Resonance scattering in waveguides: Acoustic scatterers


 Bertina Hutchinson
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1 Resnance scattering in waveguides: Acustic scatterers Philip M. Carrin and Marc Antni P. de Brit UFBA /IGEO/PPPG, Campus UniversitYri, Federaq , SalvadrBA, Brazil (Received 31 January 1989; accepted fr publicatin 29 September 1989 ) In this paper, acustic scattering in shallw and deep inhmgeneus waveguides is analyzed. The full acustic wave equatin that describes the scattered and reflected wave fields, as well as all multipathing within the scatterer and the waveguide, is emplyed. The explicit finite difference time integratin and the kw transfrm in the space dmain (pseudspectral methd) were utilized. Prperly chsen bundary cnditins enabled the authrs t mdel bth shallw and deep ceans. The pseudspectral methd was cmpared with the explicit finite difference technique (see Appendix). The pseudspectral methd can be valuable fr mdeling different underwater wave phenmena: It is characterized by much smaller numerical dispersin than the cnventinal finite difference methd. The results shw that in shallw cean, strng resnance cupling between the underwater scatterer and the waveguide may ccur. An imprtant cnclusin f this paper is that in a limited aperture experiment when the acustic reflectins are beynd the recrding aperture (due t the finite length f the recrding cable), the measured data are mainly represented by the primary diffractin arrivals and "diffractin resnances." In this paper, a detailed discussin will be given n the acustic scattering in an inhmgeneus waveguide. (Fr the sake f simplicity, an acustic scatterer f the rectangular shape was cnsidered. ) Different cmplexities (elastic scatterers, mre cmplicated structures f the index f refractin in the water, etc. ) will be cmpunded in the riginal mdel and reprted elsewhere. PACS numbers: Gv, Hw, Bp, Mv INTRODUCTION Prpagatin f sund in the cean can be treated as a waveguide prblem. The sea surface can be cnsidered with a great deal f accuracy as a pressurerelease surface, s all acustic energy generated belw the sea surface is reflected back with ppsite plarity (fr pressure). These reflectins are called ghsts. The sea surface, due t its rughness, may generate significant nise that is impsed n the returned signals. The prpagatin speed in the water can be rather cmplex: this leads t additinal scattering and diffractin n large and smallscale inhmgeneities (largescale inhmgeneities are much greater thdn the predminant wavelength, whereasmallscale inhmgeneities are f the rder r less than the characteristic wavelength). Recall that smallscale scattering is related t the scalled apparent attenuatin. As the result f apparent attenuatin, a pulse prpagating thrugh a lssfree medium with a highly discntinuusundspeed prfile is getting brader in width and smaller in amplitude. The recrded underwater signals are typically nnstatinary; their stchastic prperties are functins f time. The effects f nnstatinarity f the returned signals can be wellestimated in the tw dmain. When the generated sund reaches the sea bttm, it is reflected r scattered frm marine sediments. The backprpagating signal ges back t the sea surface and is ttally reflected by the sea surface regardless f the angle f incidence. This signal will be buncing between the sea surface and the sea bttm causing waveguide resnances. The resnance characteristics depend upn the angle f prpagatin, the reflectin cefficient at the marine sediments, and the refractin index in the cean. Carrin and Satkwiak, 2 using the cepstrum analysis, shwed recently that in the range f khz, marine sediments behave as a stack f cntinuus scatterers rather than discrete reflectrs. This means that there are n distinct interfaces in the subbttm medium that reflect the generated energy, rather the incident energy is scattered by cntinuusly distributed scatterers in the marine sedimentary envi rnment. Carrin and Wilbur, using the cherency measure between the seasurface and the seabttm reflectins, fund that fr mid and smallgrazing angles, the sea bttm behaves like the sea surface that reflects virtually all energy regardless f the frequency cntent f