0 S m,; n,0 L/2 N m e R md B n r 0 S m,; n,0n m e R md B n r 0 S m,; n,0 e R m x 2 y 2 x 0 x 2 y 2 dy ei mx 0 2x e I md. 7 In the above expresson, the scatter functon s dependent on y va the azmuth angle tan (y/x). As dscussed n Appendx A, the angular wdth of the actve area on the screen n azmuth s of the order of /x. We can therefore approxmate the scatter functon wth ts value at 0 and factor t from the ntegral above snce x s large. We also expand the exponent nvolvng the varable y accordng to x 0 x 2 y 2 x 0 x y 2 2x 0 x, x 2 y 2 x y 2 2x. 8 9 Applyng the result of the followng asymptotc ntegraton over the wdth of the screen, L/2 L/2 Sm,; n,0e R m x 2 y 2 x 0 x 2 y 2 dye R mx 0 S m,0; n,0e 2xx /4 0x, 0 R m x 0 to Eq. 7, the ntegraton of the frst term n Eq. A over the area of the screen n the wave gude becomes V * s ds Sc d 2 z 0 d04k x 0 m n m * m R m n u m *z 0 u n z 0 e R n mx 0 N m N n e R m nd S m,0; n,0n m N n e R m nd S m,0; n,0 N m N n e R m nd S m,0; n,0n m N n e R m nd S m,0; n,0 e I m nx 0 e I2 mx e I m nd. Smlarly, we can evaluate the second term n Eq. A whch gves V s * ds Sc d 2 z 0 d04k x 0 m n m * m R m n * u mz 0 u n *z 0 e R n mx 0 N m *N n *e R m nd S* m,0; n,0n m *N n *e R m nd S* m,0; n,0 N m *N n *e R m nd S* m,0; n,0n m *N n *e R m nd S* m,0; n,0 e I m nx 0 e I2 mx e I m nd. 2 When we sum Eqs. and 2, takng only the negatve of the real part of the sum followng Eq. A, we obtan the range dependent extncton E(xr 0 ) of the ncdent feld n a wave gude due to an object at the orgn measured by a screen at dstance x from the object wth source at r 0, J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 2927

5 Exr 0 d 2 z 0 d02k x 0 m n R m m I u m *z 0 u n z 0 e R n mx 0 n N m N n e R m nd S m,0; n,0n m N n e R m nd S m,0; n,0 N m N n e R m nd S m,0; n,0n m N n e R m nd S m,0; n,0 e I m nx 0 e I2 mx e I m nd. 3 From Eq. 3, we see that the total extncton s a lnear sum of the extncton of each wave gude mode. The extncton of mode m nvolves a sum over all ncdent modes n that are scattered nto that extngushed mode and s expressed n terms of the object s plane wave scatter functon n the forward azmuth and equvalent plane wave ampltudes of the modes. The extncton decreases wth source-object range x 0 n a wave gude due to geometrcal spreadng, and wth source-object and object-recever ranges, x 0 and x, due to absorpton loss n the medum. A. Effect of multple ncdent plane waves To understand the mplcatons of Eq. 3, we consder several cases and examne the resultng expresson for the extncton n each case.. Sngle mode excted by source Frst we consder a source that exctes only a sngle mode p. The ncdent feld on the object and at the screen s determned by ths sngle mode p. The trple sum n Eq. 6 reduces to a sngle sum over m n ths case snce both l and n can only take on the nteger value p. The orthogonalty relaton between the modes u l *(z) and u m (z) elmnates the sum over m leavng just a sngle term where mp n Eq. 7. Consequently, the expresson for the extncton wll have only one term correspondng to mnp, the mode excted by the source, Exr 0 d 2 z 0 d02k R p u x 0 p p z 0 I 2 p N p 2 e R2 pd S p,0; p,0n p N p S p,0; p,0 N p N p S p,0; p,0n p 2 e R2 pd S p,0; p,0 e I2 px 0 x e I2 pd. 4 Even though the scattered feld from the object s composed of multple modes m, only one of these can nterfere destructvely wth the sngle ncdent mode p on the screen and t s precsely the scattered mode that has the same elevaton angle as the ncdent mode. Mode p s made up of an up-gong and a down-gong plane wave. Two of the four terms n Eq. 4 arse from the forward scatter of the up- and down-gong plane waves of mode p, whle the remanng two terms arse from the scatter of the ncdent down-gong plane wave of mode p to an upgong plane wave of the same mode and vce versa. Ths shows that when we have multple plane waves ncdent on the object, the extncton wll depend on not only the scatter functon n the forward drecton but also depend on the scatter functon ampltudes couplng each ncdent plane wave to all other plane waves wth dstnct drectons that make up the ncdent feld. 2. Many modes excted by source For a general harmonc source that exctes many modes, the ncdent feld on the screen s a sum of the contrbuton from varous excted modes. Each of these ncdent modes on the screen wll only nterfere destructvely n the forward azmuth wth the correspondng scattered mode from the object wth the same elevaton angle. The scatterng process causes the varous ncomng ncdent modes at the object to be coupled to each outgong scattered mode through the scatter functon and ths leads to a double sum n the expresson for the extncton n Eq Large object-recever range, x Next we consder the scenaro where the screen s placed at a suffcently large dstance from the object that only the frst mode survves for both the ncdent feld on the screen from the source and the scattered feld from the object,.e., lm n Eq. 6. The feld ncdent on the object s stll comprsed by a sum over the modes n excted by the source snce the range of the source from the object s not too large. The expresson for the extncton n Eq. 3 then reduces to a sngle sum over the ncdent modes n on the object that are scattered nto the outgong mode m that survves at the screen. 