Physics Letters A 373 009) 037 043 Contents lists aailable at ScienceDirect Physics Letters A www.elseier.com/locate/pla Existence of traeling wae solutions of a high-order nonlinear acoustic wae equation Min Chen a Monica Torres a TimothyWalsh b a Department of Mathematics Purdue Uniersity 50 N. Uniersity Street IN 47907-067 United States b Sandia National Laboratories PO Box 5800 Albuquerque NM 8785 United States article info abstract Article history: Receied 9 July 008 Receied in reised form 9 Noember 008 Accepted January 009 Aailable online 7 January 009 Communicated by A.R. Bishop In this Letter we present an analytical study of a high-order acoustic wae equation in one dimension and reformulate a preiously gien equation in terms of an expansion of the acoustic Mach number. We search for non-triial traeling wae solutions to this equation and also discuss the accuracy of acoustic wae equations in terms of the range of Mach numbers for which they are alid. 009 Published by Elseier B.V. Keywords: Nonlinear acoustic waes Traeling wae solutions. Background Traeling wae solutions in the form of solitons hae been studied in detail for nonlinear wae equations of the KdV type as well as in other areas of physics. In the case of acoustic wae equations these solutions hae receied considerably less attention. In a recent series of papers Sugimoto et al. 3] demonstrated the existence of acoustic solitary waes in an air-filled tube containing a periodic array of Helmholtz resonators. In ] the problem was studied theoretically and then in 3] the results were confirmed by a set of laboratory experiments. In another series of papers Jordan 4] studied diffusie soliton solutions to Kuznetso s equation which models weaklynonlinear acoustic wae propagation and then Jordan and Puri 56] applied similar analysis techniques to the problem of traeling wae solutions in nonlinear iscoelastic media. The goerning equations in the iscoelastic case are similar to the acoustic wae equation. In another related work Jordan 7] studied finite amplitude waes in a porous medium. Rasmussen et al. 8] deried an alternatie nonlinear wae equation and then deried an analytical traeling wae solution that allowed for studying front interaction. The classical theory of nonlinear acoustics as gien in Refs. 97 0] gies the speed of wae propagation in an adiabatic fluid as c = c 0 ± γ )u ) where c 0 is the small-signal speed of sound in a linear fluid γ is the ratio of specific heats and u is the particle elocity of the fluid. We note that u aries with both space and time. Since u in Eq. ) aries with amplitude of the wae the areas of higher amplitude in a wae will propagate with a faster speed than those of lower amplitude. This will lead to shock formation. Eq. ) shows that initially smooth waes with smooth input signals) in a nonlinear lossless fluid will eentually steepen to form shocks and thus cannot propagate as traeling waes since the speed of the wae always depends on position in the waeform. The interesting question is then to consider the lossy terms in the equations of motion along with the nonlinear terms and to assess if traeling wae solutions are possible in the presence of both lossy and nonlinear terms. In this Letter we extend recent results by Jordan 4] who studied traeling wae solutions to the Kuznetso equation which models nonlinear acoustic waes in lossy fluids up to second order. In our approach we use a higher-order equation 9 ] which is alid up to higher acoustic Mach numbers than Kuznetso s equation. Since the speed of the traeling wae depends on the acoustic Mach number this high-order equation allows for a more accurate assessment of traeling wae elocities. * Corresponding author. E-mail addresses: chen@math.purdue.edu M. Chen) torres@math.purdue.edu M. Torres) tfwalsh@sandia.go T. Walsh). Sandia is a multiprogram laboratory operated by Sandia Corporation a Lockheed Martin Company for the United States Department of Energy DE-AC04-94AL85000). 0375-960/$ see front matter 009 Published by Elseier B.V. doi:0.06/j.physleta.009.0.04
