they are normally restricted to modeling short range propagation 9 10 and Collins. 11,12 Thus, for the most part, the parabolic

Transcription

1 J K N L L U Z < T 5 7 D F W < G ` a T a k l b = h U! "#$%& $%# $()&* * $,+-%+.! / 0+.$%#! & Joo Tha Goh and Hnrk chd 3Dparn of can ngnrng, aach In of Tchnology, abrdg, 4aach 039 cvd 30 Janary 996; accpd for pblcaon 4 Aprl 9966 cnly, a 8pcral pr-ln 9approach wa dvlopd for acoc odlng n fld wavgd ; H. chd, W. ong, and J. T. Goh, J. Aco. oc. A. 98>, ? 995@BA. Th 9approach a hybrdaon of fn ln, bondary ngral, and wav-nbr ngraon o olv h Hlhol qaon n a rang-dpndn ocan nvronn. I provd accra, fll wo-way olon o h wav qaon, ng hr a fll, global lpl carng olon chnq, or an ffcn ngl-car, archng olon. Th papr nd ha work o d fld-lac rafcaon. AFAI d a h bac copaonal ngn for gnrang h Grn fncon n an ocan wh arbrary fld-lac rafcaon. Th hybrd odl hrfor drcly provd h pcral dcopoon of h wav fld n r of odal coponn, had wav a wll a c nrfac wav. nv nrcal prn hav hown ha h hod gv accra rl parclarly n h forward car drcon. For coparon, w hav d a fn ln parabolc qaon odl, a bondary-ln cod, a wll a h rcnly dvlopd vral orc algorh. 996 Acocal ocy of Arca. A nbr 4.30.Bp, Dr, Gv H BI INTDUTIN In rcn yar, hr ha bn a hf fro dp ocan 9acoc o h loral or hallow war acoc and wh, h rcognon ha rang dpndnc and h lacy of h abd play an poran rol n h ovrall propagaon, parclarly n h low frqncy rg. Th hallow war Lnvronn an rly coplcad wavgd bondd abov by a rogh a rfac and blow by an nhoogno, l-layrd lac abd. Frhr, h 9acoc propr of h war coln ar dpndn on prar, prr and alny, gvng r o a gnfcan paal and poral varaon. Th lac abd addd anohr dgr of coplcaon and only rcnly ha odlr hav bn abl o accon for ffc o o dgr. Th caon and propagaon bhavor of c nrfac od, nhoogno wav, and boh hadwav and lply rflcd wav nrfrnc ar all poran phnona and h nrgy carrd by c wav no nglgbl copard o h war-born fld. For rang-ndpndn oacoc propagaon odlng, AFAI n wdprad for provdng ac rfrnc olon. nc AFAI bad on ngral ranfor of h wav qaon, no drcly applcabl V o rang-dpndn probl. L and Fln drvd an 9adabac ranforaon of h wav-nbr ngral for wakly rang-dpndn probl. Howvr, h hod work wll only for ca whr X h wav fld largly donad by dcr od. 3 I non o h lac ca alo non-rval. Thr hav bn ap o h or gnral dcr approach ch a h fn dffrnc hod [ FD\ > ] 4, h fn ln hod ^ F_. 5 Howvr, nc h hod rly on paal and poral dcraon whch ar all copard o h wavlngh, hy ar norally rrcd o odlng hor rang propagaon and carng. Th parabolc qaon c df 9algorh oday who dob h o poplar approach o odlng rangdpndn ocan wavgd. Howvr, n ryng o nd h hory o lac da, wo an probl ar. Fr, h fld dcrbd by a vcor g dplacn rahr han a calar. cond, wo dffrn wav pd n a old and n a hrogno da or a bondar, w hav conno convron fro on wav yp o anohr. Frhror, lac boo ppor a wd pcr of propagaon angl. Thrfor, vn hogh vral odl hav bn propod for wav propagaon n lac Z 6 da, only a fw of h odl wr plnd. Noabl plnaon ncld ho of Won and Brook 9 0 and olln., Th, for h o par, h parabolc hor for lac wav hav no bn adqaly d nrcally, parclarly n wo-way forlaon. In 9addon o bng ld o wak rang dpndnc, a aor drawback of h a wll a h dcr hod h fac ha h olon ar no a aly nrprd phycally. Th, h odal rcr of h fld can only b drnd hrogh po-procng. 3 In h papr, w nd h work n f. 4 o arbrary fld-lac rafcaon. Th propod odlng approach can provd fll wo-way global olon b crrnly plnd ng h or ffcn ngl-car forlaon. W prn boh forward and backcar nrcal olon o a r of bnchark probl. I. TH TAL U-LNT AAH A. rafd pr-ln In h 8pcral pr-ln 9approach, h nvronn J fr dvdd no a r of rang-ndpndn 8cor or n 409 J. Aco. oc. A. 00 (3), pbr /96/00(3)/409//$ Acocal ocy of Arca 409

