Approximate solutions for an acoustic plane wave propagation in a layer with high sound speed gradient

Transcription

1 Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia Approximate solutios or a acoustic plae wave propagatio i a layer with high soud speed gradiet Alex Zioviev ad Adria D. Joes Maritime Divisio, Deece Sciece ad Techology Orgaisatio, Ediburgh,, Australia ABSTRACT The preset authors previously obtaied a exact solutio or propagatio o the icidet acoustic plae wave i a wid-iduced bubbly surace layer where soud speed gradiet is high. Based o the solutio, the depedece o the grazig agle o the eergy lux vector o the icidet wave at the surace o the grazig agle o the icidet wave at the bottom o the layer was obtaied ad utilised or calculatios o the surace relectio loss. However, the solutio i its origial ormulatio is represeted via a iiite slowly covergig series which makes the use o the solutio i practical calculatios diicult. I this paper, this solutio is simpliied with the assumptio that the grazig agle at the bottom o the layer as well as some other parameters depedig o the requecy ad wid speed are small. The covergece o the iiite series is improved usig a special techique. Approximate solutios are obtaied or the cases where the small parameters ca be eglected ad where the terms up to the third order with respect to the small parameters are take ito accout. Zoes o validity are derived or the approximate solutios as well as or the Sell s law o ray acoustics. The results o this paper allow the predictio o the grazig agle at the surace withi the zoes o validity o the approximate solutios usig simple calculatios with oly the wid speed, the requecy ad the equilibrium soud speed as the iput parameters. ITRODUCTIO Trappig acoustic eergy withi a medium, due to a avourable soud speed gradiet, is a well-kow eect. A example o duct trappig occurs withi a isothermal surace layer i a ocea or which the soud speed icreases with depth due to the icreasig hydrostatic pressure. Formatio o such a duct may lead to a sigiicat icrease i the propagatio rage, as is well kow. At the same time, the propagatio o soud wave i the layer is sigiicatly complicated by the roughess o the ocea surace. Due to the roughess, the acoustic eergy ca be scattered i o-specular directios thus cosiderably icreasig the relectio loss ad correspodigly decreasig the propagatio rage. As a result, the roughess o the ocea surace must be take ito cosideratio i ay realistic model o soud propagatio withi the surace duct. Existig models o soud scatterig at the rough ocea surace usually take the grazig agle at the surace as a iput parameter. I some istaces the grazig agle at the surace ca be calculated by meas o Sell s law o ray acoustics. However, i situatios where the soud speed chage i the surace duct is cosiderable o the scale o the acoustic wavelegth, Sell s law may o loger be applied or calculatig the grazig agle ad a wave-based theory becomes ecessary or this purpose. For example, such a situatio ca be caused by the presece o air bubbles i the ocea ear the surace. These wid-iduced bubbles chage the compressibility o water thus leadig to sigiicat reductio o the soud speed ear the surace. Joes et al. () suggested a model or evaluatig the loss o acoustic eergy due to relectio rom the rough ocea surace. The model was called JBZ by the authors ad is based o the Hall-ovarii bubble populatio model (Hall, 989, Keier et al., 99) as well as o the calculatios o the soud speed i the bubbly layer by Aislie (). A itegral part o the JBZ model is also a ovel method o evaluatig the grazig agle at the surace. The method is ouded o a wave-based solutio o the wave equatio i a vertically stratiied layer. The solutio is based o a ormulatio provided by Brekhovskikh (9) or a trasitioal layer betwee two media with dieret values o the equilibrium soud speed. This solutio has bee cosidered i detail by Zioviev et al (). It has bee demostrated that, i the icidet grazig agle at the bottom o the bubbly layer is o the order o a ew degrees or less, the grazig agle at the surace may dier sigiicatly rom the value predicted by Sell s law. The lesser o two values or the grazig agle at the surace, oe o which was predicted by Sell s law ad the other oe by the wave-based solutio, has bee utilised i