Calculation of Sound Ray s Trajectories by the Method of Analogy to Mechanics at Ocean

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1 Open Journal of Acoustcs, 3, 3, -6 Publshed Onlne March 3 ( Calculaton of Sound Ray s Trajectores by the Method of Analogy to Mechancs at Ocean V. P. Ivanov, G. K. Ivanova Insttute of Appled Physcs, Russan Academy of Scences, Nzhny Novgorod, Russa Emal: vg@hydro.appl.sc-nnov.ru Receved September, ; revsed October 4, ; accepted October 4, ABSTRACT In ths paper, one of analoges avalable n the lterature between movement of a materal partcle and ray propagaton of a sound n lqud s used. By means of the equatons of Hamlton descrbng movement of a materal partcle, analytcal expresson of a tangent to a trajectory of a sound ray at non-unform ocean on depth s receved. The receved expresson for a tangent dffers from tradtonal one, defned under law Snelus. Calculaton of trajectores, and also other characterstcs of a sound feld s carred out by two methods: frst tradtonal, under law Snelus, and second by the analogy to mechancs method. Calculatons are made for canoncal type of the sound channel. In the regon near to horzontal rays, both methods yeld close results, and n the regon of steep slope, the small dstncton s observed. Keywords: Sound Rays; Materal Partcle; Analogy; Hamlton Equatons. Introducton The ray descrpton of a sound feld s wdely appled n problems on sound propagaton at ocean. It s because of the bg accuracy wth whch the ray theory defnes the basc characterstcs of a sound feld. Among modern computng means, the ray method of calculaton of a sound feld does not cause any dffcultes. Advantage of ths method conssts that t contans a mnmum quantty of approxmatons and can serve as the standard n an evaluaton of other methods. It proves to be true, for example, when calculaton of a sound feld of caustcs s made [,]. Unque restrcton of use of a ray method s performance of condtons of geometrcal acoustcs n non-unform spaces [3]. Along wth a tradtonal ray method of calculaton of a sound feld at the ocean, usng Snelus s law, other methods based on analogy between propagaton of a sound wave at ocean and movement of a materal partcle are wdely appled, see for example [4]. Calculaton of a trajectory of a materal partcle s made on the equatons of classcal mechancs and the equatons of Hamlton. The analogy between movement of a sound wave and a materal partcle establshes connectons between the functons descrbng ther movement. In [4] and later works wth a vew of smplfcaton, the followng assumpton s entered the horzontal spatal co-ordnate of propagaton of a sound wave s consdered smultaneously and as tme co-ordnate. Other ways of an establshment of analogy between propagaton of a sound and movement of a mechancal partcle are offered n [3]. In ths paper, all co-ordnates of a sound trajectory, tme and spatal, keep the number and functonal value at defnton of a trajectory of a sound wave on the equatons of Hamlton. It provdes vsbly demonstraton and relablty of conformty of two varous physcal phenomena and possblty of ther descrpton by means of the same mathematcal apparatus. The purpose of the gven work s to carry out one more varant of use of the equatons of the classcal mechancs, offered n [3], for calculaton of a trajectory of propagaton of a sound n a lqud. Analytcal expresson for a tangent to the ray trajectory, dstnct from gven by law Snelus s receved. Comparson of calculaton of trajectores and other characterstcs of a sound feld n a wave - gude by two methods, tradtonal under Snelus law and by analogy to classcal mechancs, [3] s spent. The short descrpton of ths method of calculaton of a trajectory by analogy to mechancs s resulted n [5]. It s known that n sotropc, homogeneous space not changng n tme plane sound waves propagate n a drecton of a wave vector k whch does not vary nether on value, nor n a drecton. At ocean, a sound speed depends on depth that makes the space not sotropc. It leads to the change of a wave vector k both on sze, and n a drecton. Therefore, the sound wave s not a plane one any more. However, representaton about a plane wave can be used and n such space f on dstances of an order of length of a wave, the value and a drecton of a wave vector vary slghtly. The method of replacement of a real sound wave wth a set of plane waves s called as a method of geo- Copyrght 3 ScRes.