038 M. Chen et al. / Physics Letters A 373 009) 037 043 In the case of linear acoustic theory it is easy to see that traeling wae solutions exist since there are no dissipatie or nonlinear effects that would distort the waes. The more interesting case is whether these solutions can exist when nonlinear and dissipatie terms are included. In most cases it depends on the physical constants inoled as we will show here. It is instructie to begin the discussion with the linear wae equation which in one dimension is gien as c 0 φ xx φ tt = 0 ) where φ is the elocity potential and c 0 is the linear speed of sound. Traeling waes exist for this equation and are of the form gien by the d Alembert solution φx ± t) where = c 0 is the wae speed. In 97 Kuznetso ] deried a nonlinear acoustic wae equation that extended the linear wae equation to include dissipatie and nonlinear effects. The equation takes the form c 0 φ xx φ tt + νγ )φ txx = φ x ) + ] β )φ t c t ) 4) 0 3) where β is the coefficient of nonlinearity γ = C p is the ratio of specific heats and C νγ ) is the diffusiity of sound. We note that the first two terms of Kuznetso s equation are the same as the linear wae equation ). The additional terms account for dissipation and nonlinear effects. Howeer Kuznetso s equation is only a second order equation in terms of the nonlinearities which means that it is only alid for certain ranges of acoustic Mach numbers. More on this will be gien later in this section when we re-write this equation in a nondimensional form. A higher order acoustic wae equation HOAWE) exists for gases 9 ]. This equation uses a more accurate equation of state rather than the Taylor series expansion used in Kuznetso s equation. Consequently it is alid for larger alues of acoustic Mach number and thus represents a more accurate model of acoustic wae propagation than Kuznetso s equation. The equation as gien by Söderholm 0] is as follows c 0 φ xx φ tt + νγ )φ txx = φ x ) ) + t φ x φx ) ) + γ ) φ x t + φ x) ]φ xx 5) where for gases γ and β are related by β = γ +. We note that this equation is a generalization of the relation for lossless gases gien by Eq. 3.6) in 9] the difference being the dissipatie term νγ )φ txx. Also as with Kuznetso s equation this equation includes the linear wae equation as a special case. Although it is clear that the two wae equations 4) and 5) reduce to the linear wae equation ) when nonlinear and dissipatie effects are neglected the arious physical constants make the relatie magnitudes of the terms difficult to interpret. Hence we show here how all three can be written in a nondimensional form thus facilitating their comparison and analysis. The dimensional analysis procedure follows one that was originally gien by Wojcik 3] andwasfollowedonbyjordan4]. Defining a characteristic flow speed V and characteristic length scale L we can define a nondimensional elocity potential as Φ = φ. We also define the nondimensional time VL T = c 0t L and nondimensional position X = x. Then the following relations can be deried between the first time and spatial deriaties L φ t = Vc 0 Φ T φ x = V Φ X. Using these relations we can also derie the following relations for the higher deriaties and nonlinear terms φ xx = V L Φ XX φ tt = Vc 0 L Φ TT φx ) ) = V c 0 ΦX ) ) t L φt ) ) = V c 3 0 ΦT ) ) T t L φ T xxt = Vc 0 L Φ XXT. 7) Substituting Eqs. 6) and 7) into the original wae equations ) 4) and 5) we obtain the following nondimensional equations Φ TT Φ XX = 0 8) Φ XX Φ TT + Re Φ XXT = ɛ ΦX ) + β )Φ T ) ] 9) T Φ XX Φ TT + Re Φ XXT = ɛ Φ X ) ) + γ )Φ ] T T Φ XX + ɛ γ Φ X ) ) Φ XX + Φ X ) Φ XX ] 0) where ɛ = V is the acoustic Mach number and Re c 0 = c 0 L νγ ) is the Reynold s number. Eqs. 8) 9) and 0) represent the nondimensional forms of the linear Kuznetso and HOAWE wae equation respectiely. Deriations of the Kuznetso and HOAWE equations from fundamental principles can be found in Refs. 9 4]. We note that the linear wae equation neglects all nonlinear effects the Kuznetso equation represents nonlinear effects to the first power in ɛ and the HOAWE includes both linear and quadratic terms in ɛ. A recent study 4] focused on searching for traeling wae solutions to the Kuznetso equation. Although traeling wae solutions were shown to exist the wae speed depended on the acoustic Mach number and in fact had a bifurcation depending on the physical constants. No traeling wae solutions were possible aboe a certain critical acoustic Mach number. Howeer since the Kuznetso equation itself is restricted to small alues of the acoustic Mach number it was not clear if the critical Mach number obtained in 4] exceeded its inherent limitations. For example Makaro and Ochmann 4] suggest that the Kuznetso equation is only applicable for ɛ < 0.. In our approach we will search for traeling wae solutions to the HOAWE equation. Since this equation is alid for larger alues of the acoustic Mach number it will allow for traeling wae solutions with a wider range of wae speeds than was obtained in 4]. Inthe limit of small Mach numbers we will show that the traeling wae speeds determined from the HOAWE equation are identical to those obtained in 4] for the Kuznetso equation. This makes sense since the two equations model the same physics for small ɛ. 6)