calculatios o the trasmissio loss usig the JBZ model (Joes et al., ). The results o the calculatios have show good correlatio with the results obtaied usig a Parabolic Equatio trasmissio code. Although the JBZ model showed its validity, i the origial ormulatio the solutio or the grazig agle at the surace cotais a slowly covergig iiite series ad depeds o parameters o the layer which eed to be obtaied rom matchig the soud speed proile i the real layer ad the trasitioal layer described by Brekhovskikh (9). Thereore, to make the wave-based method o calculatig the grazig agle at the surace more practicable, its theoretical ormulatio eeds to be simpliied. This paper is devoted to idig ad justiyig simpliied approximate solutios or the grazig agle at the surace. I the irst sectio, the exact solutio obtaied by Zioviev et al. () is reviewed. I the secod sectio, major steps i the derivatio o the approximate simpliied solutios are demostrated. The third sectio cotais justiicatio o zoes o validity o the obtaied approximate solutios ad Australia Acoustical Society

2 Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia o the Sell s law. Fially, i the ourth sectio, examples o the depedecies o the grazig agle at the surace o the icidet grazig agle or dieret requecies ad wid speeds are show ad discussed. EXACT SOLUTIO FOR SOUD PROPAATIO I THE SURFACE LAYER Soud speed proile i the surace layer Zioviev et al. () showed that the soud speed proile based o the Aislie s calculatios o soud speed i the mixture o water ad bubbles ca be well approximated by soud speed proile i the trasitioal layer cosidered by Brekhovskikh (9). This derivatio is reviewed below. Assume that the vertical coordiate, z, is directed rom the surace. I the soud speed ar below the surace layer is c, the proile o the trasitioal layer ca be determied by the ollowig equatio: cz c, z. mz e mz e I Equatio (), ad m are parameters o the layer. characterises the stregth o the layer ad is determied by the ollowig equatio: c c c s, where c s is the soud speed at the surace. As m = correspods to a iiitely wide layer, whereas at m the layer is iiitely thi, it ca be said that /m describes the thickess o the layer. It is coveiet to assume that the surace o water i the Aislie s layer does ot coicide exactly with the surace z = i the trasitioal layer, but, istead, it is located at some positive depth z = z. I this case, the ollowig equatio provides the lik betwee the parameters m ad z : z c z c c mz z l. c To determie m ad z, it is ecessary to cosider Equatio () together with aother equatio chose to make sure that the trasitioal layer proile represets the best it curve or the Aislie s proile. This procedure is explaied i detail by Zioviev et al. (). Solutio or the grazig agle at the surace () () () Brekhovskikh (9) showed that, i a plae wave with the wave vector, k, approaches the trasitioal layer rom below with the grazig agle, θ, betwee the wave vector ad the horizotal axis, x, the solutio or the acoustic pressure withi the layer ca be writte with the use o hypergeometric series. By applyig Brekhovskikh s solutio or the acoustic pressure to the trasitioal layer Zioviev et al. () derived a system o equatios or the grazig agle, θ s, o the eergy desity vector, q, at the surace. The vector q is determied via the acoustic pressure, p, ad the particle velocity, v, as q = pv. Ater some parameter substitutios the system o equatios or idig θ s takes the ollowig orm: Re pv s ta, Re pv x si Z ze ik x () p x, z, () Z z, () e mz, (7) v k si x p, (8) i p v, z g (9) g, () i, () g g, () i k m si, () k m. () I Equatios () to (), as well as i the aalysis below, p is the acoustic pressure, v ad v x are vertical ad horizotal compoets o the particle velocity, ω=π is the cyclic requecy o the acoustic wave (temporal depedece e -iωt is assumed), ρ is the equilibrium water desity, (ξ) is a iiite series origiatig rom the hypergeometric series, ad δ ad µ are auxiliary variables. The equilibrium soud speed ar below the layer, c, is assumed to be m/s. Equatios () to () represet the exact solutio or a plae wave propagatio i the trasitioal layer as o assumptios have bee made about values o ay o the parameters. DERIVATIO OF APPROXIMATE SOLUTIOS FOR THE RAZI ALE AT THE SURFACE Small parameters or approximatios It is diicult to