4. Large source to object range, x 0 If the source s placed at large dstances away from the object, the feld ncdent on the object and on the screen wll be determned by the sngle mode ln that survves whle the rest of the modes are strpped due to absorpton n the wave gude. The extncton n ths case has a sngle term 2928 J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng

6 n Eq. 3 correspondng to mn, the mode that survves at the screen. The expresson for the extncton s gven by Eq. 4 wth p. These examples llustrate the fact that t s really the nterference between the ncdent feld and the scattered feld on the screen that determnes the extncton. Only scattered feld drectons that have a fxed phase relatonshp wth the ncdent feld wll contrbute to the extncton. In the lterature, extncton s often stated to be drectly proportonal to the forward scatter ampltude of a plane wave n free space. For multple ncdent plane waves, however, the extncton s not smply a functon of the forward scatter ampltude for each ncdent plane wave but also depends on the scatter functon ampltudes couplng each ncdent plane wave to all other plane waves wth dstnct drectons that make up the ncdent feld. Guo s 0 result for the extncton of a plane wave by an object placed near an nterface between two meda can also be nterpreted n ths way. B. Effect of absorpton by the medum The extncton of the ncdent feld due to an object n the far feld of a pont source n free space wth absorpton n the medum s derved n Appendx C. Comparng the expresson for extncton n a wave gude, Eq. 3, wth that n free space, Eq. C4 n Appendx C, we see that absorpton n the medum lowers the extncton that we would otherwse measure n a lossless medum. In free space, the term due to absorpton by the medum s separable from the propertes of the object n the formula for the extncton. These terms, however, are n general, convolved n a wave gude wth multmodal propagaton. The convoluton arses because the absorpton loss suffered by each mode vares from mode to mode. Furthermore, the modes have varyng elevaton angles and they are thus scattered dfferently by the object dependng on the elevaton angle of the mode. In the wave gude, the absorpton loss term can be separated from the term due to the object only f a sngle mode s ncdent on the object as seen from Eq. 4, whch s the extncton caused by a sngle mode. One way ths arses naturally n a wave gude s when the source to object separaton s large enough that only mode survves n the ncdent feld on the object. III. TOTAL SCATTERED POWER IN THE WAVE GUIDE The total power scattered by an object n a wave gude can be obtaned by ntegratng the scattered feld ntensty V s * s around a closed control surface enclosng the object, as descrbed n Eq. A8. We let the control surface be a sem-nfnte cylnder of radus R wth a cap at the sea surface where zd. The axs of the cylnder s parallel to the z axs and passes through the object centrod. The sea surface s a pressure-release surface where the total feld vanshes. Snce the ncdent feld n the absence of the object s zero at the sea surface, the scattered feld has to vansh as well. The scattered energy flux through the cap of the cylnder at zd s zero. We need only ntegrate the scattered ntensty over the curved surface of the cylnder to obtan the total scattered power. From Eq. 2, we see that the scattered feld s expressed as a sum of four terms. The scattered ntensty at the surface of the cylnder can therefore be expressed as a sum of 6 terms, the frst of whch s 2 V s * s dzd 2 0k 2 m n p q u p z z u m *z z m *u m *z erpmr m * p R N m *N p e R p md A n *r 0 A q r 0 S* m,; n,0s p,; q,0 e I m pr e I m pd. 5 An area element on the curved surface of the cylnder s gven by ds Rd dz. Makng use of Eq. 4, the orthogonalty relaton between the modes, we ntegrate Eq. 5 over the sem-nfnte depth of the cylnder and the resultng expresson s V s * s ds 0 2 V s * s dz d D 2 m * d 2 0k 2 m n q m N m 2 A n *r 0 A q r 0 2 S*m,; n,0s m,; q,0d 0 e 2ImR e 2ImD. 6 The above ntegral