Analysis of ray stability and caustic formation in a layered moving fluid medium

Transcription

1 Analyss of ray stablty and aust formaton n a layered movng flud medum Davd R. Bergman * Morrstown NJ Abstrat Caust formaton ours wthn a ray skeleton as optal or aoust felds propagate n a medum wth varable refratve propertes and are unphysal, ther presene beng an artfat of the ray approxmaton of the feld, and methods of orretng the feld near a aust are well known. Dfferental geometry provdes a novel approah to alulatng aoust ntensty, assessng ray stablty and loatng austs n aoust ray traes when the propertes of medum are ompletely arbtrary by dentfyng ponts on the aust wth onjugate ponts along varous rays. The method of geodes devaton s appled to the problem of determnng ray stablty and loatng austs n -dmensonal aoust ray traes n a layered movng medum. Spefally, a general treatment of aust formaton n sound duts and n peewse ontnuous meda s presented and appled to varous dealed and realst senaros. PACS: 43..+g, 4.5.Dp, m, h Wrtten n 4, Work performed under a summer faulty fellowshp program at the Naval Researh aboratory, Washngton, D.C * E-mal address: davdrbergman@ess-ll.om

2 I INTRODUCTION The applaton of dfferental geometry to the problem of aoust ray theory offers a unque way to trae austs whh has dstnt advantages over tradtonal methods []. The equatons for the aoust rays n a layered movng flud medum an be solved n terms of depth ntegrals, the fnal result gvng horontal range and travel tme as a funton of depth and the ntal ondtons at the soure. Causts are loated by expltly varyng the range wth respet to an ntal ray parameter, usually the launh angle, and searhng for rtal ponts of ths varaton []. In paraxal proedures ths devaton s determned by a seond order lnear equaton for the Jaob feld along the ray [], [3]. The Jaob feld of the ray system s dental to the geodes devaton vetor assoated wth the dfferental geometr struture whh arses from applaton of the method of haratersts to the equatons of hydrodynams [4]. Ths relaton onnets the onjugate pont theorems of dfferental geometry [5] to the phenomena of aust formaton. Caust formaton s a dvergent artfat of ray theory ndatng a breakdown of the exstene of a unque soluton of the ray equaton between two ponts leadng to an nfnte value for the feld ampltude n the lmt of geometr opts or aousts. Whle t s feasble to desrbe the aoust feld solely n terms of normal modes or n some ases exat solutons to the feld equatons, thus bypassng the ourrene of austs, rays have many dstnt advantages over felds and modes n ts relatve smplty, oneptual tangblty and omputatonal nformaton. Furthermore, modern treatments of austs provde aurate orreton terms to the feld near a aust [6], [7], hene ther prese loaton n spae and tme s needed.

3 By analyng the Jaob equaton assoated wth the rays one gets mmedate nformaton about the fousng propertes of a gven medum nludng ray stablty.e. whether ray onverge or dverge from ther neghborng rays wth smlar ntal ondtons and aust loaton. Caust formaton an our as a result of: a the smooth behavor of the loal sound-sped profle SSP or wnd profle, b refleton from a boundary, ntal ondtons or, d the presene of jump dsontnutes n the dervatves of the SSP and wnd profle. Although the later ase d may be ruled out on physal grounds ts effet s of nterest sne peewse ontnuous envronmental parameters are sometmes used to model underwater and atmospher aoustal systems. In suh ases dsontnutes n the sound speed and veloty gradents leads to the presene of a Dra delta funton n the Jaob equaton leadng to speal boundary ondtons for mathng the Jaob felds of dfferent segments at the boundary between two regons. In ths artle a omplete treatment of ray stablty and the formaton of austs n the aoust feld propagatng n a dmensonal layered movng flud bakground s presented. II THEORY IIa RAY THEORY The ray equatons n a movng layered flud medum are well known, havng been presented n ts general form, derved from the theory of haratersts [3], [8], [9]. The Cartesan oordnates and the travel tme are expressed here n the form of a spae tme trajetory and parametered by an arbtrary parameter,. Consder a medum wth Strtly speakng ths ray parameter s an affne parameter. Ths desgnaton s requred to ensure that the rays are n fat geodess. 3