use Equatios () to () i practical calculatios or two reasos. First, they iclude a iiite series, (ξ), which is covergig slowly ear the ocea surace, where ξ. Secod, to id the layer parameters m, ad z, it is ecessary to kow the real soud speed proile i the bubbly layer, which is ot always easible. However, it is possible to simpliy the solutio or θ s by makig assumptios that some parameters are small ad subsequetly usig Taylor series to eglect terms o higher orders with respect to these parameters. First o all, it is kow that, durig the soud propagatio i the surace duct, the icidet agles are ormally ot larger tha a ew degrees. Thereore, Australia Acoustical Society

3 Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia it is reasoable to assume that the icidet grazig agle at the bottom o the bubbly layer,, is small:. rad. () Also, some parameters o the bubbly layer i realistic coditios ca be cosidered to be small. Table shows the parameters o the layer calculated or our values o the wid speed usig the method described by Zioviev et al. (). The wid speed i the preset aalysis reers to the height o 9. m above the ocea surace. i i. ow the iiite series (ξ) takes the orm o i a a..., (9) () Table. Depedece o the calculated parameters o the layer o the wid speed. Wid 7.. speed, m/s.... m, m z, m.... It is clearly see rom Table that, at the wid speeds o iterest, the ollowig assumptios ca be made:., () mz.. (7) Takig ito cosideratio Equatios () to (7), all terms i a Taylor series cotaiig θ, ad mz o the secod ad higher orders are eglected i this aalysis. At the same time, the parameters δ ad µ determied by Equatios () ad () cotai the small parameters multiplied by the ratio k /m, which is proportioal to the requecy ad may be sigiicatly larger tha uity at requecies o a ew kilohertz. Thereore, terms o higher orders tha the irst oe with respect to δ ad µ are required to be take ito accout. I this aalysis, the terms cotaiig δ ad µ o the orders up to the third oe are used ad the terms o higher orders are eglected. Improvig covergece o the iiite series Detailed descriptio o the calculatios leadig to the approximate solutios is cosidered to be outside the scope o this work. However, these calculatios ivolve a importat step a applicatio o a special techique to sigiicatly improve the covergece o the iiite series. This step is described i this sectio. It is clear that the iiite series (ξ) described by Equatios () to () coverges at all depths z >. Ideed, the term i the brackets i Equatio () teds to uity i. Also, the ollowig coditio is satisied or the variable ξ: mz z e z. (8) However, close to the surace, where z z, the covergece is slow due to the small value o the expoet mz. umerical experimets show that, or the series (ξ) to coverge, a large umber o terms eed to be take ito accout. Obviously, derivatio o a simpliied approximate solutio requires this slow covergece to be improved. Such improvemet is carried out here usig Kummer s trasormatio (Lito, 998). Ater expadig ito Taylor series over small parameters the term i the brackets i Equatio () ca be writte as ollows: where a i. j () j j Accordig to Kummer s trasormatio, the sum o a iiite covergig series, x, ca be represeted by the ollowig equatio: where x x x S, () S x. () Assume the sum S i Equatio () ca be oud aalytically. The iiite series i the right-had part o that equatio coverges much aster tha the series i the let-had part, as the two terms i the brackets asymptotically ted to each other ad their dierece teds to zero with icreasig. For the series (ξ) uder cosideratio, it ca be oud umerically that the coeiciets a at large ted to the ollowig expressio: a b ai, () where a ad b are costats determied as a., b.. () Applyig Kummer s trasormatio to the series (ξ) allows oe to re-write the series i the ollowig orm: i b ai a b ai. () It ca be show by umerical calculatios that the series determied by Equatio () coverges quickly. Approximate solutios or the grazig agle at the surace A ivestigatio o the series i Equatio () shows that oly a ew terms i the series are eeded to obtai a solutio with a good degree o approximatio. As successive terms o the series have opposite sigs, the total sum o the series is always betwee two successive partial sums. I this aalysis, the approximate solutios have bee obtaied or oe ad two terms i the series. The substitutio o Australia Acoustical Society