cannot be further evaluated wthout specfyng the scatter functon of the object. In general the total scattered power n the wave gude s a complex expresson wth a trple sum of 6 ntegrals. The real part of Eq. 6 gves the trple sum of just the frst ntegral. If there s no absorpton by the object, the extncton caused by the object s due entrely to scatterng. If the object s n a perfectly reflectng wave gude or a wave gude wth small absorpton loss, the total scattered power s the extncton. In that case, the complcated expresson wth trple sum of 6 ntegrals dscussed above reduces to the smple expresson of a double sum and no ntegral of Eq. 3. In a lossy wave gude, f we measure the extncton around a small control surface enclosng the object, the absorpton loss nsde the control volume s small and the above holds as well. Therefore, the extncton formula elmnates the need to ntegrate the scattered energy flux about the object n a wave gude when determnng the scattered power. IV. COMBINED AND MODAL EXTINCTION CROSS SECTIONS The rato T between the rate of dsspaton of energy and the rate at whch energy s ncdent on unt cross sectonal area of an obstacle s called the extncton cross sec- J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 2929

7 ton of the obstacle. In the wave gude, the ntensty of the ncdent feld on the object at the orgn from a source at r 0 s I 0r 0 RV *0r 0 0r 0 R d 2 z 0 d08x 0 p q u p *z 0 *z z u p z *u p *0 p z0 x u qz 0 u q 0 er q px 0 e I p qx p * q 0. 7 In our dervaton, the screen s postoned normal to the x axs and t measures the extncton of the energy flux propagatng n the x drecton. We therefore normalze ths extncton by the component of the ncdent ntensty n the x drecton to obtan the extncton cross secton T of the object n the wave gude, T xr 0 Exr 0 I 0r 0 x 4 k m n R m m I u m *z 0 u n z 0 e R n mx 0 N m N n e R m nd S m,0; n,0 n N m N n e R m nd S m,0; n,0n m N n e R m nd S m,0; n,0 N m N n e R m nd S m,0; n,0 e I m nx 0 e I2mx e I m nd q p Ru*z p 0 u*0u p q z 0 u q 0 p */ q e R q px 0 e I p qx 0. 8 Equaton 8 s due to the combned extncton of all the modes of the wave gude by the object and we defne t to be the combned extncton cross secton. Ths combned cross secton of an object depends on the propertes of the object whch are convolved wth the propertes of the wave gude, as well as the source and object locatons. For a source that exctes only a sngle mode p, the ncdent ntensty on the object n the x drecton s I 0r 0 p d 2 u z 0 d08x p z 0 2 u p 0 R* 2 p e I2 px p Dvdng the extncton of mode p by the object n Eq. 4 wth the ntensty of the ncdent feld composed solely of mode p n Eq. 9, we obtan the cross secton of the object for the extncton of mode p, p x 4 k R p u p 0 2 I p N p 2 e R2 pd S p,0; p,0n p N p S p,0; p,0 N p N p S p,0; p,0n p 2 e R2 md S p,0; p,0 e I2 px e I2 pd. 20 We defne Eq. 20 as the modal cross secton of the object for the extncton of the ndvdual modes of the wave gude. Analogous to plane waves n free space, the modes n a wave gude are the entty that propagate n the wave gude and determne the energy of the acoustc feld n the wave gude. It therefore becomes meanngful to quantfy the extncton caused by an object of the ndvdual modes of the wave gude and subsequently the cross secton of the object as perceved by the ndvdual modes of the wave gude. V. ESTIMATION OF OBJECT SIZE FROM EXTINCTION THEOREM IN AN OCEAN WAVE GUIDE The extncton formula can be used to estmate the sze of an object by measurng the extncton t causes n an ncdent beam. For nstance, n astronomy, the sze of a meteorte s estmated from the extncton t causes n the lght reachng a telescope when the meteorte s n nterstellar space between a star and the telescope, so long as the telescope s large enough to measure the entre shadow remnant. 4 For an object that s large compared to the wavelength, ts extncton cross secton n free space, accordng to Babnet s prncple, s equal to twce ts geometrcal projected area. 4 If we let T p be the projected area of the object n the drecton of an ncdent plane wave n free space, we obtan 4 k 2 IS f 2T p. 2 The sze of the object s therefore drectly related to the free space forward scatter functon of the object for objects that are large compared to the wavelength. The forward scatter functon can be determned from a measurement of the extncton caused by the object J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng

8 Extncton measurements usually nvolve ntegratng the ntensty of the ncdent and total felds over a suffcently large screen that regsters the full extncton caused by the object. We measure the ncdent power on the screen n the absence of the object and the total power n the presence of the object. The dfference between these two energy fluxes on the screen s the extncton. An ntensty measurement at a sngle pont n space n the forward scatter drecton s typcally nadequate. Ths can be seen from Eq. C3 for free space, and Eq. 6 n the wave gude, where the nterference ntensty V * s at a pont depends very senstvely on the source and recever postons whch cause rapd fluctuaton n the phase term. To determne the forward scatter functon from a sngle recever n the forward drecton then requres extremely accurate knowledge of the source, object and recever locatons. In practcal measurements, t may also be dffcult to precsely locate the pont sensor n the forward drecton. Ths s especally true for large objects as they have very narrow forward scatter functon peaks. Equaton C4 for the extncton n free space on the other hand has no phase dependence nvolvng the source or screen poston. Extncton measurement over a screen s therefore a more robust method for estmatng the forward scatter ampltude and hence the sze of an object. For measurements n a shallow water wave gude, the screen over whch the ntensty s ntegrated can be ether a suffcently large planar array, or a bllboard array whose spacng between the sensor elements satsfes the Nyqust crteron for samplng the feld n space. In a wave gude, the extncton caused by an object, Eq. 3, depends not only upon the propertes of the object through the scatter functon, but also the propertes of the wave gude and the measurement geometry. They are, n general, convolved n the expresson for the extncton and are separable only when the ncdent feld s composed of a sngle mode as evdent n Eq. 4. Ths suggests a possble scenaro for extncton measurements n a wave gude to extract the scatter functon s forward ampltude and subsequently to estmate the sze of an object. For large source to object separaton x 0, the mode that survves n the ncdent feld s mode. Mode of any wave gude propagates almost horzontally and we can approxmate ts elevaton angle as /2. In ths case, the four scatter functon ampltudes n Eq. 4 can be approxmated as S(/2,0,/2,0) and factored out of the equaton for the extncton. Usng the fact that for mode, R I we rewrte the extncton for mode as usng Eq. 5, we see that u 0 2 N 2 e R2 D 2N N N 2 e R2D e I2D. The extncton formula for mode therefore leads to Exr 0 d 2 z 0 d02k x 0 R u z 0 2 u IS/2,0;/2,0e I2 x 0 x. 25 Equaton 20 for the modal cross secton of the object for mode n the Pekers wave gude, smplfes to x 4 k R IS/2,0;/2,0e2I x. 26 Snce mode propagates close to the horzontal, R k. The cross secton of an object for the extncton of mode n a Pekers wave gude, Eq. 26, s almost dentcal to the cross secton of the object for the extncton of plane waves n free space, Eq. C6. In Eqs. 25 and 26 the propertes of the target are separated from the wave gude and geometrc parameters. If we can measure the extncton of mode caused by the object n the wave gude, we can estmate the free space forward scatter ampltude of the object and subsequently, the sze of the object. A knowledge of the wave gude propertes, and locaton of source, object and screen s necessary to correct for the spreadng and absorpton loss n the wave gude, as well as the ampltude of mode at the source and object depths. Expermentally, we can estmate the source to object range x 0 from the arrval of the back scattered feld from the object usng a sensor that s co-located wth the source. As dscussed n Eq. 2, the object sze s related to the forward scatter functon ampltude. The extncton of the hgher order modes of the wave gude, apart from mode, depend on the scatter functon ampltude n other drectons n addton to the forward. It s therefore much more dffcult to extract nformaton about the sze of the object from modes hgher than mode unless the object s compact as wll be dscussed n Sec. VI E. For objects that are bured n sedments that are faster than water, mode excted by a source n the water column does not penetrate nto the bottom due to total nternal reflecton. The above method wll therefore not be useful n estmatng the sze of objects bured n fast bottoms. Exr 0 d 2 z 0 d02k x 0 R u z 0 2 IS/2,0;/2,0N 2 e R2 D 2N N N 2 e R2 D e I2 x 0 x e I2 D. In a Pekers wave gude, 4,5 wth N N d0 2H, VI. ILLUSTRATIVE EXAMPLES In all the llustratve examples, a water column of 00 m depth s used to smulate a typcal contnental shelf envronment. The sound speed structure of the water column s sovelocty wth constant sound speed of 500 m/s, densty of g/cm 3 and attenuaton of db/. The seabed s ether perfectly reflectng or comprsed of sand or slt half spaces. The densty, sound speed and attenuaton are taken to be.9 g/cm 3, 700 m/s, and 0.8 db/ for sand,.4 g/cm 3, 520 m/s, and 0.3 db/ for slt. Calculatons are made of the combned and modal extncton, ncdent ntensty on the ob- J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 293