M. Chen et al. / Physics Letters A 373 009) 037 043 039. Existence of traeling waes solutions for HOAWE In this section we show the existence of traeling wae solutions to the HOAWE Eq. 0). Our results depend on the alue of γ which is in the range of. to.7 for most monoatomic and polyatomic gases. We seek traeling wae solutions to Eq. 0) which take the form ΦX T ) = ΦX T) = Φξ) ) where ξ = X T and > 0 is the speed of the wae. At this point is unknown since it is not known a priori what the speed of the traeling waes will be. Substituting Eq. ) into 0) weobtain Φ Φ Re Φ = ε Φ Φ γ )Φ Φ ] + ε Φ ) Φ γ ) + Φ ) Φ ]. ) The following relations Φ Φ = d dξ Φ ) Φ ) Φ = d 3 dξ Φ ) 3 lead to the equation ) Φ Re Φ + ε ) ] ) ξ Φ ) ε 6 Also we note that an integration in 3) can be performed in ξ with ξ 0 Φ dξ = Φξ) Φ0) ξ Φ ) 3 ] = 0. 3) and similarly for Φ and Φ. Collecting all integration constants in the constant c on the right-hand side of 3) we obtain ) Φ ε) Re Φ + Φ ) ε ) Φ ) 3 = c. 5) 6 If we make the substitution w = Φ and multiply through by Re weobtain w Re ) w ε Re) w + ε Re) w 3 = c. 6) 6 Eq. 6) is an Abel equation of the first kind 5]. It is a generalization of the Ricatti equation see 4) in the next section) which appears when searching for traeling wae solutions of Kuznetso s equation. In this case the w 3 term is a direct consequence of the terms of the type Φ) 3 in Eq. 3). These terms are not present in Kuznetso s equation. By denoting a := Re ) ε Re) a := > 0 a 3 := ε Re) < 0 7) 6 Eq. 6) becomes w = a w + a w + a 3 w 3 + c = pw). In order to show the existence of solutions to Abel s equation 6) we denote one of the three roots of the cubic polynomial pw) as w. Gien w wedefineh γ ε w ) as h γ ε w ) := 3γ 5) + ε)w + 8 ε )w. 9) The following theorem then holds. 4) 8) Theorem.. Let ε > 0 be the Mach number and γ >. If w are such that h γ ε w )>0 then there are one or two bounded traeling wae solutions of 6) with denoting the speed of the traeling wae. In particular this is true for any > 0 and ) 4 γ ) + 8 < w < ) 4 + γ ) + 8. 0) ε ε Proof. Let w be the solution of 6) which approaches to w at or ). We compute c = a w a w a 3 w 3 and hence 8) becomes where w = a w w ) + a w w ) + a3 w 3 w 3 ) = w w ) a + a w + w ) + a 3 w + ww + w )] = w w )gw) gw) = a 3 w + a 3 w + a )w + a + a w + a 3 w. )
040 M. Chen et al. / Physics Letters A 373 009) 037 043 If the discriminant of this quadratic form Δ is positie where ) Δ := a 3 w + a ) 4a 3 a + a w + a 3 w = a 3 w + ) ] a a 3 w 4a 3 a + a w + a 3 w + a 3 = a 4a + a w + 3a 3 w + a it follows that w = a 3 w w )w w )w w 3 ) = pw) where w w 3 are the real roots of ). Thus the cubic polynomial pw) can hae three or two different real roots. In the first case there exist two different bounded solutions of 8) and in the second case w = w or w 3 = w meaning that w is also a root of gw) there is only one bounded solution w. In the first case by relabeling the roots if necessary we can write w < w < w 3. Since the constant functions w w and w 3 sole the equation w = pw) thetheoryofode6] implies that there exist two different bounded solutions w and w of 8). Sincea 3 < 0 we hae w = pw)<0ifw < w < w and hence lim ξ wξ) = w and lim ξ + wξ) = w. Also w = pw)>0ifw < w < w 3 which yields lim ξ wξ) = w and lim ξ + wξ) = w 3. In the second case by relabeling the roots if necessary we can write w < w. If w = a 3 w w ) w w ) then w < 0 for w < w < w and the only bounded solution w satisfies lim ξ wξ) = w and lim ξ + wξ) = w.ifw = a 3 w w )w w ) then w > 0forw < w < w and w satisfies lim ξ wξ) = w and lim ξ + wξ) = w. If Δ = 0 then the cubic polynomial pw) can hae three equal roots or only two different real roots. In the first case there is no bounded solution to 8) while in the second case we hae again only one bounded solution w. We now proceed to characterize the condition Δ>0. Since ) Δ Re = ε ) 4 ) 6 ε + ε)w w ] ) + ε ) 4 we hae 6 Δ = Re ε ) 4 ) ] + ε)w ε ) w + 3 ε ) 3 = ε ) ) + 4 ) ] γ + ε)w w + ) ε = ε ) 4 5 + 3 ] γ γ + ε)w w + ) ε. Hence Re Δ = ε ) ] 8 5 + 3γ + ε)w ε )w = ε ) ] 3γ 5) + ε)w + 8 ε )w = ε )h γ ε w ). 3) From 3) it follows that Δ>0if that is h γ ε w ) = 3γ 5) + ε)w + 8 ε )w > 0; ε w εw 8 + 3γ 5) < 0. If we define θ = εw thenδ>0istrueif θ <θ= εw <θ 4) where θ are the roots of θ θ 8 + 3γ 5) = 0. Using that γ > and writing θ explicitly we obtain θ = ± 4 + 48+3γ 5) ) γ + = ± + 8 + 3γ 5) ) 4 = ± γ ) + 8 which proes the theorem.
