4 DIRECTIONAL RESPONSE OF A CIRCULAR ARRAY IN AN EMBEDDED FLUID CYLINDER

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1 4. Chapter 4 DIRECTIONAL RESPONSE OF A CIRCULAR ARRAY IN AN EMBEDDED FLUID CYLINDER 4.1 INTRODUCTION Arrays of hydrophones are housed wthn sonar domes to protect them and to prevent the flow of water drectly over them (Wate, 22). The arrays are used to detect acoustc waves radated by dstant sources and determne the drecton of arrval of the nearly plane waves. The thckness of the dome s nvarably much less than the radus of curvature of the dome and often much less than the wavelength n water at the freuency of nterest (Warren, 1988). However, the curvature of the dome gves rse to convergence or dvergence of the acoustc waves that are ncdent on t. In ths chapter, a uanttatve understandng of how well ray theory can be used to determne the drectonal response of a hydrophone array n a sonar dome s studed by usng the theory to analyze a canoncal problem. Specfcall ray theory s used to determne the nteror pressure feld when a plane acoustc wave s normally ncdent on an nfnte flud cylnder embedded n another flud of nfnte extent as shown n Fg The pressure feld s then used to determne the drectonal response of a phased crcular array. Methods based on ray theory are sutable for acoustc analyss of sonar domes. Acoustc and hydrodynamc consderatons are used to desgn the shape of the sonar dome (Loeser, 1981). The dome s usually doubly curved and has absorbng nternal structures. Further, the normalzed freuency ka, where k s the acoustc wavenumber n the water and a s the radus of curvature of the dome, s usually much greater than one. Therefore, the nteror pressure feld cannot be easly determned usng analytcal or numercal methods. Moreover, the surface of the dome and the nternal structures are desgned such that only one or at most a few rays called egenrays n an ncdent plane wave pass through any pont wthn the dome. Therefore, ray theory s used n the present analyss. Domes used n shps are flled wth fresh water and pressurzed usng an overhead tank. The speed of sound n fresh water s less than that n sea water. However, the speeds of both types of stress waves n the dome are often greater than that n sea water. Therefore, on entry nto the nteror water, the rays may converge or dverge. In 56

2 ths chapter, the effect of ths spreadng s studed usng nteror fluds wth sound speed greater than, and less than, that n the exteror flud. The approach can be extended to nclude the effect of the dome materal. The pressure feld nsde the cylnder s determned usng ray theory and the results are uanttatvely valdated by comparng them wth those obtaned usng the method of separaton of varables and seres solutons. The latter method s used to study scatterng from rgd cylnders and spheres (Morse and Ingard, 1968; Junger and Fet, 1972), flud spheres (Anderson, 195; Foote, 27), flud cylnders (Skudrzyk, 1971; Alemer et al., 1986), sold elastc cylnders and spheres (Faran Jr., 1951) flud-flled, concentrc, elastc cylnders (Akay et al., 1993), and absorbng cylnders (Mtr et al., 24). Results are often presented as a functon of ka where k s the acoustc wavenumber of the exteror flud and a s the radus of the scatterer. At low freuences, the method s very convenent but the prmary lmtaton of the method s that t cannot be appled to several other geometres of nterest. Further, at hgh freuences, where ka s much greater than one, a very large number of terms s reured for the seres soluton to converge. However, hgh freuences and other geometres are of nterest n many practcal applcatons. ka s approxmately 2 when the freuency s 5 khz and the radus of the scatterer s 1 m as well as when the freuency s 1 MHz and the radus of the scatterer s 5 mm. Further, the method of separaton of varables cannot be easly used to analyze domes because of the shape of the dome and the presence of nternal surfaces. Other methods are used even at low freuences when the geometry does not permt use of the method of separaton of varables. However, these methods are often valdated by comparng the results obtaned for cylnders or spheres wth those obtaned usng the method of separaton of varables. Boag et al. (1988) use fcttous flament sources to study the scattered cross secton of flud cylnders wth arbtrary cross secton. The number of sources reured for convergence ncreases when the freuency or radus ncreases. Numercal results are presented for a crcular cylnder wth ka up to approxmately 1. Chandra and Thompson (1992) also study a plane wave ncdent on an embedded flud cylnder. The denstes of the nner and outer fluds are the same but the speeds of sound n them are dfferent. They use Pade approxmants to mprove the convergence of the Neumann seres that s used to solve an ntegral euaton. They present accurate numercal results for the nteror and exteror pressure felds at ka = 2. It s known that, n general, ray theory s accurate at very hgh freuences. However, at freuences that are not very hgh and n cases where there are several 57

3 egenrays or caustcs n the neghbourhood, t s to be determned as done n ths chapter whether the use of ray theory yelds results that are suffcently accurate by comparng the results wth those obtaned usng some other method that s known to be accurate at low freuences. The comparson s best done for a canoncal problem that can be solved by usng both methods. Ray theory s used by Clay and Medwn (1977) who present a method that yelds exact results at all freuences when appled to reflecton and transmsson of a plane wave obluely ncdent on a mult-layered flud wth planar nterfaces. In ths method, the total pressure s expressed as a sum of pressures due to rays that are partally reflected and partally transmtted at each nterface. Marston and Langley (1983) use ray theory to study acoustc backscatterng from a flud sphere. They present results for ka = 1 and 1 and compare them wth those obtaned usng the method of separaton of varables and seres solutons. Marston (1992, 1997) has extensvely used geometrcal acoustcs and physcal acoustcs (Bowman et al., 1969) to study a varety of problems. Stanton et al. (1993a, 1993b) use a model wth only 2 rays to study backscatterng from elongated objects such as cylnders and prolate spherods wth low acoustc contrast. They use a heurstc phase term, and compare results for ka up to 1 wth the seres soluton. Another method that s used at hgh freuences s based on the method of separaton of varables and the Sommerfeld-Watson transformaton. Brll and Uberall (197) use ths method to study the porton of a hgh-freuency plane wave that s transmtted through a flud cylnder embedded n an nfnte flud. The transmtted wave s expressed as a sum of waves that are reflected, 1, 2, 3, tmes wthn the cylnder where there are n waves that are reflected n tmes wthn the cylnder. Numercal results are presented for ka = 1. Rumerman (1991) uses ths method and the Krchhoff thn shell theory to study scatterng from an elastc cylndrcal shell. He uses the resdue theory and the Sommerfeld-Watson transformaton to convert the expresson for backscattered pressure from a modal sum to a contour ntegral. The ntegral s shown to have contrbutons from specular reflecton, waves that crcumnavgate the shell, and certan poles. Includng all the contrbutons s shown to yeld results that are n good agreement wth those obtaned usng modal analyss for 1.5 <ka<2. A dfferent hgh freuency method s used by Bruno et al. (24) who study scatterng from rgd bodes usng a boundary ntegral formulaton and present results for a cylnder wth ka varyng from one to 1,. They express the surface pressure as the 58