the initially generated signal. Fr larger grazing angles, the cherence between these signals fails fr the range f high frequencies, which are attenuated in marine sediments. This means that the wave guide resnances will be especially significant if the incident angle is near r exceeds the critical angle. Fr angles belw the critical angle, the amplitudes f waveguide buncing waves decrease due t transmissin lsses in the subbttm envirnment (quasiresnances). Theretical aspects f acustic and elastic resnance scattering was studied by different authrs. Imprtant cntributins t this subject include Flax eta/., TM Gaunaurd, 5 Frisk and Uberall, 6 and Werby and Green 7 amng thers. Hackman and Sammelmann 8 have recently presented a rigru study f acustic scattering in rangeindependent waveguides based n the Tmatrix frmalism. Their mdel included an elastic spherical shell in inhmgeneus layered 1062 J. Acust. Sc. Am. 87 (3), March /90/ Acustical Sciety f America 1062 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
2 waveguides whse square f the index f refractin was linear with respect t depth. In the previus wrk, Sammelmann and Hackman 9 cmputed scattering respnses frm the elastic shell in a hmgeneus waveguide bunded by the pressurerelease sea surface and the rigid sea bttm. (The rigid sea bttm was chsen t avid the branch cuts f Green's functins. ) In a typical snar experiment, a hrizntal array f hydrphnes is twed behind a ship and used as snar receivers. lo Thus it is necessary t analyze resnance scattering waveguides detected by a twing array f snar receivers with limited recrding aperture. In this paper, we study scattering in waveguides using a string f hydrphnes bth in shallw and deep cean. The numerical algrithm used in the paper is neither limited by the chice f frequencies nr by the gemetry f the mdel. In fact, this study can be extended withut much difficulty t rangedependent waveguide resnances and mre cmplicated gemetry f the underwater bject. The methd described in the paper can be an alternative t the Tmatrix frmalism since it is rather flexible with regards t the chice f the bundary cnditins fr deep and shallw cean. Usually, the twing arrays are characterized by a limited aperture s a significant prtin f the reflectins frm the target might be missing. In this paper, we studied effects f the limited aperture and intrduced the cncept f the diffractin resnances that cnstitute a significant cntribu tin t the recrded data. An imprtant cnclusin f this paper is that in shellw cean, strng resnance cupling between the waveguide and the scatterer may ccur. Our results shw that the resnance cupling in deep cean is small and typically can be ignred. Althugh the examples presented in the paper are limited by acustic scattering, different cmplexities will be included in the mdel t make it mre realistic. In the Appendix, we will cmpare the Furier methd with the cnventinal finite difference mdeling. It will be shwn that the pseudspectral methd can be very valuable fr mdeling different underwater wave phenmena. Fr example, fr the same frequencies, numerical dispersin in the pseudspectral methd des nt depend n the angle f prpagatin and is usually much smaller than in the cnventinal finite difference techniques. I. SCATTERING THEORY Usually acustic/elastic scattering is analyzed using the Green'sBetty therem, where the scattering respnse is fund by integratin f Green's functin, pressure, and their nrmal derivatives alng the surface f the scatterer. The T matrix frmalism can then be emplyed. 4 The Tmatrix apprach can be efficiently used fr bjects f different shapes whse basis functins are nt cmplicated. Fr targets f arbitrary shapes, basis functins are n lnger simple and this can lead t cumbersme calculatins. Anther apprach t acustic/elastic scattering is based n numerical slutins f the acustic/elastic wave equatins with apprpriate initial (IC) and bundary (BC) cnditins. In this paper, we will cnsider the cmplete (full wavefrms) acustic wave equatin that gverns the prpagatin f pressure in liquids. Wave prpagatin in a 2D acustic medium can be written as fllws: P 1 2p P = 1 x p x q zz p& Pc 2 t 2 +S(t)6(x)6(z), (1) where P(x,z,t) is the pressure field, c(x,z) is the prpagatin speed, p(x,z) is the density, S(t) is the surce time functin (snar generted pulse), and 62 (.) describes the lcatin f the snar. We use the explicit discretizatin in time (explicittime scheme). The secndrder time derivative can be presented as 2p! pt+ 1 2p t1 = '... + O( t 2), (2) t 2 t 2 where n and m represent sampling in the x and z directins and l is the sampling rate in time. The time step was cnfrmed t the LCF stability cnditin. We use the FFT methd previusly discussed by Kslft and Baysal. 