2 J 7 K o I > ½ < K L K L a ü W 6 š K Z L Ô Ì H FIG.. pr-ln dcraon of rang-dpndn ocan wavgd. 8 pr-ln>, parad by vrcal bondar or o c>,a llrad n Fg.. Whn ach cor, h ocan nvronn horonally rafd and w wll allow for arbrary fld-lac rafcaon. In drvng h pcral-ln qaon w wll a h acoc fld o b plan. Th, n ordr o accon for cylndrcal pradng n ayrc cnaro, h pradng facor appld plcly o h rlng fld. Th valdy of h approach dcrbd n h appnd of f. 4. Th qaon of oon for a hoogno oropc Llac old n plan ran ar gvn by 5,6 prq wvy{ ~} / ƒ ˆŠ {Œ Ž V f V >, whr 9 and 9ar h La ` conan, h dny, ( š >, œ J w) h dplacn vcor, and f h body forc pr n a of aral. W wr h dplacn vcor n r of wo calar ponal fld, ž >,.., Ÿ 0, > >,,0 whr h fr and cond r rprn h dlaaonal roaonal par of h olon rpcvly. Th ponal ar olon of h calar wav qaon V o c V >, p o V» V >, ¾ 3 c whr o c 9 p and o c 9 ar h copronal and har wav pd rpcvly, and ar gvn by Á/ÂÄà ŠV± ³ o V c p >, c ÌÎÍ œ Ang an p( ) haronc dpndnc Ï p- Kprd hncforhð >, h ponal hn alo afy h followng Hlhol qaon Ç/ÈÉËÊ ÑrÒ Ó Ô hõö ØÚÙ Ô àâá ã 0, ÛÜ ÝÞß 0, / c ä~åâæ ã / c dno è, ¼ é *ï ó(ô ø whr h o o p h wav nbr for h copronal and har wav, rpcvly. Th fld n ach cor now prd a a prpoon of h fld prodcd n h rafd ln n h 9abnc of h vrcal bondar, *, > h fld arng fro h lf bondary ç >, and h fld arng fro h rgh bondary, ê >, ëíìíî ê >, ëðòñ ê >, ëíõí ê >, ëíù >, ú 4û whr akn o dno conrbon fro h ponal n h ca of an lac rafcaon, alo ncld conrbon fro ý. Th wav fld ar drnd ng an ndrc bond- 9ary ngral hod, 4 bad on Grn hor for h -nfn vral ln oband by lnang h ohr vrcal bondary and lng h ln conn o nfny, ÿ r G r;r r;n š H r;r ;n r! d Hr œ ( r) and ( r;n )( œ ) ar h * r coponn of h dplacn and racon on h bondary of h -nfn ln +-,. G ( r, > r. œ ) and H ( r, > r/ ;n0 œ ) ar h h coponn of h dplacn and racon of h Grn fncon a h pon r on h rfac +3 wh ogong noral > Ì d o a n forc appld n h h drcon a a pon n4, r5. W#". $ %& 5 No hr, ha h pr-ln alway hav fn dph. Th, n h prnc of a lowr half-pac, h lowr bondary of h pr-ln chon dp nogh no h half-pac o nr ha h fld af h radaon condon along h horonal bondary, n whch ca h 9aocad rfac ngral conrbon vanh. To olv h ngral qaon n q. 7 %8 5 >, w nrodc boh yrc and anyrc panl orc a h bondary. A dplacn forlaon n cobnaon wh h Galrkn approach hn d o rdc h ngral qaon no a y of lnar qaon, h nknown of whch ar h orc rngh for h panl orc. nc h nknown orc rngh ar drnd, h wavfld n ach cor can b drnd ffcnly ng fa fld progra 9 FF chnq. A n h drc global ar DG hod, w pr h fld n ach layr a a ; < = prpoon of h fld prodcd by h panl orc whn h layr n h abnc of bondar, rfrrd o a h drc panl conrbon ˆ, > and an nknown fld whch ncary o afy h bondary condon a h layr nrfac )> AB 6 Th lar fld afy h hoogno qaon of oon and rfrrd o a h hoogno olon. In ohr word, hy ar h orc-fr wav ha b 9addd o h drc panl conrbon o afy h bond- condon. Th hoogno fld govrnd by q. 9ary D wh h body forc r f qal o ro. B. Fld panon Th bondary condon o b afd bwn h pr-ln, oghr wh q. %F 5 >, now provd an ngral qaon for h fld G on h vrcal bondar of h pr-ln, h nrcal olon of whch rqr o knd of dcraon. For fld pr-ln, h 40 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 40