4 product of a slowly varyng ampltude and a hghly oscllatory exponental. Then, they use a localzed ntegraton method that s related to the method of statonary phase. They present results for scatterng from a rgd cylnder wth ka varyng from 1 to 1 5. Other authors use, at hgh freuences, the Krchhoff approxmaton for an ntegral euaton formulaton or mprovements based on t. Medwn and Clay (1998) provde detals of a method to use the Krchhoff approxmaton, convert the Krchhoff ntegral euaton to an ntegral expresson, and determne the scattered pressure. Schneder (23) summarzes the results for scatterng from a submarne obtaned by usng several methods and brefly dscusses the lmtatons of usng the Krchhoff approxmaton n the ntegral euaton formulaton. Junger (1982) presents a method to study scatterng from a rgd body of revoluton characterzed by a radus that vares along the length of the body. He assumes that t scatters sound as though each crcular element s a part of an nfnte cylnder of the same radus. Ths s less restrctve than the Krchhoff approxmaton n whch t s assumed that scatterng occurs as though each element s a part of an nfnte plane. He presents results for a prolate spherod that are n good agreement wth those obtaned usng a T-matrx approach n the low freuency Raylegh regon as well as the md freuency resonance regon. In the hgh freuency Krchhoff regon, the results are n good agreement wth those obtaned usng the Krchhoff approxmaton. Ye et al. (1997) extend the method and apply t to scatterng from flud prolate spherods. In another hgh freuency approach, the scattered pressure s expressed as a seres wth the terms contanng negatve nteger powers of the wavenumber (Bowman et al., 1969; Kravtsov and Orlov, 1993; Kaufman et al., 22). The seres s known varously as Debye seres and Luneburg-Klne seres. Retanng only the frst term n ths seres and makng an approxmaton yelds ekonal and transport euatons that can also be derved usng geometrcal acoustcs (Knsler et al., 1982). Other approxmate solutons wth fractonal powers of the wavenumber are used n the geometrc theory of dffracton and n the study of caustcs. In ths chapter, a hgh-freuency ray-acoustcs method s presented and used to determne the nteror pressure feld when a plane wave s normally ncdent on a flud cylnder embedded n another nfnte flud. The geometrcal and physcal acoustcs (Bowman et al., 1969) approxmatons are used. Geometrcal acoustcs s used to determne the pressure when the rays dverge or converge. The physcal acoustcs or Krchhoff approxmaton for scatterng s also used: the reflecton and transmsson of 59

5 each ray when t meets a curved nterface s assumed to occur as f t s from an nfnte plane nterface that s tangent to the nterface (Medwn and Cla 1998). The method s of nterest because t can be extended to study the nteror pressure feld for other bodes wth shapes that are not sutable for usng the method of separaton of varables. It s shown usng numercal results that the pressure feld computed usng ths method s n good agreement wth that computed usng the method of separaton of varables. The output from a sector of a crcular array of hydrophones n the embedded cylnder s computed and compared wth the output from an array n an nfnte homogenous flud. These outputs are of nterest as they are used to determne the drecton of arrval of the wave. The error n estmatng the drecton of arrval depends on the sgnal processng method also. 4.2 DIRECTIONAL RESPONSE OF HYDROPHONE ARRAY Consder a hgh-freuency plane wave travelng along the postve x axs as shown n Fg It s ncdent on a flud cylnder of radus a and nfnte length embedded n an outer flud. The axs of the cylnder les on the z axs. Therefore, there s no varaton n the pressure feld along the axs of the cylnder and only the (x,y) Cartesan coordnates and (r,θ) polar coordnates are used. The denstes of the nner and outer fluds are ρ and ρ o, respectvely; and the speeds of sound n the fluds are c and c o, respectvely. Followng conventon, the densty rato ρ / ρ and sound-speed rato c / c are defned as g and h, respectvely. The non-dmensonal freuenc ka, where k s the wavenumber n the exteror flud s much greater than one. ρo, c o B β β β B 1 O ρ, c B 2 Fg. 4-1 Reflecton and transmsson of rays n an embedded flud cylnder of radus a (sold lne). A crcular array of radus b nsde the flud cylnder s also shown (dotted lne). 6

6 Consder next 2H euspaced pont hydrophones. They are on the permeter of a crcle of radus b wth centre at the orgn where b<a. The outputs from 2J adjacent hydrophones n one sector are delayed and summed to smulate a lnear array where J s typcally approxmately eual to H/3. 2H such lnear arrays are smulated by usng 2H sectors where, for example, hydrophones 1 to 2J form sector 1, hydrophones 2 to 2J1 form sector 2, and so on. The effect of the embedded flud cylnder on the outputs from the smulated lnear arrays s of nterest. The outputs from the hydrophones depend on the pressure feld nsde the embedded cylnder. An expresson for the nteror pressure feld s presented. It s derved here by usng expressons for transmsson and reflecton coeffcents, pressure varaton due to dvergence and convergence of rays nsde a cylnder, and methods to trace rays and determne whch rays wll pass through a partcular pont of nterest. The nteror pressure feld computed usng ray theory s shown to be ute accurate even n the neghborhood of caustcs. It s then used to determne the outputs from the smulated lnear arrays Incdent ray n the outer flud Consder a ray travelng n the outer flud n a drecton perpendcular to the axs of the cylnder and ncdent on the flud cylnder as shown n Fg The ncdent and transmtted rays are at angles and β, respectvely to the local normal to the surface of the cylnder where s specfed and β s determned by usng Snell s law. The pressure due to the transmtted ray s of nterest. H F P t ρ o, c o P β a O Y X ρ, c E G Fg A ray travelng n the outer flud s ncdent on the nterface. The tangent to the nterface s EF. The radus of the cylnder s a. The angle of ncdence between the ray and the normal to the nterface, GH, s. The angle of transmsson s β. 61