4 sound speed and one dmensonal flud veloty of the form w x w, where s depth or heght. Rays that are ntally fred n the x plane do not turn out of ther ntal osulatng plane,.e. are torson free, and onsttute an effetve two dmensonal system. The range, depth and travel tme are gven by a frst order system, x w w, w, t w, 3 n whh dot denotes dfferentaton wth respet to, os os s the ray parameter and the ntegraton onstant s hosen suh that dt / d. Presented n ths form the rays or bharahtersts are null geodess of a pseudo- Remannan manfold 3. Ths onneton has been ponted out ndependently by Whte [9] and Unruh [] and served as a prmary nspraton for the work n ths artle. IIb JACOBI EQUATION, GEODESIC DEVIATION The spreadng or stablty equaton for the system of Eqs. 3 s θ s the ntal angle between the wavefront normal and the x axs. 3 Ths s smlar to the stuaton n general relatvty where these speal urves desrbe the trajetory of photons n a urved spae tme. 4

5 d Y d KY 5 n whh, K 3. 6 The quantty K measures the stablty of the ray system along a gven ray, labeled by. The ndvdual terms n Eq. 6 dretly affet the onvergene of neghborng rays n a predtable way. Equaton 5 has a general soluton, Y k 3 d k, 8 wth ntegraton onstants k and k set to math the ntal ondtons. To model sound from a pont soure the approprate ntal ondtons are Y and Y, where and δθ s an arbtrary ntal devaton n ray launh angle. Geometrally the devaton, Y, s tangent to the wavefront and gves a loal measure of the deformaton of the wavefront. Equaton 5 s the Jaob equaton of the ray system defned by Eqs - 3. From the nterpretaton of rays as null geodess t follows that K s a dret measure of the urvature of the manfold defned by the method of haratersts and Eq 5 s the equaton of geodes devaton, [5], []. For two dmensonal aousts problems Y an be used to alulate the ntensty of the aoust feld from the onservaton law 5

6 I Y nˆ tˆ onstant, n whh nˆ tˆ s the projeton of the ray tangent, tˆ, onto the wavefront normal, nˆ [], [3]. II CAUSTICS AND CONJUGATE POINTS A aust s the lous of ponts determned by solutons of the equaton Y. Solvng ths equaton gves,, the value of the ray parameter affne parameter at the aust loaton n terms of the ray s ntal ondtons. Insertng ths value nto the ray oordnates then gves the aust as a urve n spae, x,,,, parametered by the ntal ondtons of the ray. Based on the omments of the preedng seton, at these ponts the ntensty beomes nfnte. Ths dvergent artfat sgnals a breakdown n the valdty of ray theory and s orreted for by feld expansons near the aust. From the pont of vew of dfferental geometry the vanshng of an otherwse nontrval devaton vetor, Y, at two ponts along a geodes ndates a breakdown n the unqueness of solutons to the geodes equaton 4. Whle the general soluton gven n Eq 8 along wth the approprate ntal ondtons an be used to generate Yλ and ts behavor studed dretly, muh useful nformaton an be obtaned by studyng the Eq 5. There are three lmtng ases for whh a soluton to Eq 5 an be found n terms of ordnary funtons along wth a smple geometr nterpretaton. These ases, labeled I, II and III are desrbed by K = onstant >, K = and K onstant < respetvely. Defnng K, the soluton to Eq 5 n eah ase s 4 For a more omplete presentaton of the onjugate pont theorems, geodes devaton and Jaob theory the reader s referred to Referene [5]. 6