3 y J K D 9 a f š É U ã o r F œ L Ý U d Å Ç Þ J æ Á = š N W A ò ò ò b bondary condon ar h conny of prr and parcl dplacn,.. a h vrcal bondary parang pr-ln KJ, š)l ê >, ën b p ê >, ën š- ê >, ënt XW bu p ê >, ënv >, Z[ 7 for lac pr-ln, h bondary condon ar h conny of r and dplacn,.., š-\ ê >, ën] š-k _^ w ê >, ën` ê >, ënl X _ bdc ê >, ë w ê >, ën bhg >, ë bdp ê >, ënq. v w 8 bh >, ën ê ê A prcrp d o dnfy h pr-ln hr and n h followng. nc h fld whn ach layr n h rafcaon a ooh fncon of dph, w hr choo a Galrkn bondary ln 9approach. 7 In h Galrkn approach, h conny of h fld acro h vrcal bond- prd n h 9ar wak for {} 0 ~ ê >, ë X ê >, ënƒn ê >, ën W ˆ d Š Œ 0, 9 larly for h r. Th dplacn and r 9ar now prd a panon n r of a of ba fncon. By choong an orhogonal of panon fnc- q. Ž 9 rqr h panon coffcn n h wo on nghborng cor o b dncal. Hr w choo an orhonoral of Lgndr polynoal, norald whn Lach layr š- ê >, ë _ w ê >, ën bd ê >, ën ã Ÿ U - ê W 3 ê ž / bh ê >, ënœ ãž - T 3± ê³ ë» >½¼, 0¾ + / 3 ê whr J h ordr of panon, J h Lgndr fncon, and ž h hckn of layr Á. Th argn o h Lgndr polynoal h norald, local dph coordna ëâ ëäã Å / ã. Ç È / ( ê œ ) hrogh +Ê ( ê ) Th panon coffcn UÉ œ ar fncon of UË (0) and WÌ (0), h nknown panl orc rngh of h yrc and anyrc orc, rpcvly. In ffc, hy ar lk h Grn fncon for h panl orc. A olnd arlr, w dcopo h Grn fncon no wo coponn on corrpondng o h drc fld d o h panl orc n h layr, h ohr corrpondng o h rflcon fro h layr nrfac. W can hn wr h panon coffcn UÍ ( ê ) a UÎ 3Ï êðnñ Û Ò -Ó ê³ônõ Ū Ö 3 êø >, Ù Ú whr Û Û ( ê œ ) and Ū Ü ( ê ) œ dno h drc and hoogno conrbon, rpcvly. Th ohr coffcn can 9alo b wrn n a lar annr. Th panon coffcn ar d n h Galrkn ch whn olvng h bondary ngral qaon. W fr drv vrcal wavnbr rprnaon and bqnly h ar ranford no horonal wav-nbr rprnaon abl for h DG hod. Th lar for ar alo fl for ffcn copaon of h rlan fld.. Drc panl orc conrbon W ar by dfnng h ponal for h panl orc. Th copronal and har dplacn ponal whch afy h Hlhol qaon and h radaon condon can b wrn n r of a vrcal wav-nbr pcral rprnaon a ß àâá ûüý ê >, ënãnä ê >, åâæ ënnÿ Açâèêéë ì bí î ïñð òó {}ô õ Wø / d >, ù B b ò { õ / W d >, whr >, Ì Ì Ô h 9 and Ì"!$# Ì% & )( 9 ar h vrcal horonal wav nbr, rpcvly. Th qan A*,+ œ ( ) and B- /. œ ( ) ar h orc pcr o b drnd fro bondary condon a h vrcal nrfac.. Th yrc probl Th yrc probl characrd by wo cond- a h dconny ( ê0 on 0), b43 0, ë65 ã879<; / ë?> L š 0, ë6f6g<hi ê)k T VU 0, 3ú w ê)b 0, 4D UX Z 0 ohrw. [ V\ ] ë^ >, 0_ ë/` bon of q. 3f no q. g 5h yld a rlaonhp bwn U (0) and kl A /n œ ( ) and bqnly q. o 5p bco wyz{ šrq 0, ë66 vw } ~ A ƒ ˆ {Š õ /ˆ Œ Ž d, whr f ë6 6 š 8 T V U Z œ 0 ž ëÿ, 6 ). W dfn h Forr ranfor par a ±³ g ˆ d, 7 ±»¼6½ g râ ¾ Á f ë6ã ˆ Ä d, and akng h forward ranfor wh rpc o ë of q. 6È, w oban 4 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 4 8a 5c