2 V. P. IVANOV, G. K. IVANOVA metrcal acoustcs [3]. In those cases where we wll apply a method of geometrcal acoustcs, t s possble to enter concept about rays as about lnes, tangents to whch concde wth a drecton of propagaton of a plane wave on small regons of space. Thus, justce of a method of geometrcal acoustcs reduces a problem of propagaton of a sound n the ansotropy space to calculaton of ray trajectores.. Tradtonal Method Ray s Calculaton Let s consder brefly tradtonal way of defnton of ray s trajectores at ocean wth speed of a sound dependng on depth (a method ). We wll accept, that the ocean s a plane- layered space n whch speed of a sound vares wth change depth z and does not depend on co-ordnates x, y. Thanks to unform and sotropc space along axes x, y projectons of a wave vector k on an axs x, y kx, ky also do not depend on co-ordnates x, y, [3]. We wll accept, that a plane of fallng of a sound wave serves the plane x, z. Let θ a corner between a vector drecton k and an axs z. Then the runnng sound wave n a wavegude s descrbed by followng expresson: kr p x, z, t aexp x, z, t, xzt,, t kr x x kr z z t, k c z, k k k, k ksn, k kcos () x z x z R x x z z, R Rsn x, R Rcos z. x z Here a ampltude of a wave, x, zt, a wave phase (ekonal), k a wave vector, k ts module (length), R a vector of a trajectory of the wave, parallel to a vector k, R the module (length) of a vector, co-ordnates of the begnnng of vector R x, z, end co-ordnates (current co-ordnates of a trajectory) x, z. We wll accept, that absorpton of a sound s a lttle, the ampltude of a wave does not vary. Accordng to Snelus law the component of a wave vector kx sn c z remans nvarable for a concrete ray at non-unform ocean on depth: k x sn sn const () c z c z Here z depth of a source of a sound, θ an angle of an departure of a ray from a source, cz speed of a sound on depth of a source. Accordng to Snelus law, Equaton (), f a refracton angle r π on depth z z, where r cz cz r, const n Equaton () s equal to cz r. Apparently from Equaton (), the constant kx can be expressed through an angle of departure of a ray from a source θ and speed of a sound c z too, const sn cz. We wll accept, that on dstances of several lengths of waves the module of a wave vector vares slghtly, consderably ts drecton vares only. Therefore we wll enter a sngle vector n, concdng on a drecton wth a wave vector, k kn. Components of a sngle vector n are expressed through an angle : nx sn, nz cos. Usng for a constant n Snelus law () expresson cz r we wll receve: n x sn c z c zr, (3) n cos c z c z z r Equatons (3) defnes components of an sngle vector n along axes of co-ordnates, and also a tangent to a trajectory of propagaton of a wave tgθ, concdng on a drecton wth a vector k. From Equaton (3) t s vsble, that components n x, n z depend from z, n x has the same dependence on depth, as c z. For a components of a wave vector takng nto account Equaton (3) we wll receve: k с, k c z c z c z (4) x r z r It s vsble, that k x a constant for the chosen ray wth a turnng pont on depth z r, k z depth functon of z. Components of a wave vector, Equaton (4), are receved n the assumpton, that perodcty of a sound feld at change k z s not broken, condtons of geometrcal acoustcs are applcable. Trajectory co-ordnates x, z and length of a trajectory l, are connected by known partes: x z tgd z, (5) l dz cos, whch allow to calculate wth the bg accuracy a trajectory, and also tme of propagaton of a sound along a trajectory. Convncng acknowledgement of accuracy of calculaton of a sound trajectory by a ray method can be found n many works, for example, n [6] where the value of tme of propagaton of a sound on dstance ~7 km calculated under the ray theory and measured n experment are resulted, concded wth each other. Thus, a method of geometrcal acoustcs n partally sotropc space defnes a tangent to a trajectory of a ray n the form of analytcal functon of speed of a sound on whch the ray trajectory, tme of propagaton of a sound on a ray and the value of the sound feld are calculated along a ray trajectory. 3. The Method of Analogy to Mechancs Let s consder another method, method, calculaton of trajectores of the sound rays, based on analogy between movement of a materal partcle and sound propagaton n approach of the geometrcal acoustcs, offered n [3]. Let us conceder brefly prncpal features of the clas- Copyrght 3 ScRes.