7 Asn Y A B, 9 Asnh n whh A, B and φ are ntegraton onstants. An example of a manfold orrespondng to eah ase s: sphere K = onstant >, plane K = and pseudo-sphere K onstant <. In ase I the devaton vetor has perod eros sgnalng the onset of onjugate ponts on an atual sphere or globe ths would orrespond to the North and South poles whh are passed perodally as one travels along any longtude. A hgher value of K produes a hgher frequeny of onjugate pont formaton. onseutve onjugate ponts, measured n unts of λ, s gven by The perod between / K. In the other two ases onjugate ponts wll not form. When the urvature depends on poston we an say the followng: f K onjugate ponts wll never form along the ray, f K onjugate ponts wll form along the ray as long as t obeys the ondtons of ompleteness 5. Inompleteness an our when an absorbng boundary s present n the envronment, n whh ase the ray may smply termnate before the aust gets a hane to form, or beause the soluton to Eq -3 s not well defned for all λ. A smple example of the latter ase ours for the SSP. The exat soluton for the rays s well known n ths ase [4] and an be used to verfy that the fnal value of s fnte for any ray launhed at an angle / / from a soure plaed at a fnte value of and landng at f. Of partular mportane are ases when, due to the onavty of 5 A geodes s omplete f ts ponts exst for all,. 7

8 the envronmental parameters, rays beome trapped between vertal turnng ponts. These trapped rays wll, deally, propagate forever n the horontal dreton as they osllate n the vertal dreton. If K > everywhere along the ray the onjugate pont theorem states that the perod n λ between onseutve onjugate ponts obeys the relaton K K, where Kmn K KMax. It s presely ths senaro that / / Max mn we onsder n the next seton. III APPICATION TO SMOOTH and w SOUND DUCTS The formalsm of seton II s appled to an envronment wth a smoothly varyng 6 SSP and horontal urrent or wnd, eah a funton of depth,. To better understand the effets of the envronment on the aoust feld onsder Eq. 5 n detal. Eah term n Eq. 6 has a dstnt effet on the feld desrbed as follows 7. The last term,, governs the fousng propertes of the medum aused by an nhomogeneous sound speed. A ray propagatng n a regon wth wll eventually enounter onjugate ponts, whle rays propagatng n regons where wll dverge from one another. The seond term, proportonal to w, s always negatve ausng ray dvergene. The frst term, w, wll ause fousng of aoust rays when and w are the same sgn. When both and w depend on depth the term wt n K ouples the sound-speed gradent to the flud-veloty gradent. Consder a smple stuaton n whh a wavegude s reated by a sound-speed profle wth everywhere and above below the wavegude axs. Furthermore let the 6 Here we mean, w, and all dervatves are ontnuous. 7 Ths desrpton appears n Ref [] and s added here for ompleteness. In the followng desrpton an overall fator of s gnored. 8

9 flud veloty satsfy w and w everywhere n the wavegude. For aoust rays wth, there s a separaton of neghborng rays above the wavegude axs and an enhaned onvergene of rays below the wavegude axs. When the bakground flud moton s weak and slowly varyng, the leadng-order terms n Eq. 6 are w envronmental parameters., ndatng that the domnant effets are due to the onavty of the The sound-speed and wnd gradents affet the bendng and twstng of the aoust rays. In general rays may be unbound or bound dependng on how the envronmental parameters vary wth poston. A ommon example of ths s gven by the Munk profle whh reates a natural sound dut or hannel [5]. From the omments of the prevous paragraph t s lear that duted regons may be reated n the atmosphere or oean by the onavty of ether the wnd or sound-speed. Consder an envronment wth smooth and w suh that at some depth, : w, and w. These ondtons defne a sound dut wave gude at, near whh / and w w w /. Furthermore, sne w, a pont soure plaed on the sound dut axs, =, wll launh two speal rays desrbed by ntal ondtons p = ±, whh travel along the sound hannel axs n the ± x dretons. These two rays are desrbed exatly by: t, = and x x w t. The exat soluton to Eqs - 3 for quadrat and w and arbtrary ntal ondtons nvolve nomplete ellpt ntegrals. An exat soluton to equaton 5 may be developed along the sound hannel axs thus allowng a paraxal desrpton of the near axs rays n terms of ordnary funtons, Eq. 9. Evaluatng the setonal urvature, K, along the rays on the sound hannel axs gves the followng onstant 9