9 FIG. 2. a The combned extncton Eq. 3 of all the modes, caused by a pressure release sphere of radus 0 m centered at 50 m depth, n a Pekers wave gude composed of 00 m water wth ether sand or slt half space s plotted as a functon of x 0, ts range from a pont source of frequency 300 Hz also placed at the same depth n the wave gude. The separaton of the screen from the object s the same as that of the source from the object at each source to object range, xx 0. b The ncdent ntensty on the sphere Eq. 7. c The combned cross secton of the sphere Eq. 8. Both the coherent and ncoherent approxmaton of the quanttes are plotted n each subfgure. ject, and, the combned and modal cross sectons n varous wave gudes for dfferent objects as a functon of source, object and screen locatons. The object sze and frequency s also vared. Except for Sec. VI E, the frequency used n all other examples s 300 Hz. A. Combned extncton cross secton n dfferent wave gudes Frst, we examne how the combned extncton of all the modes, caused by a pressure-release sphere of radus 0 m, n a Pekers wave gude wth ether sand or slt bottom half space vares as a functon of source to object range at a source frequency of 300 Hz. The source and sphere centers are located at D50 m n the mddle of the water column. The combned extncton measured by the screen Eq. 3, the ncdent ntensty on per unt area of the sphere Eq. 7, and the combned cross secton of the sphere Eq. 8 n the wave gudes are plotted as a functon of source to object separaton x 0 n Fgs. 2a c, respectvely. At each x 0, the separaton of the screen from the object s the same as that of the source from the object,.e., xx 0 The combned extncton s calculated usng Eq. 3 wth the scatter functon for the sphere gven by Eqs. 8 and 9 of Ref. 3 wth f (n) replaced by () n f (n) to convert from Ingento s defnton to the standard one descrbed n Ref. 2. The combned extncton and ncdent ntensty fluctuate wth range due to the coherent nterference between the modes. The resultng combned cross secton of the sphere also fluctuates wth range. The ncdent ntensty and extncton are larger n the wave gude wth sand bottom. The fluctuatons n the felds are also greater n the sand bottom wave gude as compared to the slt bottom wave gude. The dfference arse prmarly because the number of trapped modes s larger for the sand half space due to the hgher crtcal angle of 28. for the water to sand nterface as compared wth the 9.3 of water to slt leadng to larger felds and fluctuatons n the wave gude wth sand bottom. For a screen placed at a fxed range from the object, t s the coherent extncton and cross secton that we measure expermentally. From Fg. 2c, we see that the coherent combned cross secton of the object vares rapdly wth range n the wave gude. Consequently, t s dffcult to extract nformaton about the sze of the object from a measurement of ts combned extncton of all the wave gude modes. We fnd t useful to approxmate the combned extncton measured by the screen and the ncdent ntensty on the sphere as a sngle ncoherent sum over the modes whch provdes an average trend to the curves as a functon of range. Takng the rato of the ncoherent combned extncton and ncdent ntensty, we obtan the ncoherent combned cross secton. The combned extncton, ncdent ntensty and combned cross secton of the sphere calculated ncoherently, usng Eqs 3, 7, and 8, respectvely, by replacng the double sum wth a sngle sum over the modes are plotted n Fgs. 2a c. From the ncoherent plots, we see that the extncton and the ncdent ntensty decay wth range due to geometrcal spreadng and absorpton loss n a real wave gude. In a perfectly reflectng wave gude, there s no absorpton n the wave gude. Consequently, an ncoherent approxmaton for T s ndependent of range as can be seen from Eq. 8. The decay n the extncton due to spreadng loss s compensated by spreadng loss n the flux ncdent on the object whch keeps the cross secton a constant. In ths case, the extncton measured by the screen s due entrely to the 2932 J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng

10 FIG. 3. Incoherent combned cross secton of a 0 m radus pressure release sphere at 300 Hz source frequency n a Pekers wave gude wth sand bottom half space, Pekers wave gude wth slt bottom half space, perfectly reflectng wave gude, and free space as a functon of source to object range x 0. For ths plot, xx 0. In the wave gudes, the source and sphere center are located at 50 m water depth. The ncoherent combned cross secton s calculated usng Eq. 8 by replacng the double sum over the modes wth a sngle sum. object. Fgure 3 shows T, calculated ncoherently, plotted for a pressure-release sphere of radus 0 m n a perfectly reflectng wave gude as a functon of x 0. In ths fgure x x 0. The ncoherent combned cross secton of the object n free space wth no absorpton and n the Pekers wave gude examples consdered so far are also plotted for comparson. Fgure 3 shows that ths ncoherent combned cross secton for the extncton of all the wave gude modes dffers sgnfcantly from the free space cross secton of the object. So, t s dffcult to obtan an estmate of the sze of an object from an ncoherent as well as a coherent measurement of ts combned cross secton. B. Modal cross secton n dfferent wave gudes In ths secton, we wll nvestgate how the modal extncton cross secton of the 0 m pressure release sphere vares for the ndvdual modes n varous wave gudes at 300 Hz. Fgures 4a and b show the ampltudes of the modes at the source depth of 50 m n the Pekers wave gude wth sand and slt bottom, respectvely. Only the propagatng modes are plotted because these are the modes that compose the ncdent feld on the object n the far feld. These are the mode ampltudes at the object depth because the target s also at 50 m depth. The ampltude of the modes n the perfectly reflectng wave gude are plotted n Fg. 4c. Only the even number modes are excted by the source at 50 m depth and they have the same ampltude. The extncton of each ndvdual mode n the Pekers wave gude wth sand bottom caused by the sphere and calculated usng Eq. 4 are plotted n Fgs. 5a and b at the source to object range of km and 25 km, respectvely. The screen s placed the same dstance away from the object as the source n each case. The modal extnctons n the Pekers wave gude wth slt bottom at km and 25 km are plotted n Fgs. 5c and d, respectvely. Comparng Fg. 5 wth Fg. 4, we see a dependence of the extncton of each mode on ts ampltude at the object depth, wth the more energetc modes beng extngushed the most. The extncton of the modes vary wth range due to spreadng and absorpton loss suffered by the modes. Absorpton loss suffered by each mode as a result of absorpton n a real wave gude s more severe for the hgh order modes due to ther steeper elevaton angles. The hgher order modes are gradually strpped wth ncreasng range and at suffcently long ranges, the extncton caused by the object s very much lmted to the extnc- FIG. 4. Modal ampltude at the source and target depth of 50 m n a Pekers wave gude wth sand half space, b Pekers wave gude wth slt half space, and c perfectly reflectng wave gude for a frequency of 300 Hz. J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 2933

11 FIG. 5. Extncton of the ndvdual modes Eq. 4 of the wave gude by the 0 m radus pressure release sphere at 50 m water depth wth the source separated from the sphere by a km range n a Pekers sand half space wave gude, b 25 km range n a Pekers sand half space wave gude, c km range n a Pekers slt half space wave gude, d 25 km range n a Pekers slt half space wave gude, and e km range n a perfectly reflectng wave gude, f 25 km range n a perfectly reflectng wave gude. The source depth s also 50 m and the source frequency s 300 Hz. The screen measurng the extncton s separated the same dstance from the object as the source n each case. ton of the frst few propagatng modes. For the perfectly reflectng wave gude n Fgs. 5e and f at km and 25 km, respectvely, there s no absorpton loss, so the extncton for each mode decays only wth source to object range x 0. There s no mode strppng effect n a perfectly reflectng wave gude and the relatve magntude of the extncton across the modes remans the same, ndependent of range. Fgures 6a c show the modal cross sectons of the sphere, calculated usng Eq. 20, for the extncton of the ndvdual modes n the Pekers sand, slt and perfectly reflectng wave gudes, respectvely. We set x0 n Eq. 20 to obtan the modal cross secton of the object corrected for absorpton n the wave gude. In each of the wave gudes llustrated n Fg. 6 we see that the modal cross secton of the sphere for the extncton of mode s very close to ts cross secton for the extncton of a plane wave n free space. For the hgher order modes, the modal cross secton of the object can be much larger or smaller than ts free space value dependng on the wave gude. We can calculate the forward scatter functon ampltude of the object from a measurement of the extncton of