4 ý ½ Ü ± œ ç { A _ d ê J ž 5 K L % U FIG. 3.. B onfgraon for h ngl layr bnchark. ~ A U 0 õ!. 0 " bon of h abov no q. # 4$ yld ( ) * +,.- õ0/ B& 3 4 U õ =<> ÊZË Î ÏÑÐÒÓ Ì AÍ ˆÔ ÕÖ Ø / Ù Ú VÛ akng of h rlaon, 8 Õé à 0 Vê ë ëì ˆ í æ ïî d 8ð ˆñò Õó / UÝ ZÞ 0 A. ³ a T FIG.. odfd NDA 3A ca. confgraon, b rcvr a 50,» c¼ rcvr a 0. old AFAI, dahd pcral pr-ln. ß Õá à Vâ ã ëä ˆ å æ d. 0 8è ô õ V V 8øZù, ú 9û whr Vü h phrcal Bl fncon, w oban for h yrc par of h orc pcr A,ÿ ( ),. Th anyrc probl ý wo Th anyrc probl larly characrd by condon a h dconny, bb 0, ë ãd / FHG I XW w 0, ë0[z]\^ ë` acb f g=h kl 0, ên w ët 0, Wn ko p 0 kq r ë, 0 ëv ohrw. gw, V y3 3 Afr o algbra w oban h anyrc par of h orc pcr a ~ y õ0ƒ A } 3 k Wˆ k8 Š 0 õ k g= Ž kœ, 4 B 0šœ k WŸ k 0 õ k k g r rprnaon for h drc panl fld Th nknown panl orc rngh U k (0) and W k ( 0) ar now drnd hrogh achng h rlvan bondary condon a h vrcal c. W do h by h Galrkn bondary ln approach whch ply rqr ha h panon coffcn n h wo nghborng cor b dncal. Th panon coffcn can b raghforwardly racd fro h ponal rprnaon abov by ng h orhogonaly rlaon for h Lgndr polyno- 4 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 4

5 ¾ { Á 5 Ð œ Ú Þ ß â â ò ô TABL I. arar for h r of wo-cor canoncal probl. Wav pd ar gvn n /, dn n g/c 3, and anaon n db/. Bnchark arar B B B3 B ! Lf c" p cor c# $ " p % # & gh c" p cor c# ( " p # ) al. In pracc, w nd o rnca h nfn r panon a a ffcnly hgh ordr. nc h Lgndr polynoal rprn h vrcal varaon n h fld, on can oban h rncaon l fro an a of h nbr of noral od. L condr h coffcn Û Â Ã k( ê ), L Ä Å ê, ëçè yêé gëíì ÃÏÎ k Û Ñ ÃÏÒ êôó k Ã[Õ k Ö ë. Ø 6Ù FIG. 5. olon o h ngl layr bnchark. a ca B3, 3 b B, dahd VIA, dod pcral pr-ln..04 B4, old W fr lply boh d of q. Û 6Ü by ký ( ë) œ. Ingrang ovr h layr hckn and ng h orhogonaly propr of Lgndr polynoal, Õáà à 0 kã ä ëå ÃÏæ ç ëè éëê d â k gì, 0, íïó íïð ñ k, ñ k, y!õ 7 oghr wh h dny, Õá à 0 kø ù ëú ûü ý æ éÿ d g û Õ / õ k k k g, 8 * 4. olon o h ngl layr bnchark. + a, ca B, - b FIG. old B, dahd VIA, dod pcral pr-ln..0/ ca B, and h pron for h orc pcr drvd prvoly, w oban 43 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 43

6 y ˆ Œ r Å ƒ ý ß à ó A ü é é 8 é Û6 k7 ê98; k=< Ã?> k õ ka ÃB k û U bwv c T à k DF éhg d UX à ònz;[ k 0 IKJ \ L õw]_^ ` õwdf g Wh ûn b; Wk Ãl ònn k 0 ko g p_q g ûn b ûna bb y, v 9 whr k { ( í} )~ k g /( ) and k {; k ( ). panon coffcn for h ohr fld parar can b oband n a lar fahon. Whn valang h ngral, yw can plo hr yr n ( í, ñ ƒ k) o rdc h aon of copaon. In addon, for fš ò 0, o of h ngral can b valad n clod for. In c. II, w dc o of h nrcal nvolvd n valang h ngral. 4. Horonal wav-nbr rprnaon for h drc panl fld Th dplacn ponal n h vrcal wav-nbr doan ar ŽF, ; ; š œ ž WŸN ; y õ; W ywhr FIG. 6.. chac of h lac cylndrcal aon. 9, ûn b ûn æ Õ± 霳 / d, ;;» ¼F½ ¾ y õw ÁžÂWÃÄ Ç È=É ÊWËÌ ûní b;î ûnï ÐÒÑ æó ÕÔ Õ éhö / d, yw 30 FIG. 7. lac cylndrcal aon.. cvr a 80. v aw Forward card blk r, y% b forward card har r. old VIA; dahd pcral pr-ln. žøúùû;üþý kß ë ìwínî?ïñð kò Uá kâ 0 Wô k_õ 0 ònã õ kä å kæ g ç_è y, òn kø å kù gúû. ê 3 Th DG hod for h l-layrd cor rqr h ngral rprnaon for h fr-pac Grn fncon o b prd n r of h horonal wav nbr, ü. Ung conor ngraon a dvd by Hlan, 9 h vr- wav-nbr ngral of q. ÿ cal 30 ar convrd no horonal wav-nbr ngral. Afr o algbra, h ponal n h ü doan bco <;= ywhr acb, >?,!#"%$ ü û,+*-./ û,7 d, & (*) 6 B ü #D FGH ü #T U#VLW ûx*z.[ ]^ d, ˆgnd f ghg 4 /, %klon p rq h, 9 3 { 44 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 44