10 K w, for rght/left axs rays. Fgure llustrates sample ray fans about the sound hannel axs - 36 o to +36 o, n 6 o nrements for ~ and w ~ b. Column A/B llustrates rays fred to the left/rght aganst/wth the wnd. The rows a to e orrespond to nreasng values of the parameter b =,.,.4,.8 and respetvely. In all 5 ases the flow s subson. The rghtward ray fan has enhaned onvergene due to the presene of the wnd whle the leftward ray fan shows the opposte effet. As the onavty of w s nreased the ray fans suffer more severe aberraton untl at b = the austs n the left ray fan are ompletely destroyed and rays near the axs begn to dverge lnearly. At b = the onvergene n the left ray fan s ompletely destroyed. A detaled applaton of equaton allows one to predt exatly where the tps of the aust urves on the dut axs our when, / / x n b. Fgure ompares the onvergene and dvergene ones for b = and b =, for other values of b the one boundares depend senstvely on the ndvdual ray oordnates. A strkng feature of ths behavor s that t s ompletely determned by the onavty of the wnd profle and not at all by the loal Mah number n fat for the above example the soure s plaed n a statonary regon, w = m/s, and the onavty n w ompletely destroys the aust struture for p = - and b =. Hene, n general even very weak but rapdly hangng wnds an wreak havo on the stablty of rays n a sound dut. To onnet these analyses to a more realst example onsder a senaro n the oean n whh a sound dut s desrbed by the Munk profle along wth an osllatng urrent: D a e D and w we sn d, n whh

11 D d / d, and / tan, where λ s a haraterst wavelength of the urrent varatons not the affne parameter 8. The urrent, w, s fxed so the ondtons desrbed n the prevous example hold. From these profles 4d and w w /. The SSP s held fxed wth d = 5m, = 5m/s and ε =. an exaggerated Munk profle. From Eq. the free parameters of the wnd profle may be fxed n suh a way that K = n one dreton of the wavegude. Ths has been done here by hoosng α =.m - and alulatng w for some hoe of λ. Fgure 3 shows wnd profles for b λ = d w = 6.63m/s, λ = d w = 4.6m/s and d λ = d/ w =.69m/s along wth the SSP a. Fgure 4 shows a sample ray trae of the near axs rays for the hoe n Fg. 3b and pont soure plaed at = 5m. Ray fans for a pont soure on the wavegude axs launhed nto and aganst the urrent show mmense dsparty. Clearly these urrents are neglgble ompared to the loal sound speed of 5m/s and produe mnor orreton to travel tmes along rays near the wave gude axs. In spte of ths the ombned fousng propertes of and w on the axs ause serous hanges n the full aoust feld ndatng that these mnor urrents annot be gnored n ntensty alulatons. Fgure 5 shows the exat same ray traes as n Fg. 4 wth the flud speed nreased by a fator of. Ths nrease n ampltude auses K < along the wavegude. Comparng Fg. 4 to Fg. 5 one an learly see the ntal exponental rate of dvergene for rays near the axs n Fg. 5 as ompared to the lnear dvergene llustrated n Fg Ths urrent profle s hosen for llustratve purposes sne t s easy to mplement and demonstrates the profound effet that a small urrent an have on ray stablty. Smlar profles desrbe flud flow n Ekman layers, for example see Ref [6]. Stratfed urrents of ths form have been observed n the Indan oean durng monsoon season, see Ref [7].