3 V. P. IVANOV, G. K. IVANOVA 3 scal mechancs, descrbng movement of a materal partcle by the equatons of Hamlton [7]. Poston of a materal partcle n the classcal mechancs s defned by ts generalzed co-ordnates q q x, z, t and the generalzed mpulses p x, z, t, equal m q, where m mass and a q projecton of speed of a partcle to an axs, the ndex means x, z. The generalzed co-ordnate q n a homogeneous space s a partcle radus-vector f the begnnng of ts movement concdes wth the begnnng of co-ordnates [7]. In ths case projectons of length of a vector q on an axs of co-ordnates concde wth values x, z, snce x z. The same concerns and to a vector of the generalzed speed. In the closed system where the partcle s not exposed to any nfluence, a vector q s the straght lne, and a vector p s a constant mpulse of a partcle, a constant value of speed. The bg role n the classcal mechancs plays the functon of acton S defned as ntegral along a trajectory, taken on tme from Lagrange s functon between two ponts of a trajectory q and q whch the partcle occupes durng the set moments of tme [7]. The dervatve on tme from acton S defnes Hamltonan of materal partcle H, and on the generalzed co-ordnate a partcle mpulse. The equatons of Hamlton can be receved from a condton of exstence of a mnmum value of acton S [7]. We brng these four equatons whch we use to establsh correlaton between movement of a materal partcle and propagaton of a sound wave and for calculaton of a sound trajectory: S S H H, p, q H, p (6) t q p q At known functon of acton S from frst two equatons t s defned Hamltonan H of partcle and ts mpulse p. From second two dfferental equatons whch are equatons of Hamlton, co-ordnates of a trajectory of a partcle q and components of ts speed q are calculated. The equatons of Hamlton defne a partcle trajectory, as well as Newton s second equaton. We wll accept, that expresson for ekonal x, zt, kr t n the defnton of the sound wave s feld, Equaton (), s analogue of acton S of a materal partcle. One of the functons enterng n Equaton (), s a trajectory of a sound wave (a ray on whch the sound wave propagates), vector R. It s natural to accept, that vector R s analogue of a vector of the generalzed co-ordnate q of the partcle, and projectons to axes of co-ordnates of a vector of trajectory R - analogue q. Contnung analogy between S and Ψ we wll receve from Equatons (6) the frst equaton, that analogue Hamltonan H s frequency of radaton, H, and from the second equaton analogue of an mpulse p of the partcle s the wave vector k, p - k. It s already formal analogy between partcle s Hamltonan H and frequency of a sound wave kc z and components of the generalzed vector p and a wave vector k. The equatons of Hamlton, 3, 4 n Equaton (6), for a trajectory of a sound wave look lke: dk d t r,dr dt k (7) Let s present a wave vector, as n a method, n the form of ts module that don t change on a dstance of several wave s lengths and sngle vector n, that vares ts drecton, k kn. We wll receve from the Equatons (7) at statonary propagaton of a sound n the motonless non-unform space the equatons for vectors n and R: dn d t c z,dr dt cn. (8) Authors [3] brng the equaton for vector defnton n, contanng an addtonal member, n our desgnatons nn c z. Ths member has appeared at transton from dk dt to dn dt, through dervatves d d dr dt dk dr on k R and replacement d cz d cz c z R, and dr dt on cn from Equaton (8), the second one. It s ncorrect procedure snce the generalzed functons q and p for a mechancal partcle, together wth ther analogues R and k n a sound wave, depend by ther prmary defnton only on varables x, z, t, but not from each other. Therefore the correct equaton for vector s defnton n s the frst n Equatons (8). It s easy to show, that the decson of the equaton for n, resulted n [3] wth an addtonal member nn c z, s the followng: n n x n z,.e. s defnton of an sngle wave vector, and ts components values for a consdered problem reman not defned. Let s wrte down system of the Equaton (8) for projectons of vectors R and n on an axs x, z: dx d t cnx,dz d t cnz, (8 ) dn dt dc dx,dn dt dc dz x z The decson of the thrd of Equaton (8 ) can be or nx const or nx f z. The frst decson accordng to exstng connecton n x and n z leads to n z const. Ths decson s far only n the sotropc spaces. Remans nx f z. Snce the component of a sngle vector n x can be expressed through nz f z becomes known after defnton n z. Usng a method of the decson of system of the ordnary statonary dfferental equatons of st order, [8], after excepton dfferental dt from Equatons (8 ) we wll receve two equatons: x z z z dx d z n n, n dn dc z с z (9) Solvng the second of them, we wll defne a projecton of a sngle vector n z as functon of depth z: nz lnc c, where с an ntegraton constant. A constant с we wll accept equal speed of a sound n a turnng pont of a ray, с = с r. As a result for n x, n z, we wll receve followng expressons: Copyrght 3 ScRes.