M. Chen et al. / Physics Letters A 373 009) 037 043 04 From the proof of Theorem. we see that it is also possible to prescribe two roots of the polynomial pw) sayw and w and sole for the alues w w ε). One can then test the existence of non-triial traeling wae solutions with Theorem.. We now describe the procedure in detail. Let w = Φ x t) be a solution of 6) obtained in this way with lim wξ) = w and lim wξ) = w. ξ ξ 5) Thus from 6) and 5) we obtain Re ) w ε Re) This expression can be simplified as follows w w ) ε) w + ε Re) 6 w w ) ε ) + 6 w 3 = Re ) w ε Re) w + ε Re) w 3 6. 6) w 3 w3 ) = 0. 7) Since w w w = w implies constant solution) we can factor the term w w out of the preious expression. Upon doing this and multiplying through by we arrie at ) ε) + w + w ) ε γ 6 + ) w + ) w w + w = 0. 8) Multiplying through by we obtain εγ + ) w + w ) + ε γ 6 + ) w + ) w w + w = 0. 9) Therefore we can compute from the equation ε) = w + w ) ± ε ) w + w ) 4 4 3 ε ) w + w w + w) + 4. 30) It is interesting to note that the solutions for are independent of the acoustic Reynolds number Re. Since the HOAWE is a generalization of Kuznetso s equation we should expect the traeling wae elocity to reduce to the one obtained in 4] in the limit of small ɛ. Indeed using that + x + x for small x in the preious formula for and choosing the positie sign we obtain: ε) + w + w ) + O ε ) 3) 4 which shows that when ε 0 our wae speed w w ) agrees with the wae speed w w ) of Kuznetso s equation gien in 4 Eq. 3)] up to the order Oε). We consider the discriminant δ of the quadratic form under the square root in 30) as a function of the Mach number ε: δɛ) := aɛ + 4 3) where ) a = 4 w + w ) ) γ 3 + ) w + w w + w) = w + w ) ) w 4 3 + w w + w w + w ) )]. 33) When a > 0 there is always at least one positie solution for any alue of ɛ. Whena < 0 there is a critical Mach number ɛ c aboe which there are no real-alued elocities corresponding to traeling wae solutions to the HOAWE. This critical Mach number is gien by δε c ) = aε c + 4 = 0; that is ε c = a. 34) We note that if γ < 5 3 and w w < 0thena < 0because w +w w +w > and a <w w +w ) + w ) γ +) ) ] < 0. Specifically if 4 3 a < 0 which is the case when γ < 5 3 and w w < 0 there is no solution when ε > ε c. 3. Traeling wae solutions of Kuznetso s equation We also seek traeling wae solutions to Eq. 9) which take the form ΦX T ) = ΦX T) = Φξ). 35) Proceeding as in Section and letting wξ) = Φ ξ) we obtain the equation w b w b w c = 0 36) where c is the constant of integration as in 4) and 5) and
04 M. Chen et al. / Physics Letters A 373 009) 037 043 b := Re b := ε Re + β ) ]. 37) 38) If w is a solution of 36) by defining w := lim wξ) and w := lim wξ) ξ ξ + 39) and assuming lim ξ ± w = 0 then from 36) it follows that c = b w b w = b w b w. 40) Therefore 36) becomes w b w b w + b w + b w = 0. 4) For solutions of 4) of the form wξ) = A + B tanhλξ) we hae the following theorem. Theorem 3.. If the following equations hold w + w = b b λ = b w w ) for some constants w w λb and b then wξ) = b b λ b tanhλξ) 4) 43) 44) satisfies Eq. 4) and w := lim wξ) and w := lim wξ). ξ ξ 45) Proof. We look for solutions of the form w = A + B tanhλξ) and thus w = Bλ tanh λξ) ] w = A + ABtanhλξ) + B tanh λξ). 46) 47) If we substitute 46) and 47) into 4) and use the notation y = tanhλξ) we obtain Bλ Bλy b A + By) b A + ABy+ B y ) + b w + b w = 0. From this and 40) we get the system of equations: Bλ b A b A + b w + b w = 0 b w + b w = b w + b w b B b AB = 0 Bλ + b B = 0 48) 49) 50) 5) and we add to the system the two equations A = w + w ) B = w w ). 5) 53) Since we want B 0 and λ 0 we must hae w w.using5) and 53) Eqs. 50) and 5) simplify to: w + w = b b λ = b w w ). We note that Eq. 49) reduces to exactly Eq. 54) and thus it does not add any information. Thus since λ = b B A = b and B b = b b w we only need to show that Eq. 48) is satisfied: 54) 55)