4 Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia Equatio () ito Equatios () to (), expadig the resultig equatios ito the Taylor series where ecessary ad eglectig terms o higher orders leads to the ollowig simple equatios or the grazig agle at the surace or oe ad two terms i the series respectively: where s.9, (7) s.9, (8) m, m -... Best it curve or m, Power = Calculatios Best it. (9) ote that the parameter µ ca be cosidered as odimesioal requecy. Due to the opposite sigs o successive terms o the iiite series, the ull solutio or θ s should () be oud betwee θ s ad θ () s. The validity o Equatios (7) ad (8) is discussed urther i this aalysis. Determiig the layer parameters by polyomial curve ittig The obtaied approximate solutios (Equatios (7) ad (8)) iclude the layer parameters ad m. ote that the parameter z is abset rom these solutios. As show above, the layer parameters ca be oud by matchig the soud speed proile o the bubbly layer to the soud speed proile o the trasitioal layer. It is clear, however, that this method is ot useul or practical purposes. It is demostrated below that the parameters ad m ca be also oud by polyomial curve ittig usig oly the wid speed, W, as a iput parameter. Figures ad show values or ad m calculated usig the soud speed matchig algorithm as described i Zioviev et al () as well as by meas o the curve ittig (best it curve) method. The equatios determiig the best it curves are as ollows: W.8 W. W., (). i W 8 () m. W.7W i W 8.W. Best it curve or, Power = Calculatios Best it 8 Wid speed, m/s Figure. Parameter calculated by meas o the soud speed matchig algorithm i compariso with the best it curve. 8 Wid speed, m/s Figure. Parameter m calculated by meas o the soud speed matchig algorithm i compariso with the best it curve. It is clear rom Figures ad that polyomial curve ittig o the third power approximates the depedecies o the parameters ad m o the wid speed withi the give rage to such a extet that the origial curves ad the best itted oes are virtually idistiguishable. ote that m is costat below the wid speed o 8 m/s due to a peculiarity i the Hall- ovarii bubble populatio model. umerical calculatios show that, withi the rage o realistic values o c, m does ot deped o the equilibrium soud speed, c, ad depeds o c oly slightly. Thereore, Equatios () ad () provide a tool or determiig the parameters o the trasitioal layer acoustically equivalet to the bubbly surace layer with oly the wid speed as a iput parameter. AREAS OF VALIDITY OF THE SOLUTIOS FOR THE RAZI ALE AT THE SURFACE Equatios (7) ad (8) show that, i µ is small eough, the grazig agles at the ocea surace, θ s, ad at the bottom o the surace layer, θ, ca be cosidered equal ad, cosequetly, the reractio i the layer ca be eglected. Assume that the parameter µ ca be cosidered to be small or this purpose i the ollowig coditio is satisied:.. () Takig ito cosideratio Equatio (9), the criterio o the absece o reractio i the layer ca be writte as., () where ad m are determied via the wid speed, W, by Equatios () ad (). Figure shows the areas o validity o the approximate solutios o the requecy/wid speed plae. The case o o reractio correspods to the lowest (gree) curve, so that reractio ca be eglected at ay poit below this curve. it is reasoable to assume that the ext term i Equatios (7) ad (8) with respect to the parameter µ will be o the order o µ ad, thereore, the coditio o validity o Equatios (7) ad (8) ca be writte as., () Australia Acoustical Society

5 , khz Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia which leads to the ollowig coditio or the correspodig requecy rage:.9. () I Figure, the area o validity o Equatios (7) ad (8) is below the cya lie. It ca be see that these approximate solutios are valid at higher requecies ad wid speeds tha the solutio which eglects the iluece o bubbles o reractio altogether. Equatios () ad () together with Equatios () ad () represet a simple method o determiig the requecy rages where the reractio i the bubbly layer ca be eglected altogether ad where the approximate solutios are valid. The iput parameters are the wid speed, which ca be easily measured, ad the equilibrium soud speed, which, i real coditios, is close to a costat value o