4 4 V. P. IVANOV, G. K. IVANOVA n nz ln cr c z, nx z () The frst equaton from system Equeton (9) defnes a tangent nclnaton to a trajectory of a sound ray tg nx nz. For calculaton of a trajectory of a ray t s necessary to substtute value tgθ, expressed through n x, n z n Equaton (5). From Equaton () also, as well as Equaton (4) t s vsble, that the component of a sngle vector along an axs z becomes zero n a ray s turnng ponts at czczrcr. Thus, at analogy use between the descrpton of propagaton the sound along a ray and movement of a materal partcle (method ) analytcal expressons for components of a sngle vector along a trajectory of the ray, Equaton (), dstnct from the expressons n Equaton (3) receved by method () wth use of Snelus law, Equaton (), are receved. 4. Calculaton Ray s Trajectores and Sound Feld along Them wth Two Methods Let s spend comparson two ways of ray s trajectory calculaton for a profle of a sound s speed of a canoncal form. We wll accept, that the axs of the sound channel lays on depth of km, depth of a wave- gude s 4 km, the sound source s on a channel axs. It s easy to show, that rato nx c z s not a constant. However, calculaton of ths rato for varous rays shows, that t dffers from a constant on small sze. We wll compare horzontal components of sngle vectors n x, n x trajectores of the rays, calculated under formulas (3) and (). In Fgure values of n x, n x show as functons of depth z for the rays havng turnng ponts (ТP) below an axs of a wave gude on depths z =.75 km and z = km. In Fgure on a horzontal axs values n x, n x, are postponed, on vertcal r r r all 4 curves have the dentcal values equal, n x = n x =, see Equatons (3) and (), on correspondng depths. Numbers 3, axs depth z. In ТP π, c cz 4 desgnate two curves concdng on the scale of drawng havng ТP on depth z. Apprecable dfference n x, n x s observed near to a wave gude axs, for steep rays, curves,, π, wth ТP on depth z. Fgures n drawng near to these curves specfy, what method calculates curves. Dfference of curves s caused, obvously, not only dstncton of the functons descrbng a ray trajectory (a vector n) n methods,, but also dstncton of angles of departure at whch rays have the same depth. On Fgure the length of a cycle of rays as functon of an angle of departure θ, calculated n two methods s shown. The dvergence of curves on the scale of drawng also s observed only for steep rays (the least θ ). The lower curve s calculated by a method, upper on a me- thod. It s vsble, that length of cycle ) more than that calculated by a method, D D (a method. So, at depth of a tunng pont a ray, z =.5 km, angles θ concde for two methods to wthn 5 th sgn, a dfference of cycle s lengths D D D3.m, at z km,.6, D 3 m. Let s consder vertcal structure of a sound feld z(θ) n the canoncal wave gude, calculated n two ways. On Fgure 3 angular dstrbuton of vertcal co-ordnates of the rays whch have left a source towards a bottom, on, const s resulted. An axs of abscses angles of ray s departure θ, ordnate depth z arrval of a correspondng ray on dstance x from a source. An ntal (greatest) θ = Extremes of curves s co-ordnates of the caustc s centers on depth on the gven dstance x. The contnuous curve on Fgure 3 s receved by a method, shaped a method. It s seen, that wth decrease of a θ the curves concdng at the bg angles θ, dverge a lttle, caustc s co-ordnates do not concde. Smlar curves wth the caustc s centers along an axs z only dependng on tme of arrval of rays, horzontal axs, are resulted n a number of works. On Fgure 4 depth of dstance from a source x = 5 km, z x z (km) 55 D(θ ) (kм) 3, 4 Z Z 5 3 θ (degree) n x, n x. 45 Fgure. Dependence of projectons of sngle vectors n sn, n sn, from depth of a wave gude z for rays, x x curves 3, 4 wth turnng ponts at a bottom on depths z =.75 km and z = km, curve,, θ an angle along a trajectory Fgure. Dependence on an angle of departure of length of a, calculated n two ways, the bottom curve D ray s cycle a method. Copyrght 3 ScRes.