mode as dscussed n Sec. V whch allows us to estmate the sze of the object. C. Dependence of modal cross secton on object depth The modal cross secton of an object depends on the depth of the object n the wave gude. We nvestgate how the modal cross secton of the 0 m pressure release sphere vares when we lower ts depth by half a wavelength dstance to 52.5 m n the Pekers slt, sand, and perfectly reflectng wave gudes. We also lower the source depth to 52.5 m so that all the modes n the perfectly reflectng wave gude are excted by the source. The source frequency s 300 Hz. Fgure 7 shows the ncoherent combned cross secton of the sphere n the three wave gudes. In the perfectly reflectng wave gude, the ncoherent combned cross secton of the sphere s now larger than ts free space value. Fgures 8a c show the modal ampltudes n the three wave gudes and 2934 J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng

12 FIG. 6. Modal cross secton Eq. 20 at 300 Hz of the 0 m radus pressure release sphere at 50 m water depth for the extncton of the ndvdual modes n a a Pekers sand half space wave gude, b Pekers slt half space wave gude, and c perfectly reflectng wave gude. We set x0 n Eq. 20 to remove the effect of absorpton by the wave gude. The modal cross secton of the sphere for mode n each wave gude s almost equal to ts free space cross secton. Fgs. 9a c show the modal cross sectons, Eq. 20. Inthe perfectly reflectng wave gude Fg. 9c, all modes that exst n the wave gude are scattered by the object to form the scattered feld when t s at the shallower depth of 52.5 m, unlke n the prevous example of Fg. 6 where t was at 50 m depth and only the excted odd number modes were scattered by the object. Comparng Fg. 9 wth Fg. 6, we see that the modal cross secton of most of the modes vary wth object depth. For mode, however, n all the three wave gudes, the modal cross secton of the object remans close to ts free space value. D. Modal cross secton for varous object types The cross secton of the 0 m pressure release sphere s compared to that of a rgd or hard dsk of radus 0 m n the FIG. 7. Incoherent combned cross secton of a 0 m radus pressure release sphere at 300 Hz source frequency n a Pekers wave gude wth sand bottom half space, Pekers wave gude wth slt bottom half space, perfectly reflectng wave gude, and free space as a functon of source to object range x 0. For ths plot, xx 0. In the wave gudes, the source and sphere center are located at 52.5 m water depth. The ncoherent combned cross secton s calculated usng Eq. 8 by replacng the double sum over the modes wth a sngle sum. wave gude. In free space, wth the plane of the dsk algned normal to the drecton of propagaton of the ncdent waves, t s well known that ts plane wave extncton cross secton s equal to twce ts projected area, whch s m 2 n ths example. The cross secton of a sphere n free space depends on the crcumference of the sphere relatve to the wavelength of the ncdent waves,.e., ka2a/ where a s the radus of the sphere. The dependence of the extncton cross secton of a pressure release or hard sphere on ka, n free space s plotted n Ref. 6. For a large pressure release sphere, hgh ka, the extncton cross secton s roughly twce the projected area whch s the same for both the sphere and the dsk. For a compact pressure release sphere, small ka, the cross secton of the sphere begns to exceed twce ts projected area. For the present example, at 300 Hz source frequency, ka2.6 and the extncton cross secton of the sphere n free space s m 2. The ncoherent combned cross secton of the 0 m hard dsk n the three dfferent wave gudes s plotted n Fg. 0. In free space, the cross secton of the sphere at 300 Hz s only a lttle larger than that of the dsk of the same radus. Comparng Fgs. 3 and 0, we see that n the perfectly reflectng wave gude, the ncoherent combned cross secton of the sphere s much larger than that of the dsk. The elevaton angle of each mode of the wave gude ncreases wth the mode number. Snce the dsk s algned wth t s plane parallel to the y z plane, the projected area of the dsk perceved by each mode decreases as the elevaton angle of the mode ncreases. For the sphere, however, each mode sees the same projected area, regardless of the elevaton angle of the mode. Therefore the combned extncton of the modes by the sphere s much larger than by the dsk. In the real wave gude, absorpton by the wave gude alters the ampltude of each mode wth the hgher order modes J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 2935