m/s. The well-kow Sell s law o ray acoustics, i applicatio to the layer uder cosideratio, ca be ormulated as coss cs. cos () c As c s < c or the surace bubbly layer, Equatio () shows that the grazig agle θ s at the surace is ot zero eve i the icidet grazig agle θ approaches zero. This directly cotradicts Equatios (7) ad (8), which show that θ s i θ. To resolve this cotradictio, it is ecessary to determie the area o applicability o the Sell s law i the bubbly layer. As show above, Equatios (7) ad (8) have bee obtaied usig the assumptios that some parameters are small ad subsequetly cosiderig oly low-order terms with respect to these parameters. It is clear, however, that takig ito cosideratio terms o higher orders i Equatios (7) ad (8) will ot lead to the Sell s law, as the higher order terms will also vaish at θ. Thereore, it ca be cocluded that the parameters assumed small i the aalysis leadig to Equatios (7) ad (8) are ot small whe Sell s law is valid. umerical calculatios show that the ull solutio or the grazig agle at the surace teds to the predictios o the Sell s law with icreasig θ (Zioviev et al., ). Out o the small parameters described earlier i this paper oly oe parameter, δ, which is determied by Equatio (), depeds o θ. Thereore, it is reasoable to assume that the Sell s law is valid i δ is large eough or the Taylor series over this parameter ot to be valid. Assume that the ollowig coditio or δ is satisied i this case:.. (7) Cosiderig that is small, Equatio (7) ca be used to derive the ollowig coditio o requecy to determie the validity o the Sell s law i the applicatio to the surace bubbly layer:.79. (8) I Figure, the areas o validity o the Sell s law are show or the icidet agle θ o. ad degrees. The Sell s law is valid at ay poit above the correspodig curves. Validity o approximate solutios A o reractio Approximatio Sell's law ( =.) Sell's law ( = ) B C F D H 8 W, m/s Figure. Areas o validity o approximate solutios. The solutios correspodig to the case o o reractio ad to the approximatio with respect to the small parameters are valid below the respective curves. The Sell s law is valid above the respective curves. Circles deote the poits or which graphs are show i Figure. It ca be see rom Figure that the reractio i the layer ca be eglected i the wid speed ad/or the requecy are ot too high. I this case the acoustic wavelegth is large i compariso to the depth o the layer, so that the layer caot aect sigiicatly the acoustic wave propagatio. It is also clear that the obtaied approximate solutios (Equatios (7) ad (8)) are valid at larger itervals o the wid speed ad requecy accordig to the assumptio that the odimesioal requecy µ is small but o-zero. Equatio (8) demostrates that the area o validity or the Sell s law depeds o the icidet grazig agle θ. The lowest requecy where the Sell s law is valid is iversely proportioal to θ. This leads to a importat coclusio that, or each requecy, there is a critical icidet grazig agle such that the Sell s law is ot valid at ay agle smaller tha this critical value. I geeral, the approximate solutios derived ad ivestigated i this paper ca be used i the correspodig poit o the wid speed/requecy plae is below the gree lie i Figure. I the icidet agle is larger tha the agle determied or a give requecy by Equatio (8) Sell s law ca be used to determie the grazig agle at the surace. I other cases the ull solutio eeds to be used or this purpose. DEPEDECE OF THE RAZI ALE AT THE SURFACE O THE ICIDET RAZI ALE I Figure, depedecies o the grazig agle at the surace, θ s, o the icidet grazig agle, θ, are show or the combiatios o values o the wid speed ad requecy marked by circles i Figure. The letters ear the circles i Figure poit to the correspodig graphs i Figure. Figure D correspods to the poit i Figure below the gree lie. It is clear that, i this case, the ull solutio ad the solutio or the case o o reractio are very close, which meas that, as expected, the iluece o bubbles o soud propagatio ca be eglected at this wid speed ad requecy combiatio. E Australia Acoustical Society