5 V. P. IVANOV, G. K. IVANOVA 5 z (км) 5 3 θ (degree) Fgure 3. Angular dstrbuton of vertcal co-ordnates of rays at dstance x = 5 km, calculated by two methods; a contnuous curve a method, shaped a method, θ an angle of departure. 3 z (км) t (с) Fgure 4. Vertcal co-ordnates of rays as functons of tme of arrval at dstance x the = 5 km calculated by two methods; a contnuous curve a method, shaped a method. arrval s rays whch have left the source towards a bottom, dependng on tme of ther propagaton, horzontal axs, s shown. By a shaped lne the co-ordnates of rays calculated by a method are shown. Dstncton of the curves s vsble only n that range of tme and depth, n whch the rays calculated by a method, are absent. In other ponts curves on the scale of drawng concde. In co-ordnates tz, the centers of caustcs are tops of ponts of bend of curves z t. Comparson of numercal values of tmes of ray s arrvals on the same depth shows, that tmes of propagaton of the near horzontal rays, calculated by both methods, are very close to each other, consderable dstncton s observed for steep rays. So, for corners θ 88 tmes of ray s rrvals dffer on ~ c, tme of arrval of the most steep rays calculated on a method on ~.3с more than calculated on method. Let s consder structure of the module of an acoustc feld along a vertcal, calculated by two methods, Fgure 5, an abscse an axs z, ordnate the ampltude of feld. The feld calculates on frequency of 33.6 Hz. Contnuous lnes calculaton by a method, shaped a method. 5 z (км) 3 Fgure 5. Structure of the module of an acoustc feld along a vertcal axs of a wave-gude at dstance x = 5 km, calculated by two methods; a contnuous lne a method, shaped a method. Dstncton n caustc s poston and n value of ampltudes s vsble. 5. Conclusons As a result of comparson of two methods of obtanng of analytcal expressons for a tangent to a ray trajectory, t s possble to draw the followng conclusons. Practcal concdence of calculaton of trajectores by two methods s the confrmaton of legtmacy of the decson of a problem on sound propagaton n the non-unform space wth the decson n ray approach. When the general structure of an acoustc feld n a wave gude wthout quanttatve comparson of separate characterstcs s requred, both methods of the account are equvalent. If t s necessary to compare calculated value wth experment, t s necessary to gve preference to a tradtonal method. Under the physcal concept, t s more proved than the nd method usng formal analogy between acton of a mechancal partcle and ekonal of a sound wave. The method of analoges s of nterest as possblty of varous comparsons by the nature of the physcal phenomena, n whch the general laws that promotes ther deeper understandng can be found. In ths case t s possblty to receve one more analytcal expresson for a tangent vector to a ray trajectory n approach of the geometrcal acoustcs, dstnct from tradtonal, under Snelus s law. REFERENCES [] V. A. Zverev, V. P. Ivanov and G. K Ivanova, Caustcs n the Underwater Channel and Ther Connecton wth the Wave-Front of Pont Source, Proceedngs of the 3th L. M. Brekhovskkh s Conference Ocean Acoustcs, Moscow,, pp [] V. A. Zverev, V. P. Ivanov and G. K Ivanova, Caustcs n the Underwater Channel and Ther Connecton wth the Wave-Front of Pont Source,. [3] L. D. Landau and V. M. Lfshts, Gdrodnamka, Sc- Copyrght 3 ScRes.

6 6 V. P. IVANOV, G. K. IVANOVA ence, Мoscau, 988. [4] J. Smmen, S. M. Flatte and G.-Y. Wang, Wavefront Foldng, Chaos, and Dffracton for Sound Propagaton through Ocean Internal Waves, Journal of the Acoustcal Socety of Amerca, Vol., No., 997, pp do:./.498 [5] G. K. Ivanova, Two Methods of Calculaton of Sound Ray s Trajectores at Ocean, Proceedngs of the th Conference of the Russan Acoustc Socety, Moscow, 8, pp [6] G. J. Heard and N. P. Chapman, Heard Island Feasblty Test: Analyss of Pacfc Part Data Obtaned wth a Horzontal Lne Array, Journal of the Acoustcal Socety of Amerca, Vol. 96, No. 4, 994, pp do:./.4 [7] L. D. Landau and V. M. Lfshts, Mechancs, Phys- Math. Lt. State Publcaton, Moscau, 958. [8] V. I. Smrnov, The Course of a Hgher Mathematcs, Phys-Math. Lt. State Publ, Moscau, 958. Copyrght 3 ScRes.