7 It s assumed that the physcal acoustcs or Krchhoff approxmaton (Medwn and Cla 1998) s vald and that the transmsson coeffcent s eual to that when a ray (or plane wave) s ncdent on a plane nterface that separates the two fluds. The plane nterface s tangental to the cylnder at the pont of ncdence. Therefore, consder a ray travelng along the X axs and ncdent on an nclned surface, EF, separatng two semnfnte fluds as shown n Fg The normal to the surface, GH, passes through the orgn. The dstance, a, between the orgn and the surface, along the normal, s eual to the radus of the cylnder. The pressure due the ncdent ray of unt ampltude s P ( x, ) = exp( jk ) ox where ω s the angular freuency and k = ω / c s the wave number n the outer flud. o o The tme factor, exp( jωt), where t denotes tme s suppressed for convenence. The pressure due to the transmtted ra neglectng spreadng effects that are ncluded later, s expressed as P [ j( k d k d )] ( x, ) = To ( )exp o. (4-1a) t t Here, the overbar ndcates that the pressure s due to a unt wave n the exteror flud, the subscrpt ndcates that the ray has not been reflected at a concave nterface, and ( ρ c cos ρ c cos β ) T o ( ) = (2ρc cos) / o o (4-1b) s the transmsson coeffcent (Medwn and Cla 1998) when a plane wave s obluely ncdent on a plane nterface that ncludes the orgn and separates two sem-nfnte fluds, and β are the angles that the ncdent and transmtted waves, respectvely make wth the normal to the nterface, k = ω / c s the wave number n the nner flud, d = a cos s the addtonal dstance that the wave n Fg. 4-2 has to travel before the wavefront reaches the orgn, and d t s the dstance between a feld pont M ( x, ) and y B ( ) where B ( ) = ( a cos, asn ) s the pont at whch ray s ncdent on the flud cylnder. Ths expresson for Pt can also be obtaned by usng wave theory (Mathew and Ebenezer, 29) and d t = ( x a cos)cos( β ) ( y asn)sn( β ) Incdent ray n the nner flud The ray that s transmtted nto the embedded cylnder s reflected when t meets the nterface at B1 as shown n Fg The ray s travelng at an angle β to the normal 62

8 as shown n Fg. 4-1 and explaned n the next sub-secton. The ray that s transmtted to the outer flud travels to nfnty and does not re-enter the cylnder. Only the reflected ray s of nterest. It s agan assumed that the Krchhoff approxmaton s vald. Therefore, consder the wave of unt pressure ampltude n Fg. 4-3 that s ncdent on an nclned nterface EF separatng two sem-nfnte fluds. The wave s travelng n a flud of densty ρ and speed of sound c. The normal to the nterface, GH, forms an angle γ 1 wth the x axs. As shown n Fg. 4-3, the ray s travelng at an angle β to the normal and at an angle γ β 1 to the x axs and s expressed as { jk [ x cos( γ β ) sn( γ )]} P1( x, ) = exp 1 y 1 β. The pressure due to the reflected ra correspondng to the unt ncdent ray and neglectng spreadng effects that are ncluded later, s expressed as Pˆ ( x, ) = R ( ) exp [ jk ( d d )] (4-2a) r1 o 1 1r where the hat over the pressure ndcates that the pressure s due to a unt wave n the nteror flud, the subscrpt 1 ndcates that the ray has been reflected once at a concave nterface, and R ( ) = ( ρ c cos β ρ c cos) /( ρ c cos β ρ c cos) (4-2b) o o o o s the reflecton coeffcent when a plane wave s obluely ncdent on a plane nterface that ncludes the orgn and separates two sem-nfnte fluds, β and o are the angles F H ρ c ρ o c o Y P β β a G γ 1 P r X E P t Fg A ray travelng nsde the flud cylnder s ncdent on the nterface. The tangent to the nterface s EF. The angle of ncdence between the ray and the normal to the nterface, GH, s β. The angle of transmsson s. 63

9 that the ncdent and transmtted waves, respectvely make wth the normal to the nterface, d1 = a cos β s the addtonal dstance that the wavefront n Fg. 4-3 has travelled after t has passed through the orgn, and d 1 r s the dstance between a feld pont M 1( x, y) and B 1( ) = ( a cosγ 1, a sn γ 1). Ths expresson for Pˆr 1 can also be obtaned by usng wave theory (Mathew and Ebenezer, 29) and d r = x a cosγ )cos( γ β ) ( y asn γ )sn( γ ). 1 ( β The pressure due to the ray reflected once at a concave nterface, correspondng to the unt ray travelng n the exteror flud and ncdent on the flud cylnder, neglectng only spreadng effects that are ncluded later, s expressed as P ( x, ) = Pˆ r1 = T r1 o ( x, ) P ( ) R o t ( x, ) / P where, as shown later, γ = 2β 1 ( x, ) ( ) exp[ j( k a cos k { x cos( γ β ) y sn( γ β ) 3a cos β} )] o (4-3) In general, the pressure due to the ray reflected tmes at a concave surface, ( 1), correspondng to the unt ray travelng n the exteror flud and ncdent on the flud cylnder, neglectng spreadng effects, s expressed as where Pˆ r [ jk ( d d )] ( x, ) = R ( ) exp (4-4) o d s the addtonal dstance that the wavefront has travelled after t has passed r through the orgn, and d s the dstance between a feld pont ( x, y) and r M B ) = ( acosγ, asnγ ). ( The pressure due to the ray reflected tmes at a concave nterface, correspondng to the unt ray travelng n the exteror flud and ncdent on the flud cylnder, neglectng only spreadng effects, s expressed recursvely as P ˆ r ( x, ) = Pr ( x, ) Pr ( 1) ( x, ) / P ( x, ) = To ( ) Ro ( ) exp[ j( koa cos k{ x cos( γ β ) y sn( γ β ) (2 1) ka cos β} )] where P ( x, ) exp{ jk [ x cos( γ β ) y sn( γ β )]} = ncdent ray and t s shown later that γ = π [ ( π 2β )]. (4-5) s the pressure n the unt Ray tracng nsde the flud cylnder The path of each ray that enters the cylnder s of nterest. Some rays wll undergo total nternal reflecton n the exteror flud. Each ray that enters the cylnder s reflected 64