2 This methd requires fewer number f grid pints t achieve the same accuracy than ther techniques and cnsists f the fllwing steps: (a) apply the FFT t the wave field (e.g., pressure); (b) multiply the results by ik and ik ; (c) apply the FFT, giving 8P/Sx and 8P (d) multiply 8P/Sx and 8P/& by 1/p; (e) apply the FFT t the prducts and multiply them by ikx and ik ; (f) apply the FFT i and btain (8/Sx) ( 1/p) (SP/&) and (8/&) (l/p) (SP/&). Remark: Due t the finite discretizatin f the mdel space, finite difference r finite element numerical schemes are characterized by the apparent numerical dispersin that is expressed by the difference in the phase and grup velcities. Fr pseudspectral methds, this dispersin can be negligibly small. Initial cnditins evlve the causality principle, whereas the bundary cnditins at the sea bttm fr the deep cean are chsen t be absrbing and fr shallw cean, express the cntinuity f pressure acrss the bund ary. Usually, acustic scattering is cnsidered in the WKBJ apprximatin. In the highfrequency apprximatin, the large prtin f data can be described in terms f the WKBJ (gemetrical ptics) apprximatin using the phase and the amplitude functins. The phase functin satisfies the eiknal equatin, whereas the amplitude functin is gverned by the transprt equatin. Fr the refractin index f different cmplexity, slutins t the transprt equatin are typically singular due t ccurrence f caustics in the medium f cnsideratin. In this study, we d nt perate in the WKBJ apprximatin and thus we d nt expect that caustics d ccur. We will cnsider limited aperture twing arrays that d nt detect the reflectins frm the scatterer that are beynd the recrded aperture. We advcate an interesting result, that in the case f limited aperture array, the recrded data 1063 J. Acust. Sc. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brit: Waveguide scattering 1063 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
3 mainly cntain diffractin and diffractin resnance arrivals. In the next sectin, we will give the descriptin f the acustic numerical experiment in deep cean where the resnance cupling is small. surce Twing rry f hydrphnes II. RESONANCE SCATTERING IN INHOMOGENEOUS DEEP OCEAN In this sectin, we will cnsider deep cean. In rder t better understand resnance scattering, the mdel was chsen in such a way that the reflectins frm the bundaries f the underwater bject placed in the waveguide were missing due t the limited recrding aperture. The sea surface is presented by the free surface (pressurerelease surface). At the freesurface pressure vanishes at all pints. In rder t eliminate the influence f the sea bttm, absrbing bundary cnditins were incrprated. Absrbing bundary cnditins were set up in the cat layer. The lssfree acustic wave equatin in this "layer" was substituted by the viscacustic equatin with a damping parameter. When this parameter is large, and the cat layer is small, we may anticipate sme reflectins frm the "absrbing" bundaries. The best results turn ut if the cmprmise between the value f the damping parameter and the size f the cat layer is achieved. Prperly chsen damping parameter and the thickness f the cat layer enable ne t suppress unwanted reflectins frm the sea bttm and thus t accurately analyze the deepcean respnses. This can be dne using the trial and errr methd. In ur case, 30 grid pints were assigned t the cat layer. The pressure field Pc in the cat layer is presented by Pc exp[  a(n i) ], (3) where N is the ttal number f the grid pints in the cat layer and a is a damping parameter. Let us intrduce the dimensinless quantity Ka (where a is the size f the scatterer). Fr ur mdel, Ka = Alng with Ka, we will cnsider anther