12 IV PIECEWISE CONTINUOUS PROFIES, GENERA TREATMENT Consder a peewse model of an oean or atmospher envronment n whh the medum s dvded n depth nto a fnte number of regons, see Fg. 6. Ray segments are ndexed n order of nreasng ray parameter wthout referene to the orrespondng regon. The Jaob feld s a funton of affne parameter evaluated along the ray path. Eah ray segment n spae orresponds to an nrement of affne parameter. The SSP and wnd profle take the form N N N, N N N D w D D w D w w wth and D w D w. The frst and seond dervatves of and w are N N N N N N D w D D w D w w N N N N

13 w w N N D w D D wn DN w D respetvely, n whh f f f. Equaton 5 s solved n eah regon wth the urvature term determned by the Heavsde funtons Θ appearng n the above expressons. The boundary ondtons at eah ourrene of an nterfae determne the onstants of ntegraton for the Jaob feld n terms of the ntal ondtons. The Jaob feld s ontnuous and the dsontnuty n Y s determned by ntegratng Eq. 5 along the ray as t passes aross a boundary. Two dstnt ways of analyng ths problem arse that requre a slghtly dfferent treatment. In the frst ase t s assumed that Eq. 5 may be solved to gve Y and boundary ondtons are appled to ponts along the axs as desrbed n Fg. In the seond ase t s assumed that the soluton s n the form Y gven by Eq. 8. Boundary ondtons are appled at a depth where an nterfae dfferent layers ours. IVa CASE, BOUNDARY CONDITIONS IN λ Passng from one regon to another n spae orresponds to boundary pont along the λ axs, see Fg. 7. The frst boundary ondton s Y B Y B. Integratng Eq. 5 aross a boundary leads to the followng ondton on the dsontnuty n Y, Y Y QY, 3 B B B Q, 4 B B B B 3

14 where the subsrpt B means evaluated at the boundary and, are the dsontnutes n the flud veloty and sound speed respetvely at the boundary. Sne Y s a soluton to a seond order equaton the values of Y and Y ompletely B B determne the soluton along the -th segment of the ray. One the rays are mathed up the parameter value, tme of flght and range for eah segment are found usng solutons to the ray equaton. IVb CASE, BOUNDARY CONDITIONS IN f We frst express the soluton Y k f k g, where g and 3 d. ayers are ndexed aordng to heght along the axs from bottom to top, the soluton n eah regon beng Y k f k g. A gven ray,, wll pass through one regon more than one, perhaps an nfnte number of tmes. The bare soluton along eah of these segments s the same but the onstants of ntegraton wll be dfferent. Applyng boundary ondtons at a layer nterfae, wth Y Y, gves the oeffents f the Y n terms of those of Y. k k,, D f g D f g q f f q f f g D D g g q g f g f g q k f g k,, 5 n whh g, q D Q and the denttes f f D f g f B g g and evaluated at the boundary have been used to smplfy as many terms as possble. The 4

15 oeffents of Y along any ray segment are expressed n terms of the ntal ondtons by repeated applaton of Eq. 5. IV EXAMPE Peewse lnear profles are partularly easy to deal wth sne the rays and devaton vetor may be expressed n losed form n terms of ordnary funtons. The effet of the jump dsontnutes that these profles produe n K on the aoust feld desrbed n the last setons s llustrated here for two ases where sound from a pont soure plaed n a homogeneous statonary medum, = onstant, w = for > s ndent on an nhomogeneous half spae: ase A, /, w = and ase B, = onstant, w w /, for. The rays n eah nhomogeneous half are well known. The devaton vetor for < s Y A and for Y A for ase A, B B Y A osh B snh, n whh w, for ase B. Fgure 9 llustrates a ray fan and austs for ase A wth = and =. The aust urve n the homogeneous spae s gven by p, x 4 p 6 or smply x 7 8 5

16 for the aust urve n the upper half spae [8]. The aust urve n the lower porton of spae s determned by the roots of the followng equaton for p 6 4 b p bb p b p a 8 n whh b and a x for fxed s. The aust as a parametered urve x p, p 3 p 3 x, p p p p. 9 p A usp forms n the lower half spae whh an be loated by fndng values of ntal launh angles for whh the tangent vetor of the aust urve vanshes. Dfferentatng Eq. 9 gves the followng for the aust tangent p p 3 3 p 3/ p dx, a dp d dp pp p 5/ p p. b Both vansh when p 3 3. Ths usp wll always exst n the nhomogeneous spae as long as. Ray traes and austs are shown for a soure plaed at m n 6