10 and transmtted each tme t encounters the nterface between the two fluds. Of these, only the reflected ray s of nterest because the transmtted ray travels to nfnty and does not re-enter the cylnder. Consder a ray n the exteror flud that s ncdent on the cylnder such that the angle between the ray and the normal to the surface of the cylnder at the pont of ncdence s as shown n Fg It s ncdent on the cylnder at the pont B ( ) = ( a cos, a sn ) n Cartesan coordnates. It s assumed that the surface of the cylnder s locally plane and the expressons derved n the earler secton are used. Therefore, the ampltude of pressure n the ray that s transmtted nto the cylnder st o. The transmtted ray travels at an angle β to the normal to the nterface as shown n Fgs. 4-1 and 4-2. Let O be the centre of the cross-secton of the cylnder. The axs of the cylnder passes through t. Let B 1 be the pont at whch the ray transmtted nto the cylnder meets a concave nterface for the frst tme. It s seen from Fg. 4-1 and the sosceles trangle B OB1 that B B 1 O = β and B OB1 = π 2β. A ray from B to B 1 s labeled as an n = ray because t has only been transmtted at a convex nterface but not reflected at a concave nterface. The chords traversed by several travelng n = rays that have just entered the cylnder are shown n Fg. 4-4a and n Fg. 4-4b for the cases when g=h= 1.1 and when g=h=.9 respectvely. Total nternal reflecton occurs when the angle of ncdence s greater than the crtcal angle, θ = sn 1 ( c / c ) that n ths case s deg for the case c o where g=h= 1.1.The n= rays reaches everywhere nsde the flud cylnder when h>1 whle there are regons nsde the flud cylnder where there are no n= rays when h<1. Fg n= rays nsde the flud cylnder: a) when g=h=1.1 and b) when g=h=.9. 65

11 Fg n=1 rays nsde the flud cylnder: a) when g=h=1.1 and b) when g=h=.9. The chords traversed by n = 1 rays after one nternal reflecton at a concave nterface are shown n Fg. 4-5a and Fg. 4-5b for the cases when g=h= 1.1 and when g=h=.9, respectvely. Only one n=1 ray reaches most parts of the cylnder. However, there s a regon n whch there are three n=1 rays. When an n = 1 ray meet the concave nterface agan an n = 2 ray s reflected, and so on Dvergence, Convergence, and Caustcs The plane wave that s travelng along the x axs and ncdent on the flud cylnder s consdered to consst of an nfnte number of parallel rays. Each ra n the z = plane, that s ncdent on the cylnder, s ncdent at a unue angle. The paths of neghborng rays that are parallel n the exteror flud are of nterest. When c > c o, the rays dverge after enterng the cylnder. However, f c < c o, the rays converge and some of them ntersect wthn the cylnder, and then dverge. Irrespectve of whether c > c o or c < c o, the rays that are travelng wthn the cylnder are reflected at the concave nterface and then converge. Some of them ntersect at a caustc (Kravtsov and Orlov, 1993) wthn the cylnder whereas others ntersect at an magnary pont outsde the cylnder. After ntersecton, they dverge. The detals are of nterest. Consder a ray n the exteror flud ncdent at an angle to the local normal to the surface as shown n Fg It meets the cylnder at B ( ) = ( a cos, a sn ) and later at B ( ) = [ a cos(2β ), a sn(2β )]. Consder, next, an adjacent ray ncdent at 1 an angle = d. It meets the cylnder at B ) and B ). The slopes of the chords B ) B ( ) and B ) B ( ) are m ( ) and m ), respectvely. The chords ( 1 ( 1 meet, n (x,y) coordnates, at the focal pont ( ( 1 ( 66

12 a m F ( ) = a [ ( ) cos( ) m ( ) cos( ) sn( ) sn( ) ] ( ) [ m ( ) sn( ) m ( ) sn( ) m ( ) m ( ) ( cos( ) cos( ))] m ( ) m m ( ) m ( ),. (4-7) The focal pont of the flud cylnder may be nsde or outsde of the flud cylnder, dependng on h and the ncdence angle. When h>1, the n= rays wll dverges as shown n Fg. 4-4a and the chord of n= rays extended to meet the focal pont on the left sde of the flud cylnder s shown n Fg When h< 1, the n= rays wll converges as shown n Fg. 4-4b. For hgher angle of ncdence, the chord of n= rays ntersect nsde and for lower angle of ncdence, the chord of n= rays extended and meet the focal pont on the rght sde of the flud cylnder. It s mportant to note that the rays ncdent at the angles and = d ntersect at a pont F ( ) that s not the same as F ( ). The spreadng factor of the rays, S ( x, y), at a pont M that les between B ( ) and B ( ) s determned by usng the prncple of energy conservaton. There s 1 no varaton n pressure along the axs of the cylnder and t s assumed that the varaton of ntensty along a ray s nversely proportonal to the dstance from a focal pont. S ( x, y) s determned usng the focal ponts F ( ) and F ( ), and expressed, n general, as an average value: S ( x, ) =.5.5 [ B ( ) F ( ) / M F ( )] [ B ( ) F ( ) / M F ( )] / 2. (4-8a) where the overbar denotes the dstance between two ponts. However, there are two Fg Sold lne: n= rays. Dashed lne: rays extended to meet the focal pont. 67