7 Ÿ y ß ò ò y W ˆ N h l l œ \ ñ 9 V + 9 f ~ - X ˆ! T Ž ˆ Á š f f Z é Inrng h fld panon n q. ^ 0_ no h wak for of h bondary condon n q. ` acb 9 lad 9 o 9g h con- ncvy qaon bwn pr-ln and. Th nbr of qaon for ach layr dpnd on h nbr of panon r 9 d 9 k a wll a h yp of da n pr-ln and. In gnral, h conncvy qaon ar d r ˆ ˆŽ h }o~, ƒ Š Œ,, and and ar gvn by q. 3. Th ponal gvn by q. 3 ar vald only for f 0 and š, b ll allow for h applcaon of h global ar hod whn afyng h horonal nrfac bondary condon nc a h nrfac fœ 0 and f šhž of ach layr ha h fld ar bng achd. Howvr, drabl o hav a horonal wav-nbr ngral rprnaon ha vald whn h layr o ha ffcn FF chnq can b ployd for h rlng fld copaon. In f. 0 w drv a dcrd qvaln pon orc horonal wavnbr rprnaon for h ponal ha vald for ò*ÿ 0 šh. W now ply q. c 3 a h orc conrbon n h AFAI cod o drn h aocad hoogno olon, h panon of whch ar dcrbd n h n con. D. Hoogno olon. cvr a 80. a Back FIG. 8. lac cylndrcal aon. card blk r, b back card har r. old VIA; dahd pcral pr-ln. Th horonal wav-nbr ngral rprnaon for h hoogno olon can b oband by a drc n- of h qaon prnd n chd and Jnn. Th ˆon copl hoogno olon gvn by h ovr all fn layr a wll a ovr all ordr of panon. Ung h orhogonaly rlaon of Lgndr polynoal, panon coffcn can b racd a V < # í± ³ ¼<½ N ¾ r» à n k ÈoÉ whr Ù V Ú <ÛÜoÝ ŪÞ ç Âà ÄÅ Ǣ K JÇ UËÊ ò,ì ÍÎ nk 0 A,nk Ï WËÐ òñ ÒÓ nk 0,nk ÕÔ àá âã,ä T, é ê ë Wß è T ìˆí í Âå,æ K îðï ñðò ôó õ, # ø #ù ú û ü ý Ä,ÿ dag û, û, û, û, Í A,nk A,nk A,nk B,nk #" $&% B,nk T, Ò,nk () *,nk,.0/.0354,nk D,nk D,nk T, ;=< > Ǣ J6 87 dag 9 ;BD 8A, ;=FHG 9 ;KL 8J. û,ö d4, cø 33 ach cobnaon of ndc N n and ñ k rprn a ngl A- FAI rn. Hr A,nk and B,nk ar rpcvly h p/ down gong copronal and har wav n layr U d o h kh ordr yrc orc n layr N n. Th corrpondng qan fro h anyrc orc ar,nk and D[,nk. Howvr, h DG can ra lpl rgh-hand d lanoly and hnc all h apld of h p/ down gong wav can b fond wh a ngl global ar nvron. Th ak h algorh vry ffcn ] vn for probl wh a larg nbr of layr and hgh ordr of panon.. ln conncvy U on prq W o pr àwv T o pry â{ o pr} p U o pr W oƒ pr àw T o pr â ˆ oš pr,...,n, p Œ Ž,,...,. 34 Hr N h nbr of layr and h nbr of panon r d whn ach layr. By yacally achng bondary condon along h vrcal c, w oban a lnar y of qaon for 9 h 9 ž nknown panl orc rngh for pr-ln and. nc h 45 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 45

8 W ³ ¾ Â Ã Å Á ¼ ½ W Ô Ö Õ Ø Ò nknown orc rngh ar drnd, h rlng fld can hn b drnd ffcnly wh DG ng h horonal wav-nbr pcral rprnaon. Th radr rfrrd o chd al. 4 for a dcon on plnng an ffcn archng algorh fro h y of conncvy qaon. II. NUIAL ILNTATIN Th nrcal plnaon of h pcral prln hod rqr carfl ran of h nfn n- gral aocad wh h panon coffcn of h fld parar. I clar fro q. ± 9 ha a larg wav nbr, h ngrand hghly ocllaory wh an rrglar frqncy drnd by h rlav ordr of h phrcal Bl fncon. In addon, a larg wav nbr, w rqr propr cancllaon of h conrbon arng fro h yrc and anyrc orc n ordr o arrv a a fn val for h panon coffcn. Th ar frhr aggravad by h low dcay of h ngrand. To vala h ngral, w adapv ngraon ron fro h UADAK lbrary. In addon, w hav plnd a br forc hod n whch paral of h ngrand ar fr oband by ngrang n bwn h ro of h ocllang ngrand. An acclraon chnq hn d o pd p h convrgnc of h paral. Fnally, w alo ploy a odfcaon of h andard Ga hbyhv qadrar dvlopd by r-jorda al. 3 and r-jorda and an-faban. 