parameter KL (L is the range fr the snar experiment). In ur case, KL is 61.83, s KL >> Ka (this defines the farfield regin). Figure 1 is the mdel where the surce is lcated at depth 90 rn belw the sea surface. The scatterer that has the shape f a square is placed inside the sund layer at a depth f 330 rn belw the sea surface. The side f the scatterer is 30 m. The width f the sund layer is 190 m. The receivers were placed at the same depth with the surce. The nearest ffset (the distance between the surce and the first hydrphne) was 60 m, s the length f the twing array was 150 m. The surce signature is presented in Fig. 2 that shws the zerphase Rickertype wavelet. The high cutff frequency was 245 Hz. Figure 3 shws the discretizatin f the mdel scatterer. The target has 10 grid pints per side. Figure 4 shws the numerical result f scattering in deep cean recrded by 50 hydrphne spaced 3 rn apart. The first arrival (marked by "1") is the direct wave traveling frm the surce t the receivers. The secnd arrival is the ghst (reflectin frm the sea surface), the third arrival is the reflectin frm the tp f the sund layer. The furth arrival is the secndrder (firstrder multiple) reflectin V = p= t.o V = t p=t.o V = p=t.o V= 2 000,00 4 p= 5.0 target absrbing bundary cnditins FIG. 1. Mdel f the snar experiment in deep cean. Twing array f hydrphnes is placed abve the target. Absrbing bundary cnditins are impsed n the sea bttm t suppress reflectins frm the sea flr. FIG. 2. Zerphase Ricker wavelet used in the numerical mdeling. FIG. 3. Discretizatin in the mdel space. The size f the cell was 3 m: 10 grid pints per side f the scatterers J. Acaust. Sac. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brita: Waveguide scattering 1064 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
4 0.4 (a) FIG. 4. Respnse f the twing array (at the 50th hydrphne). Arrws indicate the resnance scattering. Letter d indicates diffractin arrivals frm the crners. 0.4 (b) frm the tp f the sund layer. Number "5" marks the reflectin frm the bttm f the sund layer. Number "6" stands fr the diffractin curve frm the first crner f the scatterer and the seventh arrival is the diffractin arrival cming frm the secnd crner f the scatterer (see Fig. 1 ). Number "8" depicts the diffractin event frm the third crner and finally number "9" is the diffractin event frm the furth crner. It is imprtant t nte that the scatterer shws strng resnance pattern (events between arrws in this figure). Figure 5(a) depicts the first hydrphne reading. It shws that all events belw the 0.55 s crrespnd t internal resnances within the scatterer. Figure 5 (b) is the respnse recrded by the last hydrphne lcated at 210 m away frm the surce. Events belw 0.5 s are the resnances within the scatterer. Our results cmputed fr the deepcean mdel indicate that the resnance cupling between the scatterer and the waveguide is negligibly small. III. DIFFRACTION RESONANCES When the generated waves impinge n the bundaries f the scatterer, the scatterer excites the scalled Huygens' secndary surces. The transmitted energy can be decmpsed int the vertically and laterally prpagating pressure fields. This means that internal resnances can be cnsidered as a cmbinatin f hrizntal and vertical resnating waves. In accrdance with the chsen gemetry presented in Figs. 1 and 3, ne can realize that due t the limited aperture f the recrding cable, internal vertical and hrizntal resnances will be nt recrded by the twing array f hydrphnes. This means that nly nndirectinal perturbatins will be recrded. The nly areas that generate nndirectinal energy are crners f the scatterer. Thus internal resnances within the scatterer will be detected due t pint diffractrs (crners). These events we call the diffractin resnances that are well seen in Fig. 4 where the gemetry f the diffrac FIG. 5. (a) Recrded signal at the first hydrphne shws rather cmplicated resnance signature (marked between arrws). (b) Recrded signal at the 50th (the furthest) hydrphne shws the resnant signature f bunced waves. tin respnse cincides exactly with the gemetry f the diffractin resnances marked by tw arrws. This suggests that the resnance pattern recrded by the twing array cmes frm the resnating crners f the target. Figure 6 demnstrates this cncept. One can bserve that recrded resnances have large amplitudes and a steady pattern in time. These are characteristics f standing waves. The respnse frm underwater bjects in a limited aperture experiment with missing reflectins is therefre mainly characterized by the diffractin arrivals and the internal resnance pattern. IV. SHALLOW OCEAN SCATTERING IN INHOMOGENEOUS A. Resnance cupling WAVEGUIDES Remark: By definitin, resnances are determined by buncing waves between tw bundaries. A perfect example 1065 J. Acust. Sc. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brit: Waveguide scattering 1065 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
5 INCIDENT ENERGY Suppse nw that the multiple generatr crrespnding t the scatterer is DECOMPOSED INTERNAL RE SONANCES where rn is an integer, the internal resnances and the resnances in the waveguide will then be tangled. Our results shw that this cupling can be substantial, since the amplitudes f the internal resnances match the amplitudes f the FIG. 6. Internal resnances are nt detected due t the limited aperture f waveguide resnances. In rder t untangle these resnthe twing array. Respnses frm the resnating crners have the same travel time gemetry as primary diffractin arrivals. ances, the waveguide ringing can be suppressed. 13 Fr multichannel snar experiments, the resnances can be decupled using the differences in the gemetries f the detected respnses. The prblem f decupling needs separate treatf the resnance is the wave traveling between the sea surface ment and thus is beynd the scpe f the paper. and the sea bttm. In fact, the number f resnance cycles One f the mst imprtant prblems in underwater depends n the reflectin cefficients. If the reflectin ceffi acustics is t deduce acustic surces Of energy and t decient f the sea bttm is small, ne deals with damping termine the prpagatin characteristics f acustic waves. resnances. The resnance cupling is an eventhat is a sum Fr example, the sea bttm is transparent fr acustic f tw resnances. Its signature cntains peridic wave waves. Hwever, fr near critical reflectins, it resembles the frms f different perids. In the frequency dmain, resn sea surface. The seabttm respnse is nt nly a functin f ances crrespnd t singular slutins fr the amplitude the angle but als depends n frequency: namely high frefunctin (regular ples) fr frequencies in the neighbrhd quencies can be absrbed in marine sediments at rather shalf the natural frequency f the waveguide. Resnances can lw depths s the reflectin respnse will nt cntain the be expressed in terms f the multiple generatr peratr. Let highfrequency cmpnent f its spectrum. As the first apus cnsider the multiple generatr that crrespnds t nr prximatin, we will mdel the sea bttm as a twlayer mal incidence. Buncing energy in the waveguide can be acustic medium. expressed by the fllwing peridic functin: Let us cnsider the fllwing mdel. The scatterer f the square shape was placed in the sund layer whse width was M= ( 1)Jaj 5(t  rj.), (4) 60 m. The snar was placed abve the sund layer s the j=l range was 300 m. The depth f the cean was 120 m. The seawhere bttm structure was mdeled by tw sediment layers with the sund speed being 2000 and 2500 m/s, respectively. The Tj=2/ f a c(z) dz (5) thickness f the first sediment layer was 45 m. Figure 7 deis the twway travel time and a are the reflectin ceffi scribes the mdel. Figure 8 is the reflectin respnse recrdcients. Fr example, a = R, where R is the reflectin cefficient frm the sea bttm and a 2  R 2 and s n. S (4) can be rewritten as M= ( lyr,(t (6) j=l Mx = ( 1) b &(t  Tj), (7) j=l where b is the reflectin cefficient frm the target bundaries and T is the perid f internal resnances. If Tj  m T., (8) 475 D V = p=t p= p=5 V=i482.25, p=t V = 2 50O p=5 absrbing bundary cnditins FIG. 7. Mdel f the shallw cean with the target placed in the sund layer. FIG. 8. Acustic respnse frm the shallwcean waveguide with the target. Letters SL mark the reflectins frm the sund layer. This figure shws strng cupling between the waveguide and the target resnances J. Acust. $c. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brit: Waveguide scattering 1066 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