13 exceptons. Frst, f the magntude of ether of the two terms used to determne the average spreadng factor s greater than the other by more than 5%, then S ( x, ) s y defned as the lesser of the two terms. Ths s because S ( x, ) s nfnty at a caustc and y very large close to t. Second, f s eual to the crtcal angle, the spreadng factor s expressed as [ B ]. 5 ( ) F ( ) / M F ( S. (4-8b) ( x, ) = ) The two rays, after reflecton at B ( ) and B ), respectvel are labeled as n 1 = 1 rays and meet the cylnder agan at B ( ) and B ), respectvely. The chords 2 B 1( ) B2 ( ) and B1( ) B2 ( ) ntersect at a pont F 1( ) that les nsde or outsde the cylnder. In general, the n = rays meet at [ m ( ) c ( ) m ( ) c ( ) s ( ) s ( ) ] a, m ( ) m ( ) F ( ) = (4-9) a{ m ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( )]} s m s m m c c, m ( ) m ( ) n (x,y) coordnates, where c ( ) = cos[ ( π 2β )], c ( ) = cos[ ( π 2β )], s ( ) = sn[ ( π 2β )], s ( ) = sn[ ( π 2β )] and c o sn( β ) = c sn( ) respectvely. For the n = 1 rays, the focal ponts le nsde the cylnder when h = 1.1and the loc of these ponts s a closed curve that s shown n Fg It s obtaned by usng E. (4-9) when =1. 1 ( 2 ( Fg The loc of focal ponts of n=1 rays. 68

14 The spreadng factor at a pont M 1 that les between B ( ) and B2 ( ) s S 1( x, y). In general spreadng factor at a pont 1 M that les between B ( ) and B ( ) s [ B ( ) ( ) / ( )] [ ( ) ( ) / ( )] F M F B F M F / 2 S (,, ) x y = (4-1) Alternatve expressons to E. (4-1) are used for S ( x, y), as done for S ( x, ), when y ether of the two terms used to determne the average spreadng factor s greater than the other by more than 5%, or f s eual to the crtcal angle Pressure along a Ray For the n = ra the pressure at the pont (x,y), ncludng spreadng effects, s expressed as P x, ) = Pt ( x, ) S ( x, )exp[ jµ ( x, )] (4-11) ( y where P t ( x, ) s defned n E. (4-1), µ s when M s between B ( ) and F ) and π / 2 when M s between F ) and B ( ) (Bowman et al., 1969). ( Smlarl the pressure at M 1, ncludng spreadng effects, s P x, ) = Pr ( x, ) S ( x, y, )exp[ jµ ( x, y )] S ( x, )exp[ jµ ( x, )] (4-12) 1( y where µ s when 1 M1s between B1 ( ) and F ) and π / 2 when M1 s between F1( ) and B 2( ). In general, the pressure at a pont ncludng spreadng effects s expressed as ( 1 ( 1 M that les between B ( ) and B 1( ), P ( x, ) = P ( x, ) S ( x, )exp[ jµ ( x, y)] S x y jµ r n= 1 n 1 ( n, n, )exp[ n 1( n, n)] x y (4-13) where µ s when M s between B ( ) and F ( ) and π / 2 when M s between F ( ) and B ( ), and µ n s when F ( ) s between Bn ( ) and B n 1 ( ) and 1 π / 2 when F ( ) s not between B ( ) and B ( ). n Interor Pressure Feld n n n 1 In order to determne the total pressure at a pont nsde the cylnder, t s necessary to frst fnd all the egenrays rays that pass through the pont of nterest. The n = ray meets the nterface between the exteror and nteror fluds at the 69

15 ponts (a, π-) and (a, 2β-) n polar coordnates. The ray passes through the pont (r, θ) f the pont les on the straght lne that jons B ( ) and B ( ). Therefore, the angle at whch the n= ray should be ncdent on the cylnder n order for t to pass through (r, θ) s determned by solvng 1 [ sn{ θ π} sn{ θ 2β } ] a sn{2β} = r (4-14) after usng Snell s law to elmnate β and by usng numercal methods. It s noted that there s only one soluton to the above euaton when c > c. In general, L rays pass through the pont and the angle of ncdence of these rays when they are n the exteror flud s l, l = 1,2,3,... L. Smlarl the n = rays that passes through the feld pont are dentfed by solvng r [ sn{ θ 2β ( 1) π} sn{ θ 2( 1) β π} ] asn(2β ) = l (4-15) fnd, l = 1,2,3,... L.The number of solutons, L, to the above euaton depends on the locaton of the feld pont, the value of, and the relatonshp between c and For example, when / c =1. 1, only one n = 1 ray (dashed lne) passes through c the feld pont (.3, 9 o, ) deg s shown n Fg. 4-8a whle three n = 1 rays pass through the pont (.9, o,) and the solutons to E. (4-15) when =1 at the pont r=.9 and θ=, are = , 2 =, and 3 = deg s shown n Fg. 4-8b. c o. Fg. 4-8a. Rays passng through r =.3, θ = 9 o. n = (sold lne) and n =1 (dashed lne). Fg. 4-8b. Three n = 1 rays passng through r =.9, θ = o. 7

16 After fndng all the rays that pass through the pont of nterest, the complex pressure due to each of them s added to determne the total pressure, P ( x, y),at the pont. N n= L P( x, y) = P ( x, ) (4-16) l = 1 n l where P x, ) s the pressure due to the lth ray and s obtaned by usng E. (4-13). n ( l The value of N s chosen to be large enough to ensure convergence of the seres to a desred accuracy. When the acoustc contrast s small, t s suffcent to use a small value of N because the reflecton coeffcent s small Feld Theory Consder a plane wave of unt ampltude s ncdent on the flud cylnder and t can be expressed n cylndrcal coordnates (r,θ,z) n seres form as P = n= n ε ( j) J ( k r)cos nθ (4-17a) n n o where J n(.) s the nth order Bessel functon of the frst knd and Newmann coeffcent ε n = 1 2 n = n > The radal component of dsplacement assocated wth ncdent wave s gven by u 1 = 2 ρoω P r Where denotes the partal dervatve. u ko n ' = ε ( ) ( ) cos 2 n j J n kor nθ ρ ω o n= (4-17b) (4-18a) (4-18b) where denotes the dervatve. When the ncdent wave meets the flud cylnder t scatters and the scattered pressure s expressed n seres form as = 2 Ps An H n ( kor)cos nθ (4-19) n= 2 where H (.) s the nth order Hankel functon of the second knd, and A n s the unknown n coeffcent of scattered pressure. The radal component of dsplacement assocated wth the scattered wave s gven by 71