4 Thr forlaon parclarly d for aoac qadrar. Th radr rfrrd o f. 0 for or dal abo h varo qadrar ch. III. NUIAL XAL In h followng w llra how h prn approach provd accra olon o canoncal propagaon and rvrbraon bnchark probl. W copar or olon wh rl oband fro h bondary ln cod by Grof and chd, 5 h vral orc algorh» VIA by chd 6 and h fn ln parabolc qaon odl F of olln. Unl ohrw nod, h prln olon ar oband ng only for ordr of - panon n h fld parar whn ach layr. In addon, n ach of h apl, w ak h war coln o b lol Ä wh a ond pd of 500 / and a dny of g/c 3. A. odfd NDA bnchark apl A bad on ca 3A d n h NDA arabolc qaon Workhop. 7 Th probl wa fr odfd for a a ca for lac by Won and ³ Brook 0 and w rn a lghly dffrn vron hr. Th wavgd, llrad n Fg. Ç ÉÈ a Ê, con of a war layr wh a hckn of 00, ovr a old layr wh a hckn Ä of 00, a dny of. g/c 3 Ê, a copronal pd of 590 /, and a har pd of 500 /. Th old ha a copronal anaon of 0. db/ë and a har anaon of 0.5 db/ì. A 5-H ln orc placd a a danc of 5 k fro an arfcal ranparn nrfac. Th prary hr o how wll nrgy copld hrogh a ranparn vrcal nrfac and rprn h r ca of a low-conra vrcal p. oparon bwn A- FAI and or olon for rcvr dph of 50 and 0 ar hown n Fg. Í cî b and Ï Ð c. For clary w hav hown h olon fro o 8 k and w ha h pr-ln olon agr wll wh AFAI. For rang l han 5 k, h pr-ln forlaon rdc o AFAI acly and w prfc agrn n h olon. For rang byond h arfcal nrfac, h agrn ll q good for boh rcvr, ndcang propr coplng acro h nrfac. B. ngl layr bnchark Th n bnchark con of a of wo-cor probl hown n Fg. 3. Th wavgd bondd a h op and boo by a prr rla bondary. A 5-H ln orc placd a a dph of 5 and a a danc of k fro h vrcal dconny. By bondng h wavgd by prr rla bondar, h bnchark rqr h propagaon cod o proprly conrv nrgy bfor on can arrv a h corrc anwr. In addon, by varyng h aral propr on boh d of h dconny, w can a h nvy of a parclar cod o conra n h prary drcon of propagaon. Tabl I how h for dffrn confgraon ha w hav chon. Th B cod 5 pcd o prodc good rl for h of bnchark and hrfor akn a h rfrnc olon. olon for h noral r Ñ a a rcvr dph of 35 ar hown n Fg. 4 and 5 and w gnrally hav good agrn aong h hr olon. Ó. lac cylndrcal aon apl, hown n Fg. 6, condr an lac aon n a cylndrcally yrc ocan nvronn. Th fld vron of h probl fr appard n Glbr and van. 8 A 5-H orc locad n h ddl of h wavgd. Th dph of h war coln a h orc rang 00. A 35--hgh aon ha nnr rad 5 k and or rad 0 k. Th boo a hoogno half-pac wh a copronal ond pd of 700 / Ä and a har pd of 700 /. Th dny.5 g/c 3 and h copronal and har anaon n h boo 0. and 0.5 db/ rpcvly. W olvd h probl ng only hr rang cor and gh layr down o a dph of 400. W how boh forward and backcard dlaaonal and har r a a rcvr dph of 80. oparon bwn h vral orc algorh and h pr-ln hod ar hown n Fg. 7 and 8. In forward car, w hav clln agrn bwn h wo olon. Thr alo good agrn n h backcar olon. Th ncra n backcar nrgy a h orc rang n Fg. 8Ù Ú a d o h fac ha for a pon orc n a cylndrcally yrc wavgd h backcard nrgy foc a h orc rang. Th hgh-frqncy ocllaon n n h backcar VIA olon ar d o nrcal no crpng no h rly low fld val copd. 46 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 46