M. Chen et al. / Physics Letters A 373 009) 037 043 043 Bλ b A b A + b w + b w = b B b A b A + b w + b w = b w + b = b w b w b + b b + b w + b w 4b b 4b = 0 which implies our desired result. Remark 3.. From the conditions 4) and 43) we conclude that: w + b b 4b ] + b b b b b 4b + b w + b w i) There is a traeling wae solution for any prescribed alues of ε γ λ) but once these parameters are prescribed they imply certain w and w. ii) If we look for a solution with prescribed behaior at or + ); that is if we prescribe w or w ) then there is a solution for any ε γ ). Remark 3.. If we look for a solution with prescribed behaior both at and + ; that is if we prescribe both w and w then the possible alues of and ε for which there is solution are restricted. This situation was analyzed in 4] where solutions to Kuznetso s equation were shown to exist only for Mach numbers less than a critical alue ε c. 4. Conclusions In this Letter we presented an analysis of traeling wae solutions to a high-order acoustic wae equation. We showed that there exist non-triial traeling wae solutions for any wae speed and any alue of the acoustic Mach number. We also showed that if the alues of the solution at ± are prescribed then there is a critical Mach number aboe which there is no traeling wae solutions. Although traeling wae solutions were shown to exist for Kuznetso s equation waes with higher Mach number may not be accurately modeled by this equation. Since the Kuznetso equation itself is restricted to small alues of the acoustic Mach number the authors inoked the high-order acoustic wae equation as a more accurate equation to model nonlinear acoustical waes. We also showed that in the limit of small acoustic Mach number the traeling wae speeds obtained here reduce to those obtained in 4]. This makes sense since the high-order wae equation is a generalization of the Kuznetso equation considered in 4]. Acknowledgements Monica Torres s research was supported in part by the National Science Foundation under grant DMS-0540869. The authors would like to thank the anonymous referee for helpful comments. References ] N. Sugimoto J. Fluid Mech. 44 99) 55. ] N. Sugimoto M. Masuda J. Ohno D. Motoi J. Acoust. Soc. Am. 99 996) 97. 3] N. Sugimoto M. Masuda J. Ohno D. Motoi Phys. Re. Lett. 83 999) 4053. 4] P.M. Jordan Phys. Lett. A 36 004) 77. 5] P.M. Jordan A. Puri Phys. Lett. A 335 005) 50. 6] P.M. Jordan A. Puri Phys. Lett. A 36 007) 59. 7] P.M. Jordan Phys. Lett. A 355 006) 6. 8] A. Rasmussen M. Sorensen Y. Gaididei P. Christainsen arxi: 0806.005 physics.flu-dyn]. 9] M. Hamilton D. Blackstock Eds.) Nonlinear Acoustics Academic Press 998. 0] L.H. Söderholm On the Kuznetso equation and higher order nonlinear acoustics equations in: ISNA 5 5th International Symposium AIP Conference Proceedings ol. 54 000 pp. 33 36. ] L.H. Söderholm Acta Acustica-ACUSTICA 87 00) 9. ] V.P. Kuznetso So. Phys. Acoust. 6 97) 467. 3] J. Wojcik J. Acoust. Soc. Am. 04 998) 654. 4] S. Makaro M. Ochmann Acustica 83 ) 997) 97. 5] G.M. Murphy Ordinary Differential Equations and Their Solutions D. Van Nostrand Company Inc. 960. 6] W. Boyce R. DiPrima Elementary Differential Equations Wiley 005. 7] R.T. Beyer Nonlinear Acoustics Department of the Nay Sea Systems Command 974. 8] B.O. Enflo C.M. Hedberg Theory of Nonlinear Acoustics in Fluids Kluwer Academic Publishers 00. 9] K. Naugolnykh L. Ostrosky Nonlinear Wae Processes in Acoustics Cambridge Uni. Press 998. 0] A.D. Pierce Acoustics McGraw Hill New York 98.