17 u s = k ρ ω o ' o (2) A ( ) cos 2 nh n kor n= nθ (4-2) The nteror pressure P at a pont ( r, θ,) of a flud cylnder s expressed n seres form as (Skudrzyk, 1971) n= ( k r) cos( ) P( r, θ ) = B J nθ (4-21) n n where B n s the unknown coeffcent of nteror pressure. The radal component of dsplacement assocated wth the transmtted nteror wave s gven by u t = k ρ ω At the nterface ' B ( ) cos 2 n J n kr n= r = a nθ (4-22), radal components of pressure and dsplacement n the outer flud s eual to radal components of pressure and dsplacement n the outer flud. Applyng the contnuty condtons at the nterface, the coeffcent B n can be evaluated as B n = ko πρ a o H ρo (2) ' n 2 ( j) n 1 ε n k (2) ' ( k a) J ( k a) H ( k a) J ( k a) o n n o n ρ (4-23) Drectonal Response Consder, next, 2H euspaced pont hydrophones as shown n Fg H sectors, each wth 2J adjacent hydrophones, are formed by groupng these hydrophones. The outputs from the hydrophones n a sector are delayed to smulate a lnear array and summed, and the outputs from a few adjacent sectors that gve hgh outputs are used to C D h B E φ h Φ s O X A Fg Schematc sector of a crcular array. The hydrophones are n the arc ACB. Each hydrophone s delayed by an approprate amount to smulate a lnear array from the chord AB. 72

18 determne the drecton of arrval of the wave. In practcal applcatons, the regon r<b s occuped by a structure on whch the array s mounted and the output from the sector that s drectly llumnated by the wave s much more than that from the sector that s dametrcally opposte t. In the mnor arc AB, shown n Fg. 4-9, there are 2J adjacent hydrophones. They form one sector and are used to smulate a lnear array on the chord AB. The angle between the normal to the lnear array and the x axs s pont D h =(x h, y h ) and the lne OD h makes an angle Φ s. The hth hydrophone s at the φ = π ( 2h 1) / H, h= 1, 2, 3,... 2H, h 2 wth the x axs. A sector s used to smulate a lnear array by applyng a phase delay that corresponds to the dstance ED h, n Fg. 4-9, to the hydrophone at D h. For convenence, an addtonal delay that corresponds to the dstance OF s appled to all the 2J hydrophones n that sector. Therefore, the total delay appled to the hth hydrophone corresponds to d = bcos( φ Φ ). The delayed and summed voltage output from the sth sector s expressed as h V s h = M s 2J s 1 h= s P( x h, y )exp( jk d ) h h (4-24) where M s the recevng acoustc senstvty of each hydrophone expressed n V/Pa and P x h, y ) s obtaned by usng E. (4-16). ( h Drectonal response s defned here as the output from the sth sector when the wave s travelng along the x axs. For a partcular arra t s dependent only on the angle between the normal to the sth sector and the drecton n whch the wave s travelng. When the nteror and exteror fluds are the same (g = h = 1) and the ncdent wave s travelng along the x axs, the output from the (H-J1)th sector s maxmum because t s symmetrc about the x axs. In general, when g = h = 1, the sector wth the maxmum output s the one wth a normal most closely algned wth the drecton n whch the wave s travelng. Here, wth the wave travelng along the x axs, the outputs from the sector whose centre les on the negatve x axs and the outputs from neghborng sectors are large. Numercal results are presented to uanttatvely llustrate the effect of the embedded cylnder on the outputs from the lnear arrays. 73

19 4.3 NUMERICAL RESULTS AND DISCUSSIONS Numercal results are presented for several cases to llustrate the agreement between the results obtaned usng the present method and feld theory and to llustrate the effect of the propertes of the embedded cylnder. Unless otherwse specfed, c o = 15 m/s, and ρ o = 1 kg/m 3. The radus of the cylnder s 1 m n all the cases. Several results are presented for freuences of 5 khz and 2 khz and correspond to ka 2.9 and 83.8, respectvely. The number of terms used to obtan the feld theory results depends on the freuency and vares from 3 at 5 khz to 25 at 2 khz and the convergence of the results s tested n all cases. At even hgher freuences, the accuracy of the ray theory results ncreases but more terms are reured when usng feld theory. Unless otherwse specfed, all feld theory results are shown usng a sold lne, ray theory results are shown usng dots, and only the n = and 1 rays are used. The effect of usng an ncreasng number of terms n ray theory s llustrated n some fgures. Ray theory results are shown usng varous symbols. In Fgs. 4-1a and 4-1b, contour plots are presented of the magntudes of the nteror pressure feld computed usng ray theory and feld theor respectvely to llustrate the overall good agreement. The freuency s 2 khz. c = 165 m/s and ρ = 11 kg/m 3 ; that s, g = 1.1 and h = 1.1. Evanescent rays are generated when the angle of ncdence s greater than the crtcal angle, θ = sn 1 ( c / c ). θ c s deg when g=h= 1.1. The ncdent wave s travelng along the postve x axs. Therefore, as expected, the fgures are symmetrc about the dameter contanng θ =. It s seen from the Fgs. that the agreement between the two methods s good at most ponts and that ntrcate patterns match well. However, on r = a, and θ near 9 and 18 deg, t s seen from the feld theory results that the pressure changes rapdly and the agreement s not as good as t s at other places. If g = h = 1, then the magntude wll be one everywhere n the cylnder. It s seen from the contour plots n Fg. 4-1 that the pressure patterns nsde and outsde the caustc curve n Fg. 4-7 s dfferent. The dfference s expected to be greater when the acoustc contrast between the nner and outer fluds s greater because more of the n = ray wll be reflected when the n = rays are ncdent on the concave surface. The real and magnary parts of the pressures for g = h = 1.1, at 2 khz, on a few dameters, are also presented to llustrate the agreement between the results obtaned c o 74