9 à Â â Ê ê ï FIG. 9.. D bddd p dconny. D. bddd p dconny apl D, akn fro olln and hown n Fg. 9 nvolv wo old layr and a p dconny n layr hckn. A 5-H orc placd a a dph of 50 n h ppr layr, whch 500 hck for rang l han 7 k and 50 for rang byond 7 k. Th copronal and har pd n h ppr layr 500 and 700 /, rpcvly and h d ad o b lol. Th lowr layr a half-pac wh copronal and har pd qal o 600 nd 750 /, rpcvly. Th anaon n h lowr d 0.5 db/û for boh wav yp. Th Ä dny n h ppr and lowr d and. g/c 3, rpcvly. Th parclar probl ha a vry low conra î cvr a 300. old B; dahd FIG.. bddd lac p è. Dé. Back card noral r olon. aì cvr a 00. í ë b VIA; dod pcral pr-ln. acro h vrcal nrfac and w prn forward and backcar olon a wo rcvr dph. In h forward Ü drcon Fg. 0Ý Ê, w hav good agrn bwn h hr olon. In h backcar Þ Fg. ß Ê, hr o dagrn, parclarly nar h carng rfac. W blvd h d o naccrac aocad wh h larg dynac rang bwn h forward and backcard fld.. lac AA wdg apl, hown n Fg. ca 3 fro h arabolc qaon Workhop II. 9 Th an lac vron of h andard AA wdg bnchark probl. A 5-H pon orc placd a 00- dph. Th ocan dph d- Dã. Toal noral r. ä aå cvr FIG. 0. bddd lac p á. a 00. æ ç b cvr a 300. old B; dahd VIA; dod pcral pr-ln. FIG... nvronn for AA lac wdg. 47 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 47

10 ù ð Ã Ä û ù ü. ý a cvr FIG. 3. AA lac wdg ú. a 30, ÿ b cvr a 50 ; old F, dahd pcral pr-ln. h boo. Th bhavor o clarly n n h conor plo of Fg. 6. Th probl provd a good apl of nrgy lo d o boo roghn. IV. ùfig. 4.. F chac of h prodc roghn pach. NLUIN W hav ndd an arlr pblhd pr-ln approach for wav propagaon n l-layrd rangdpndn fld nvronn o handl arbrary fld-lac rafcaon. Th approach capabl of copng boh h forward card and h rvrbran fld, parclarly cra lnarly wh rang fro 00 a h orc rang o ro a ñóò ô r 4 k. Th ocan boo ha a copronal ond pd of 700 Ä / and a har pd of 800 /. Th dny.5 g/c 3 wh h copronal and har anaon a 0.5 db/õ. Th nvronn dcrd no 7 layr, ach of abo a wavlngh n dph, and 56 rang cor. In Fg. 3 w prn olon fro h parabolc qaon odl and h pr-ln hod. Thr good agrn for h hallow rcvr and for h rcvr n h boo, h agrn ll q good and h dffrnc ar prarly d o h parclar annr n whch h nvronn bng dcrd. F. p prodc roghn pach apl F condr a p prodc roghn pach hown n Fg. 4. A lar fld apl wa fr rad by van and Glbr. 30 Th pach nd fro 5 0 k and h dph varaon con of p whch ar 0 hgh, 00 long and 00 apar. Hnc hr ar 5 p pr klor. Th orc, a a frqncy of 50 H, a a dph of 8, and Fg. 5 how h blk r for rcvr dph of 50 and 50. Th pcral ln olon copd ng gh ordr of panon. Th ranon lo copard o h rfrnc olon for a fla-boo wavgd ø AFAI. For h boo rcvr, w an ncra n h fld blow h roghn pach and h a rl of nrgy bng dpd fro h war coln no û FIG. 5. p prodc roghn pach. F. Forward dlaaon a cvr a 50. b cvr a 50. old AFAI; dahd pcral pr-ln. 48 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 48

11 Â FIG. 6. p prodc roghn pach. F. Forward dlaaon a roghn pach. AFAI olon who h pach, b pcral pr-ln olon wh h abl for rang rvrbraon fro larg cal ocanc far and canoncal bnchark probl. Th nrcal ffcncy of h approach oband by ng AFAI o cop all nflnc fncon for ach pr-ln wh on global ar nvron. Wav-nbr ngraon alo d for valang h fld whn ach pr-ln onc h bondary panl orc rngh ar fond. vn hogh h non raghforward concpally, nrcal plnaon nonrval. In parclar, h panon coffcn ar rprnd by nfn ngral ha ar lowly convrgn a wll a hbng rapd rrglar ocllaon a nfny. Howvr, nv nrcal copaon hav donrad h prforanc of h prn plnaon, parclarly n h forward car drcon. AKNWLDGNT Th ahor ar grafl o Dr. r Grof of h ALANT Undra arch nr for fl dcon and for provdng h la vron of h bondary ln cod. W wold alo lk o acknowldg hlpfl dcon wh Dr. k olln of Naval arch Laboraory. Th work wa parally ppord by h ffc of Naval arch, n par by h Hgh Lad Dynac and h can Acoc progra. H. chd, AFAI o-acoc Fa Fld Algorh for ang Indpndn nvronn. Ur Gd. 3 ALANT AW arch nr, La pa, Ialy, 987. I. T. L and L. B. Fln, Adabac ranfor for pcral analy and 49 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 49