6 FIG. 11. Resnance cupling fr the first 10 channels. 0.6 FIG. 9. Resnance cupling between the target and the waveguide fr time greater than 0.4 s. 0.2 ed by the twing array f 90 hydrphne spaced 3 m apart. The direct wave was eliminated. Arrivals marked by letters SL crrespnd t the reflectins frm the sund layer. Since the sund layer is a lwvelcity zne and thus des nt have critical and supercritical reflectins, the energy that crrespnds t the critical reflectins is missing. This figure shws strng resnance cupling between the waveguide and the internal scatterer resnances (waveguide resnances are marked by letter W and the scatterer resnances are marked by letter T). Figure 9 is the recrded signal that shws the resnance cupling belw 0.4 s. Figure 10 is the resnance pattern fr the channel N. 70 shwing the resnance capling fr times later than 0.3 s. Figure 11 demnstrates the resnance cupling marked by letter R between the target and the waveguide fr the first 10 channels, whereas Fig. 12 depicts the resnance cupling between the scatterer and the waveguide fr the last 20 channels. One can see that this cupling is rather strng and bscures the scatterer resnances (cmpare with the deepcean results, Fig. 4). It is imprtant t nte that fr a single channel, the waveguide.9, FIG. 10. Resnance cupling fr time greater than 0.3 s. FIG. 12. Resnance cupling fr the last 20 channels J. Acust. Sc. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brit: Waveguide scattering 1067 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
7 and the.scattereresnances are difficult t separate. They can be better identified by a multichannel recrding since the gemetry (mveuts) f the scattereresnances is typically different frm the gemetry f the waveguide resnances (see Fig. 12). V. CONCLUSION In this paper, we presented the results f the resnance In the previus sectin, we cmputed the resnance rescattering inhmgeneus rangeindependent waveguides spnses frm the acustic scatterer using the pseudspectral fr bth shallw and deep cean. It was shwn that acustic methd. Let us cmpare this technique with the cnventin scatterers can exhibit strng resnance patterns. We intrduced a new cncept f diffractin resnances that are caused by resnating pint diffractrs f th scatterer. The diffractin resnances ccur despite the cmplexities f all multipathing within the target and have the same gemetry in the tx dmain as the primary diffractins. In rder t mdel deep cean, we used absrbing bundary cnditins at the sea bttm. The numerical methd used in the paper is fast and allws ne t increase the accuracy withut increasing the number f grid pints in the mdel space. Our results shw that there is strng cupling between the internal scatterer resnances and the resnances wave phenmena in waveguides. We thank the annymus reviewer wh suggested that we cmpare the pseudspectral and the finite difference techniques. APPENDIX: FINITE DIFFERENCE AND PSEUDOSPECTRAL METHODS al finite difference methd based n the explicit timespace integratin. The finite difference technique has been well develped in underwater acustics. 14 Let us cnsider the stability and numerical dispersin f bth schemes. The stability cnditin is usually expressed as At <]?c/h, ( A 1 ) where c is the minimum prpagatin speed, His the grid size, and At the time step. Alfrd et al.'5 shwed that fr finite difference schemes/3 3x/3, while Kslff and Baysa112 shwed that/3 = x / r fr the Furier methd. This means that a finer discretizatin is needed in the psuedspectral in shallw waveguides. methd t btain the same stability. In the Appendix, we cmpared the pseudspectral Let us cnsider the numerical dispersin effects f the methd and the finite difference technique. The main cn pseudspectral methd. Figure A 1 (a) shws the numerical clusin is that althugh the finite difference methd can be, dispersin fr/ The dtted curve shws the grup cmputatinalwise, mre ecnmical, the numerical disper velcity, whereas the slid line is the phase velcity. Figure sin effects in the pseudspectral methd are much less b A 1 (b) shws the numerical dispersin fr/3 = 0.2. One can served. see that in this case