20 Fg Magntude of nteror pressure at 2 khz when g=h=1.1 a) usng Ray theory wth n= and n =1 rays b) usng Feld theory wth 25 terms. usng the two methods. The real and magnary parts of the pressures are shown n Fgs. 4-11a and 4-11b, respectvel on the dameter formed by the by the θ = and 18 deg rad. Smlarl the pressures are shown n Fgs. 4-12a and 4-12b, on the dameter formed by the by the θ = 3 and 21 deg rad; and n Fgs. 4-13a and 4-13b on the dameter formed by the by the θ = 9 and 27 deg rad. In the fgures, the poston on the dameter s shown to vary from r/a = -1 to 1; where ponts on the θ =, 3, and 9 deg deg rad are assumed to have non-postve values. The wavelength nsde the cylnder, at 2 khz, s 82.5 mm and there are nearly 25 wavelengths n one dameter. It s seen from Fgs. 4-11a and 4-11b, where the pressure on the θ = and 18 deg rad s shown, that nearly 25 deep spatal oscllatons occur wth the real and magnary Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = o and 18 o rad at 2 khz when g=h=1.1. Sold lne: Feld theory wth 25 terms. Damonds: Ray theory wth n = and 1 rays. 75

21 pressure varyng from a lttle less than -1 to a lttle more than 1. Ths happens because the wavelength nsde the cylnder, at 2 khz, s 82.5 mm and there are approxmately 25 wavelengths n one dameter. If g = h = 1, then the oscllatons wll be unform and exactly between -1 and 1. In these fgures, a sgnfcant change n the pattern s seen near r/a =.48 the pont at whch the focal pont of the n = 1 ray les on the θ = lne. As r/a approaches.48 from below, the peaks n the absolute real pressure, n Fg. 4-1a, ncrease and then, after crossng r/a =.48, decrease suddenly. The peaks n the absolute magnary pressure, n Fg. 4-11b, decrease and then ncrease suddenly. It s seen from Fg. 4-1 that the magntude of the pressure on ths dameter vares from about.2 to 1.3. Even though there s good agreement between the results obtaned usng the two methods at all ponts, the dfference n the real pressure s a lttle greater near r/a =.48. In Fg the nteror pressure feld computed usng ray theory and feld theory on the dameter formed by θ = 3 and 21 deg rad are presented. The agreement s good. The caustc pont for the n = 1 ray les on r/a.78 when θ = 3 deg and the dfference between the ray and feld theory results s a lttle more n ts neghborhood. In Fgs. 4-13a and 4-13b, the pressure s presented on the dameter formed by θ = 9 and 27 deg rad. The spatal oscllatons correspondng to 2 khz are seen as perturbatons on a curve that slowly vares between 1 and -1. If g = h = 1, the real part of pressure wll be 1 and the magnary part wll be zero because ths dameter les on x = and the ncdent pressure s exp( jkx). Therefore, a hydrophone kept on ths dameter n the embedded cylnder wll sense a pressure that s consderably dfferent Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 3 o and 21 o rad at 2 khz when g=h=1.1. Sold lne: Feld theory wth 25 terms. Damonds: Ray theory wth n = and 1 rays. 76

22 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 9 o and 27 o rad at 2 khz when g=h=1.1. Sold lne: Feld theory wth 25 terms. Damonds: Ray theory wth n = and 1 rays. from the free-feld pressure that would have exsted f the embedded cylnder had not been present. The caustc pont for the n = 1 ray les on r/a.986 when θ = 9 deg and the dfference between the ray and feld theory results s greater n ts neghborhood. In Fgs. 4-14a and 4-14b, contour plots are presented of the magntudes of the nteror pressure feld computed usng ray theory and feld theor respectvely to llustrate the overall good agreement for a freuency of 5 khz at whch ka 2. The real and magnary parts of the pressures are shown n Fgs. 4-15a and 4-15b, respectvel on the dameter formed by the by the θ = and 18 deg rad. Smlarl the pressures are shown n Fgs. 4-16a and 4-16b, on the dameter formed by the by the θ = 3 and 21 deg rad; and n Fgs. 4-17a and 4-17b on the dameter formed by the by the θ = 9 and 27 deg rad. Even though the value of ka s now one fourth of the earler value, the Fg Magntude of nteror pressure at 5 khz when g=h=1.1 a) usng Ray theory wth n= and n =1 rays b) usng Feld theory wth 25 terms. 77

23 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = o and 18 o rad at 5 khz when g=h=1.1. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 3 o and 21 o rad at 5 khz when g=h=1.1. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 9 o and 27 o rad at 5 khz when g=h=1.1. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 78

24 Fg Magntude of nteror pressure at 2 khz when g=h=.9 a) usng Feld theory wth 25 terms b) usng Ray theory wth n= and n =1 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = o and 18 o rad at 2 khz when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 3 o and 21 o rad at 2 khz when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 79

25 observatons regardng the results at ka 8 are vald for ths case also. The caustc ponts depend only on h and are ndependent of freuency. It s seen that observatons regardng the agreement near the caustc ponts at 2 khz are vald for 5 khz also. In Fgs. 4-18a and 4-18b, contour plots are presented of the magntudes of the nteror pressure feld computed usng ray theory and feld theor respectvely to llustrate the overall good agreement when c = 135 m/s and ρ =9 kg/m 3 ; that s g =.9 and h =.9. It s seen from the fgure that the magntude of nteror pressure s lower n the regon where there s no n= ray. The real and magnary parts of the pressures are shown n Fgs. 4-19a and 4-19b, respectvel on the dameter formed by the by the θ = and 18 deg rad. The caustc pont for the n = 1 ray les on r/a.529,.9276 and.934 when θ = deg and the dfference between the ray and feld theory results s a lttle more n ts neghborhood. Smlarl the pressures are shown n Fgs. 4-2a and 4-2b, on the dameter formed by the by the θ = 3 and 21 deg rad and the caustc pont for the n = 1 ray les on r/a.7534, and.787 when θ = 3 deg. Smlarl the pressures are shown n Fgs. 4-21a and 4-21b on the dameter formed by the by the θ = 9 and 27 deg rad. In Fgs. 4-22a and 4-22b, contour plots are presented of the magntudes of the nteror pressure feld computed usng ray theory and feld theor respectvely to llustrate the overall good agreement for a freuency of 5 khz at whch ka 2. The real and magnary parts of the pressures are shown n Fgs. 4-23a and 4-23b, respectvel on the dameter formed by the by the θ = and 18 deg rad. Smlarl the pressures are shown n Fgs. 4-24a and 4-24b, on the dameter formed by the by the θ = 3 and 21 deg rad; and n Fgs. 4-25a and 4-25b on the dameter formed by the by the θ = 9 and27 deg rad. Even though the value of ka s now one fourth of the earler value, the observatons regardng the results at ka 8 are vald for ths case also. Next, results are presented for hgher values of g and h at whch the reflecton coeffcent for rays travelng nsde the embedded cylnder are hgher. For g = 2 and h = 1.5, the n = 2, 3, rays can be expected to have a greater effect on the error than for the earler case of low g and h. For a freuency of 2 khz, real parts of the pressures on three dameters are shown n Fgs. 4-26a, 4-27a, and 4-28a; and the magnary parts are shown n Fgs. 4-26b, 4-27b, and 4-28b. It s seen that ncludng more rays yelds better agreement between the two methods. 8