12 % ; 8 6? B, J F 8 b, B N [ 8 B ynh of wakly rang-dpndn hallow ocan Grn fncon, J. Aco. oc. A. 8, J. T. Goh and H. chd, Valdy of pcral hor for wakly rangdpndn ocan nvronn Nrcal rl, J. Aco. oc. A. 95, A. phn, A rvw of fn-dffrnc hod for o-acoc probl a h a floor, v. Gophy. 6, J.. rphy and. A. hn-bng, A o-acoc fn ln odl for ndrwar acoc propagaon, n har Wav n arn dn, dd by J.. Hov,. D. chardon, and. D. oll " Klwr, Dordrch, 99!. 6 J. A. Hdon, A parabolc approaon for lac wav, Wav o-, 07 4 # 980$. êon 7 J.. oron, B. DFaco, and. J. Krgr, arabolc approaon êo h -ndpndn lac wav qaon, J. ah. hy. 3, 577 ( 586 & Wal and J. J. coy, A coparon of parabolc wav hor + for lnarly lac old, Wav oon 5, 99 3 ) 983*. 9.. Grn, A hgh-angl on-way wav qaon for c wav propagaon along rogh and lopng nrfac, J. Aco. oc. A. 77, B. T.. Won and G. H. Brook, n-way wav qaon for oacoc propagaon n lac wavgd, J. Aco. oc. A. 87, / D. olln, A wo-way parabolc qaon hod for lac da, J. Aco. oc. A. 93, D. olln, An nrgy conrvng parabolc qaon for lac da, J. Aco. oc. A. 94, ù 3 F. B. Jnn and H. chd, pcral dcopoon of fld n a wdg-hapd ocan, n rogr n Undrwar Acoc, dd by H.. rklngr 7 ln, Nw ork, H. chd, W. ong, and J. T. Goh, pcral pr-ln approach o rang-dpndn ocan acoc odlng, J. Aco. oc. A. 98, < 995=. 5 K. F. Graff, Wav oon n lac old > Dovr, Nw ork, 99@. 6 A. D. Achnbach, Wav ropagaon n lac old A Norh-Holland, Arda, 975. ù 7 F. B. Jnn, W. A. Kpran,. B. orr, and H. chd, opaonal can Acoc D Arcan In of hyc, Woodbry, N, I.. Gradhyn and I.. yhk, Tabl of Ingral, r and rodc Acadc, Nw ork, 980G. 9. A. Hlan, adaon fro a cylndrcal orc of fn lngh, J Gophyc 8, H 953I. 0 J. T. Goh, pcral pr-ln approach for wav propagaon n rang dpndn lac d, h.d. h, aach In of Tchnology, Jn 996. H. chd and F. B. Jnn, A fll wav olon for propagaon n llayrd vcolac da wh applcaon o Gaan ba rflcon a fld old nrfac, J. Aco. oc. A. 77, K 985L.. n al., UADAK A bron ackag for Aoac Ingraon prngr-vrlag, Brln, J.. r-jorda,. an-faban, and F. ocardo, A pl, rlabl and ffcn ch for aoac nrcal ngraon, op. hy. J on. 70, J.. r-jorda and. an-faban, A pl, ffcn and or rlabl ch for aoac nrcal ngraon, op. hy. on. 77, Grof and H. chd, A bondary ln approach o oacoc fac rvrbraon, J. Aco. oc. A. 89, T 99U. J 6 H. chd, archng wav-nbr-ngraon approach o rangdpndn, wo-way oacoc propagaon odlng, J. Aco. oc. A. 97, 336 V XWX J A 995Z. 7 J. A. Dav, D. Wh, and.. avanagh, NDA arabolc qaon J Workhop \ NDA Tch No 43, 98]. 8 K.. Glbr and. B. van, A Grn fncon hod for on-way wav propagaon n a rang dpndn ocan nvronn, n can o-acoc, dd by T. Akal and J.. Brkon ^ ln, Nw J ork, 986_. 9. A. hn-bng, D. B. Kng, J. A. Dav, and. B. van, Workhop II rocdng of h cond arabolc qaon Workhop ` Naval arch Laboraory, Wahngon, D, 993a. 30. B. van and K.. Glbr, Th prodc non of pw copld od, J. Aco. oc. A. 77, c 985d. 40 J. Aco. oc. A., Vol. 00, No. 3, pbr 996 J. T. Goh and H. chd opld wav-nbr ngraon 40