6 Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia A) W =. m/s, = 9. khz E) s, s, W =. m/s, = 9. khz B) s,..., W =. m/s, =. khz F) s,..., 8 W =. m/s, =. khz...,..., C) W =. m/s, =. khz ) W =. m/s, =. khz 8 s, s, D)..., W =. m/s, =. khz H)..., W =. m/s, =.8 khz 8 s, s,...,..., Figure. Depedece o the grazig agle at the surace o the icidet grazig agle below the surace layer. Red lie ull solutio, top ad bottom cya lies approximate solutios with two ad oe terms i the series respectively, gree lie o reractio, blue lie Sell s law. The letter otatios correspod to the poits i Figure. Australia Acoustical Society

7 Proceedigs o Acoustics Victor Harbor 7- ovember, Victor Harbor, Australia The poits C ad H i Figure lie o the boudary o the area o validity o the o reractio case. It ca be see i the correspodig graphs i Figure that the gree lie or the o reractio case is slightly below the red lie or the ull solutio. At the same time, it ca be see that the approximate solutios are valid, as the red lie lies betwee the two gree straight lies or the approximate solutios. The same is applied also to Figure B. The poit i Figure is o the boudary o the area o validity o the approximate solutios. I the correspodig Figure, the curve or the ull solutio lies very close to the lie or the approximate solutio with two terms i the series (the top gree lie). The poit F i Figure lies i the area where the approximate solutio is ot valid. This is clearly see i the correspodig Figure F. The maximum o the ull solutio curve is caused by the proximity o the requecy to oe o the resoaces described by Zioviev et al. (). The poit A i Figure is located above the lie which marks the area o validity o the Sell s law at θ = º. The correspodig igure A shows that, ideed, the ull solutio coicides with the Sell s law at θ > º. Similar coicidece betwee the ull solutio ad the Sell s law ca be observed i Figure E at θ >.º. khz', J. o Acoust. Soc. Amer., vol. 8, o., pp. -. Joes, AD, Duca, AJ, Bartel, DW, Zioviev, A & Maggi, A, 'Recet Developmets i Modellig Acoustic Relectio Loss at the Rough Ocea Surace', Proceedigs o the Acoustics, old Coast, Australia. Brekhovskikh, LM 9, Waves i layered media, Academic Press, ew York. Hall, MV 989, 'A comprehesive model o wid-geerated bubbles i the ocea ad predictios o the eects o soud propagatio at requecies up to khz', J. o Acoust. Soc. Amer., vol. 8, o., pp. -7. Keier, RS, ovarii, JC & orto, V 99, 'The impact o the backgroud bubble layer o reverberatio-derived scatterig stregths i the low to moderate requecy rage', J. o Acoust. Soc. Amer., vol. 97, o., pp. 7-. Lito, CM 998, 'The ree's uctio or the twodimesioal Helmholtz equatio i periodic domai', J. Eg. Math., vol., pp Zioviev, A, Bartel, DW & Joes, AD, 'A Ivestigatio o Plae Wave Propagatio through a Layer with High Soud Speed radiet', Proceedigs o the ACOUSTICS, Frematle, Australia. Overall, the graphs depicted i Figure coirm the criteria o validity determied by Equatios (), () ad (8). COCLUSIOS I this paper, a exact solutio or plae wave propagatio i a bubbly surace layer is simpliied usig assumptios that some parameters characterisig the layer are small. A iiite series preset i the equatio or the acoustic pressure i the layer is trasormed usig Kummer s techique to improve its covergece. Approximate solutios are obtaied or oe ad two terms i the series. It is show that the ull solutio lies betwee the two approximate solutios i the uderlyig assumptios are valid. Areas o validity with respect to the requecy, the wid speed, the soud speed, ad the icidet grazig agle are obtaied or the approximate solutios as well as or the Sell s law. A criterio o validity is also obtaied or the case where o sigiicat reractio occurs i the layer. It is show that the parameters characterisig the layer ca be obtaied by meas o polyomial curve ittig o the third order with oly the wid speed as a iput parameter. Depedecies o the grazig agle at the surace o the icidet grazig agle at the bottom o the layer are calculated or several values o the wid speed ad requecy. The obtaied depedecies coirm the criteria o validity or the obtaied approximate solutios. Overall, the results o this paper demostrate that, whe a bubbly layer with high soud speed gradiet is preset ear the ocea surace, the grazig agle at the surace ca be predicted by meas o a ew simple equatios, i the parameters o propagatio satisy the obtaied criteria. REFERECES Aislie, MA, 'Eect o wid-geerated bubbles o ixed rage acoustic atteuatio i shallow water at Australia Acoustical Society 7