26 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ =9 o and 27 o rad at 2 khz when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg Magntude of nteror pressure at 5 khz when g=h=.9 a) usng Feld theory wth 25 terms b) usng Ray theory wth n= and n =1 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = o and 18 o rad at 5 khz when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 81

27 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 3 o and 21 o rad at 5 khz when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ =9 o and 27 o rad at 5 khz when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 82

28 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = o and 18 o rad at 2 khz when g=1.4 and h=1.4 Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Green suares: Ray theory wth n =, 1 and 2 rays. Yellow damonds: Ray theory wth n =, 1, 2 and 3 rays. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 3 o and 21 o rad at 2 khz when g=1.4 and h=1.4. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Green suares: Ray theory wth n =, 1 and 2 rays. Yellow damonds: Ray theory wth n =, 1, 2 and 3 rays. 83

29 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 9 o and 27 o rad at 2 khz when g=1.4 and h=1.4. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Green suares: Ray theory wth n =, 1 and 2 rays. Yellow damonds: Ray theory wth n =, 1, 2 and 3 rays. The effect of ncreasng the number of rays s more pronounced at 5 khz. Ths s seen n Fgs. 4-29, 4-3, and 4-31 where the real and magnary parts of the pressures are shown on three dameters. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = o and 18 o rad at 5 khz when g=1.4 and h=1.4. Sold lne: Feld theory wth 25 terms. Damonds: Ray theory wth n= and 1 rays. Suares: Ray theory wth n=, 1 and 2 rays. Dots: Ray theory wth n=, 1, 2 and 3 rays. 84

30 Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 3 o and 21 o rad at5 khz when g=1.4 and h=1.4. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Green suares: Ray theory wth n =, 1 and 2 rays. Yellow damonds: Ray theory wth n =, 1, 2 and 3 rays. The pressures, as a functon of freuenc at three ponts on the plane of symmetr are shown n Fgs for g = h = 1.1. The ponts are r=a/2 and θ = 18 deg; r= and θ = deg; and r=3a/4 and θ = deg. In the Fgs., ka vares from 2 to 1 and only the n = and 1 rays are used. There s good agreement n all cases between results obtaned usng the ray and feld theores. Fg a) Real part and b) magnary part of nteror pressure feld on dameter formed by θ = 9 o and 27 o rad at 5 khz when g=h=1.4. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Green suares: Ray theory wth n =, 1 and 2 rays. Yellow damonds: Ray theory wth n =, 1, 2 and 3 rays. 85

31 Fg a) Real part and b) magnary part of nternal pressure feld at the pont r=a/2 and θ = 18 o from ka = 2 to 1 when g=h=1.1. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg a) Real part and b) magnary part of nternal pressure feld at the pont r= and θ = o from ka = 2 to 1 when g=h=1.1. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. Fg a) Real part and b) magnary part of nternal pressure feld at the pont r=3a/4 and θ = o from ka = 2 to 1 when g=h=1.1. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 86

32 Fg a) Real part and b) magnary part of nternal pressure feld at the pont r=a/2 and θ = 18 o from ka = 2 to 1 when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. The pressures, as a functon of freuenc at three ponts on the plane of symmetr are shown n Fgs for g = h =.9. The ponts are r=a/2 and θ = 18 deg; r= and θ = deg; and r=3a/4 and θ = deg. In the Fgures, ka vares from 2 to 1 and only the n = and 1 rays are used. There s good agreement n all cases between results obtaned usng the ray and feld theores. When the acoustc contrast ncreases, a hgher value of N n E. (4-16) s reured for convergence. The effect of ncreasng N on the nteror pressure computed usng ray theory s shown n Table I and the results are compared wth those obtaned usng feld theory. In feld theor 3 terms are used and the results have converged. The pressures are shown at r =.5 m and θ = 9, 12, 15, and 18 deg. at 2 khz for g = h = 1.4. It s Fg a) Real part and b) magnary part of nternal pressure feld at the pont r= and θ = o from ka = 2 to 1 when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 87

33 seen from the Table 4-I that the ray theory results converge to those obtaned usng feld theory when N s ncreased. Table 4-I. Magntude of nteror pressure at 2 khz when g= h=1.4 at r=.5 m. Angle Magntude of Pressure (degree) Feld Ray Theory theory N = N = 1 N = 2 N = The outputs from a crcular array wth 2H=32 hydrophones and 2J=12 hydrophones n each sector are shown n Fgs and The outputs, V s, are computed usng ray theory and E. (4-24) wth M = 1, and are shown for s = 3 to 17 because the normal to the lnear array smulated usng the 11th sector makes an angle of 18 deg wth the x axs. The angles that the normals to the smulated lnear arrays make wth the x axs of the coordnate system are shown on the abscssas of Fgs and Fg a) Real part and b) magnary part of nternal pressure feld at the pont r=3a/4 and θ = o from ka = 2 to 1 when g=h=.9. Red sold lne: Feld theory wth 25 terms. Blue dots: Ray theory wth n = and 1 rays. 88

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