17 Fgure 9. The austs, appearng on top of the ray trae, were derved from the solutons presented here. One an see the development of the usp n the lower half spae. The tal of the aust approahes ero as x. In the lmtng ase where the soure n plaed on the x axs the usp moves rght up to the soure and the tal merges wth the x axs. Both ases, A and B, desrbed above produe a usp n the nhomogeneous half spae. In ase B ths only ours for rays launhed n the same dreton as the wnd whle those launhed n the opposte dreton eventually turn n the dreton of the wnd and do not return to the homogeneous regon. Applyng that same proedure on Yλ for ase B leads to a very lengthy expresson for the aust urve whh s omtted here n the nterest of brevty. V DISCUSSION AND CONCUSION In ths artle a new method of determnng aust formaton n layered meda s presented. The method presented here generales to three dmensonal ray trang and four dmensonal spae-tme ray trang wth SSP and wnd dependng on all three Cartesan oordnates and tme []. A sgnfant feature of the applaton to layered meda s the exstene of a soluton to the devaton equaton Jaob equaton, Eq. 8. Ths soluton may be nluded wth the standard range and travel tme ntegrals used n oean and atmospher aousts towards onstrutng a full ray theoret verson of the aoust feld. The applaton dsussed here s ts use n determnng austs as parametered urves n spae and judgng ray stablty whh has dstnt advantages to other approahes n ts oneptual tangblty and omputatonal use. The author has 7

18 mplemented Eq. 5 n a numer dynam ray trae proedure wth the result that omputaton tme was redued ompared to the tehnque of numerally dfferentatng the ray paths. From the dentfaton of the stablty parameter K wth the Gaussan urvature the onjugate pont theorems provde mmedate omputatonal value and oneptual nsghts nto the behavor of austs. ACKNOWEDGEMENTS The author thanks the Offe of Naval Researh and the Ameran Soety for Engneerng Eduaton for hostng a summer faulty fellowshp at the Naval Researh aboratory NR n Washngton DC for the summer 4, durng whh tme most of ths work were ompleted. REFERENCES [] D. R. Bergman, Applaton of Dfferental Geometry to Aoust: Development of a Generaled Paraxal Ray-Trae Proedure from Geodes Devaton, NR/MR/ , Naval Researh aboratory, Washngton DC, January 8, 5 [] Y. A. Kravtsov and Y. Orlov, Causts, Catastrophes and Wave Felds, nd ed., Sprnger, New York, 999 [3] V. Cerveny, Sesm Ray Theory, Cambrdge Unversty Press, Cambrdge, [4] R. Courant and D. Hlbert, Methods of Mathematal Physs, Vol. II, John Wley & Sons, New York, 96 [5] S. Kobayash and K. Nomu, Foundatons of Dfferental Geometry, Vol. II, John Wley & Sons, New York, 963 8