13 FIG. 8. Modal ampltude at the object depth of 52.5 m n a Pekers wave gude wth sand half space, b Pekers wave gude wth slt half space, and c perfectly reflectng wave gude for a frequency of 300 Hz. sufferng greater absorpton losses than the lower order modes. The hgher order modes are less mportant n determnng the combned extncton n the real wave gude. Consequently, n a real wave gude, the ncoherent combned cross secton of the sphere s only slghtly larger than that of the dsk. The modal cross secton Eq. 20 of the dsk for each mode of the Pekers sand, slt, and the perfectly reflectng wave gude s plotted n Fgs. a c, respectvely. From Fg., we see once agan that the modal cross secton of the object for the extncton of mode s almost equal to ts free space cross secton. In the present example, the cross secton of the dsk s equal to twce ts projected area. Ths example further llustrates that we can obtan a measure of the sze of an object from the extncton of mode n a wave gude. FIG. 9. Modal cross secton Eq. 20 at 300 Hz of the 0 m radus pressure release sphere at 52.5 m water depth for the extncton of the ndvdual modes n a a Pekers sand half space wave gude, b Pekers slt half space wave gude, and c perfectly reflectng wave gude. We set x0 n Eq. 20 to remove the effect of absorpton by the wave gude. The modal cross secton of the sphere for mode n each wave gude s almost equal to ts free space cross secton J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng

14 FIG. 0. Incoherent combned cross secton of a hard dsk of radus 0 m at 300 Hz source frequency n a Pekers wave gude wth sand bottom half space, Pekers wave gude wth slt bottom half space, perfectly reflectng wave gude, and free space as a functon of source to object range x 0.For ths plot, xx 0. In the wave gudes, the source and dsk center are located at 50 m water depth wth the dsk algned n the y z plane. The ncoherent combned cross secton s calculated usng Eq. 8 by replacng the double sum over the modes wth a sngle sum. E. Dependence of modal cross secton on object sze and frequency Here we nvestgate how the modal cross secton Eq. 20 of a pressure release sphere at 50 m water depth compares wth ts free space cross secton when we vary the sze of the sphere and the frequency of the ncomng waves. Fgures 2a d show the result n a Pekers sand wave gude, plotted as a functon of ka. The correspondng result n the Pekers slt and perfectly reflectng wave gudes are plotted n Fgs. 3 and 4, respectvely. For a large sphere wth the hgh ka of 62.8, we see from Fgs. 2 4d that the modal cross secton of the sphere for the hgh order modes fluctuates and departs drastcally from the free space cross secton for most of the modes. The modal cross secton for mode, however, remans nearly equal to the free space cross secton of the large sphere n each wave gude. For the compact sphere wth the small ka of 0. on the other hand, Fgs. 2 4a, the modal cross secton of most of the modes are farly close to the free space cross secton of the object. Fgures 5a d show the scatter functon ampltude plotted as a functon of elevaton angle of the modes at varous ka. Compact objects scatter lke pont targets and they have an omndrectonal scatter functon S 0. In Eq. 20, we see that the modal cross secton depends on not only the forward scatter ampltude, but also the scatter functon ampltude n nonforward drectons. For a compact object, snce the scatter functon ampltude s a constant, ndependent of azmuth or elevaton angles, we can factor t out n Eq. 20. Furthermore, n a perfectly reflectng wave gude, snce 4,5 N p N p d0 2H, 27 N p can be factored out of the equaton as well. Consequently, for a compact object n the perfectly reflectng wave gude, Eq. 20 for the modal cross secton reduces to p x 4 k R p IS 0e 2I px 28 whch resembles the expresson for the free space cross secton of the object n Eq. C6. The modal cross secton of the compact object n the wave gude wll, however, be slghtly larger than the free space cross secton because of the dependence on the horzontal wave number of the mode p n the denomnator of Eq. 28 nstead of k as n Eq. C6 for free space. The real part of the horzontal wave number decreases as the mode number ncreases. We see a gradual FIG.. Modal cross secton Eq. 20 at 300 Hz of the 0 m radus hard dsk at 50 m water depth for the extncton of the ndvdual modes n a a Pekers sand half space wave gude, b Pekers slt half space wave gude, and c perfectly reflectng wave gude. We set x0 neq.20 to remove the effect of absorpton by the wave gude. The modal cross secton of the dsk for mode n each wave gude s almost equal to ts free space cross secton. J. Acoust. Soc. Am., Vol. 0, No. 6, December 200 P. Ratlal and N. C. Makrs: Extncton theorem for object scatterng 2937