the numerical dispersin effects are almst missing. When we use the finite difference FD tech ACKNOWLEDGMENTS nique, the dispersin curve [Fig. A2(a) ] fr/ shws We wuld like t thank PPPG/UFBA, CNPq (Bra significant dispersin fr kh > 2. It is interesting t see what zilian Natinal Science Fundatin), Finep, Petrbras fr happens when we use/3 = 0.2. Figure A2 (b) shws that the supprt and encuragement. We thank Guenther Schwe difference between the grup and the phase velcity remains dersky fr his participatin at the initial state. PMC is grate almst the same whereas the adverse effects f dispersin ful t Rger Hackman f the Naval Castal Systems Center appear at lwer frequencies. It is very interesting t under (Panama City, FL) fr valuable discussins. We thank Dan scre that in the FD methd, the numerical dispersin is a Kslff f the TelAviv University fr the use f his pseud functin f angle f prpagatin while the pseudspectral methd des nt have this drawback. spectral sftware subrutines that were helpful fr mdeling e" ee e" n, d O Z.. ee"ø! '!..79,'.s?. e., 0 0 LAJ z (5'... O0 0' '.:56 FIG. AI. (a) Dispersin curves in the pseudspectral methd fr/3 = Dtted line is grup velcity. Slid line is the phase velcity pltted versus KH, where H is the grid size. (b) Dispersin curves in the pseudspectral methd fr/3 = 0.2. Dtted line is grup velcity. Slid line is the phase velcity pltted versus KH. KM KM (a) (b) 1068 J. Acust. Sc. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brit: Waveguide scattering 1068 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see
8 . C3 t,:.. U.J _J, N 13:: z FIG. A2. (a) Dispersin curves in the finite difference methd fr/3 = Dtted line is grup velcity. Slid line is the phase velcity pltted versus KH. (b) Dispersin curves in the finite difference methd fr/3 = 0.2. Dtted line is grup velcity. Slid line is the phase velcity pltted versus KH ! ,I J 4, KM (a) (b) IP. Carrin, and J. Wilbur, "Wideaperture reflectin acustics: Snar experiments in shallw water," J. Acust. Sc. Am. 86, (1989). 2p. Carrin, and L. J. Satkwiak, "Acustic scattering frm marine sediments in shallw water experiments," submitted t J. Acust. Sc. Am. 3L. Flax, G. Gaunaurd, H. Uberall, "The thery f resnance scattering," in Physical Acustics, edited by W. P. Masn and R. N. Thurstn (Academic, New Yrk, 1981 ), Vl. XV, Chap. 3, pp L. Flax, L. R. Dragnette, V. K. Varadan, and V. V. Varadan, "Analysis and cmputatin f the acustic scattering by an elastic prlate spherid btained frm the Tmatrix frmulatin," J. Acust. Sc. Am. 71, (1982). 5G. Gaunaurd, "Resnance thery f elastic and viscelastic wavescattering and its applicatins t the spherical cavity in absrptive media," in Mdern Prblems in Elastic Wave Prpagatin, edited by J. Miklwitz and J. Achenbach (Wiley, New Yrk, 1977). 6G. V. Frisk and H. Uberall, "Creeping waves and lateral waves in acustic scattering," J. Acust. Sc. Am. 59, (1976). 7M. F. Werby and L. H. Green, "A cmparisn f acustical scattering frm fluidladed elastic shells and sund sft bjects," J. Acust. Sc. Am. 76, (1984). 8R. H. Hackman and G. S. Sammelmann, "Lngrange scattering in a deep ceanic waveguide," J. Acust. Sc. Am. 83, (1988). 9R. H. Hackman and G. S. Sammelmann, "Acustic scattering in a hmgeneus waveguide," J. Acust. Sc. Am 82, (1987). IøW. A. Kuperman, M. F. Werby, and K. E. Gilbert, "Twed array respnse t ship nise: a nearfield prpagatin prblem," presented at NATO Cnference n Underwater Acustics Signal Prcessing, Luneberg, West Germany (1984). K. Marfurt, "Elastic wave equatin migratininvertin," Ph.D. thesis, Clumbia University, New Yrk (1987). 2D. D. Kslff and E. Baysal, "Frward mdeling by a Furier Methd," Gephysics 47 (10), ( 1982 ). i3p. Carrin, "A layerstripping technique fr suppressin f waterbttm multiple reflectins," Gephys. Prspect. 34, (1986). 14R. A. Stephen, "Geacustic scattering frm seaflr features in the ROSE area," J. Acust. Sc. Am. 82, (1987). SR. M. Alfrd, K. R. Kelly, and D. M. Bre, "Accuracy f finitedifference mdeling f the acustic wave equatin, Gephysics 39, (1974) J. Acust. Sc. Am., Vl. 87, N. 3, March 1990 P.M. Carrin and M. A. P. de Brit: Waveguide scattering 1069 Dwnladed 14 Jan 2013 t Redistributin subject t ASA license r cpyright; see