19 [6] D. udwg, Unform Asymptot Expansons at a Caust, Comm. Pure and Appled Math., Vol. XIX, p [7] M. M. Boone and E. A. Vermaas, A new ray-trang algorthm for arbtrary nhomogeneous and movng meda, nludng austs, J. Aoust. So. Am., 9 4, Pt., p [8] R. W. Whte, Aoust Ray Trang n Movng Inhomogeneous Fluds, J. Aoust. So. Am. 53, No. 6, p [9] R. J. Thompson, :Ray Theory for an Inhomogeneous Movng Medum, J. Aoust. So. Am. 5, No. 5 Part, p [] W. G. Unruh, Expermental blak hole evaporaton?, Phys. Rev. ett. 46, [] M. Spvak, A Comprehensve Introduton to Dfferental Geometry Volume Fve, Thrd Edton, Publsh or Persh, INC. Houston, Texas,999 [] E. S. Eby and. T. Ensten, General Spreadng oss Equaton, etter to the Edtor, J. Aoust. So. Am. 965 [3] D. Blokhntev, The Propagaton of Sound n an Inhomogeneous and Movng Medum I, J. Aoust. So. Am., Vol. 8, No., p [4] ] I. Tolstoy and C. S. Clay, Oean Aousts, Theory and Experment n Underwater Sound, MGraw-Hll, New York, 966 [5] W. H. Munk, Sound hannel n an exponentally stratfed oean wth applaton to SOFAR, J. Aoust. So. Am. 55, p [6] G. F. Spooner, Stablty of Free Surfae Ekman ayers, Journal of Physal Oeanography, Vol. 3, No. 4, p

20 [7] M. Tomak and J. S. Godfrey, Regonal Oeanography, Pergamon, New York, 994 [8]. M. Brekhovskkh, Waves n ayered Meda, Aadem Press, New York 96

21 FIGURE CAPTIONS. Sample ray fans for and w quadrat profles desrbed n the text for a pont soure plaed at =. Column A/B orresponds to rays launhed aganst/wth the flow of the flud whle the rows a e orrespond to nreasng values of b =,.,.4,.8,... Maps of the onvergene/dvergene ones and urves of ero urvature determned by the seton urvature, K, for a ase a and b ase e of fgure. The bold lnes are urves of ero urvature. These ones depend on the soure plaement. D 3. a Canonal Munk profle, e D, D = - d/d wth 5m/s, 5m d and.. b through d Osllatng urrent w we sn d, α =.m - and b β = π/d and w = 6.63m/s, β = π/d and w =.69m/s, d β = 4π/d and w = 4.6m/s. 4. Ray fans from a pont soure plaed n an oean envronment, desrbed by SSP and urrent form fg 3 a and b, at = 5m, wave gude axs. a aganst the urrent, wth the urrent, note the onjugate ponts along the wavegude axs n. b Enlarged lose up of the rays n fg a near the soure showng the ntal lnear dvergene. Note the aust formaton n a and b whh does not our n fg., Ae. Ths s due to the fat that far from the wavegude axs the onavty of w hanges. d Enlarged verson of for easy vewng.

22 5. Same stuaton as n fgure 4 wth the exepton that w = m/s whh nreases the onavty. The effet of ths on fgure s an nreased number of onjugate ponts and fg b learly llustrates ntal exponental dvergene. 6. Example of an envronment wth peewse lnear and w along wth a sample ray. The medum s dvded horontally, eah horontal seton labeled as Regon, =,, et. Eah pee of the ray whh rosses two onseutve boundares between layers s labeled segment, =,, et. 7. Sample plot of the devaton vetor, Y, as a funton of affne parameter, λ, for a peewse lnear SSP and w =. In ths ase the devaton vetor depends lnearly on λ along eah segment of the ray. The dotted lnes separate ndvdual segments. 8. Ray fan ndent on an nhomogeneous half spae desrbed n seton IV EXAMPE. The omplete ray fan n both the homogeneous and nhomogeneous regons s llustrated. The aust, dsplayed n bold, was plotted from the exat soluton presented n ths seton.

23 a A B b d e - 5 Depth Depth - - Range - 5 Range Fgure 3

24 5 b K > K < Depth K < K < K > K > K < Range 5 a Depth K > K > Range Fgure 4

25 a b -4 w w d w..5. Fgure 3 5

26 5 a 4 5k 4k b k d 5k 5 Fgure 4 6

27 5 a 4 5 5k 4k b 4 5k 5 5 5k d Fgure 5 7

28 Regon 6 segment segment 8 segment 5 Regon 5 5 Range Fgure 6 8

29 Y Y 7 Y Y 6 C C B Fgure 7 9

30 x Fgure 8 3