EXISTENCE OF TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER NONLINEAR ACOUSTIC WAVE EQUATION Min Chen Department of Mathematics Purdue Uniersity 150 N. Uniersity Street 47907-067 chen@math.purdue.edu Monica Torres Department of Mathematics Purdue Uniersity 150 N. Uniersity Street 47907-067 torres@math.purdue.edu Timothy Walsh 1 Sandia National Laboratories PO Box 5800 Albuquerque NM 87185 tfwalsh@sandia.go Abstract. In this paper we present an analytical study of a high-order acoustic wae equation in one dimension. We 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. Keywords: nonlinear acoustic waes traeling wae solutions 1. Background Traeling wae solutions also known as 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. [14 15 16] demonstrated the existence of acoustic solitary waes in an air-filled tube containing a periodic array of Helmholtz resonators. In [14 15] the problem was studied theoretically and then in [16] the results were confirmed by a set of laboratory experiments. In another series of papers Jordan studied diffusie soliton solutions to Kuznetso s equation which models weakly-nonlinear acoustic wae propagation in [5] and then in [6 7] applied similar analysis techniques to the problem of 1 Sandia is a multiprogram laboratory operated by Sandia Corporation a Lockheed Martin Company for the United States Department of Energy DE-AC04-94AL85000). 1
TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION traeling wae solutions in nonlinear iscoelastic media. The goerning equations in the iscoelastic case were similar to the acoustic wae equation. Classical theory on nonlinear acoustics as gien in reference [4] gies the speed of wae propagation in a lossless polyatomic gas as 1) c = c 0 + βu where c 0 is the speed of sound in a linear fluid β is a parameter of nonlinearity and u is the particle elocity of the fluid. Since u aries with intensity of the wae equation 1) gies a speed of sound that is nonconstant and will lead to shock formation since the areas of higher intensity in a wae will propagate with a faster speed than those of lower intensity. Equation 1) shows that traeling wae solutions in which the waeform does not change shape during propagation) are not possible in a nonlinear lossless fluid 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 paper we extend recent results by Jordan [5] 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 [13 4] 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. 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 when nonlinear and dissipatie terms are included and whether these solutions can exist in the presence of these extra terms. 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 0φ 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 3) φx ± t) where = c 0 is the wae speed. In 1971 Kuznetso [8] deried a nonlinear acoustic wae equation that extended the linear wae equation to include dissipatie and nonlinear effects. The equation takes the form 4) c 0φ xx φ tt + νγ)φ txx = [ φ x ) + 1 ] t c β 1)φ t ) 0 where β is the coefficient of nonlinearity γ = Cp C is the ratio of specific heats and 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
TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION 3 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 polyatomic gases [13 4]. This equation uses the exact 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 Soderholm [13] is as follows 5) c 0φ xx φ tt + νγ)φ txx = φ x ) ) t + 1 φ xφ x ) ) x + γ 1)[φ t + 1 φ x) ]φ xx where for gases γ and β are related by β = γ+1. For liquids there is no γ and thus in that case β has no relation to γ. We note that this equation is a generalization of the exact relation for lossless gases gien by equation 3.6 in [4] 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 [17] and was followed on by Jordan [5]. Defining a characteristic flow speed V and characteristic length scale L we can define a nondimensional elocity potential as Φ = φ V L. We also define the nondimensional time T = c 0t L and nondimensional position X = x L. Then the following relations can be deried between the first time and spatial deriaties 6) φ t = V c 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 7) φ xx = V L Φ XX φ tt = V c 0 L Φ T T φ x ) ) t = V c 0 L Φ X) ) T φ t ) ) t = V c 3 0 L Φ T ) ) T φ xxt = V c 0 L Φ XXT. Using the results from equations 6) and 7) substituting into the original wae equations ) 4) and 5) we obtain the following nondimensional equations 8) Φ T T Φ XX = 0 9) Φ XX Φ T T + 1 Re Φ XXT = ɛ T [ Φ X ) + β 1) Φ T ) ]
4 TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION 10) Φ XX Φ T T + 1 Re Φ XXT = ɛ[φ X ) ) T +γ 1)Φ T Φ XX ] +ɛ [ Φ X ) Φ XX + ] γ 1) Φ X ) Φ XX where ɛ = V c 0 is the acoustic Mach number and Re = c 0L νγ) is the Reynold s number. Equations 8) 9) and 10) represent the nondimensional forms of the linear Kuznetso and HOAWE wae equation respectiely. Deriations of the Kuznetso and HOAWE themseles based on the physics can be found in references [4 9 13]. We note that the linear wae equation does not represent nonlinear effects at all 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 [5] 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 seeral bifurcations 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 [5] exceeded its inherent limitations. For example Makaro [9] suggests that the Kuznetso equation is only applicable for ɛ < 0.1. 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 [5]. In the limit of small Mach numbers we will show that the traeling wae speeds determined from the HOAWE equation are identical to those obtained in [5] for the Kuznetso equation. This makes sense since the two equations model the same physics for small ɛ.. Existence of traeling waes solutions for HOAWE In this section we show the existence of traeling wae solutions to the HOAWE equation 10). Our results depend on the alue of γ which is in the range of 1.1 to 1.7 for most monoatomic and polyatomic gases. We seek traeling wae solutions to equation 10) which take the form 11) 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 equation 11) into 10) we obtain 1) Φ Φ Re Φ = ε[ Φ Φ γ 1)Φ Φ ] We make the following obserations Φ Φ = 1 d dξ Φ ) Φ = 1 d 3 dξ [ +ε Φ ) Φ + Φ ) Φ ) 3. γ 1) Φ ) ] Φ.
TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION 5 This leads to the equation 13) 1 )Φ [ ) Re Φ + ε ] ξ Φ ) ε [ ) 6 Also we note that an integration in 13) can be performed in ξ with 14) ξ 0 Φ dξ = Φξ) Φ0) ξ Φ ) 3 ] = 0. and similarly for Φ and Φ. Collecting all integration constants in the constant c on the right hand side of 13) we obtain 15) 1 )Φ ε) Re Φ + Φ ) ε ) Φ ) 3 = c. 6 If we make the substitution w = Φ and multiply through by Re we obtain 16) w Re1 ) εre) w w + ε Re) w 3 = c. 6 Equation 16) is an Abel equation of the first kind [10]. It is a generalization of the Ricatti equation see 41) 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 equation 13). These terms are not present in Kuznetso s equation. By denoting 17) equation 16) becomes a 1 := Re1 ) εre) a := > 0 a 3 := ε Re) < 0 6 18) w = a 1 w + a w + a 3 w 3 + c = pw). In order to show the existence of solutions to Abel s equation 16) we prescribe one of the three roots of the cubic polynomial pw) which we denote as w 1. Let 19) h γε w 1 ) := 3γ 5) + ε)w 1 + 8 ε )w 1 the following theorem follows. Theorem.1. Let ε > 0 be the Mach number and γ > 1. If w 1 are such that h γε w 1 ) > 0 then there are one or two bounded traeling wae solutions of 16) with phase elocity. In particular this is true for any > 0 and 0) 1 ε 4 γ 1) + 8 ) < w 1 < 1 ε + ) 4 γ 1) + 8. Proof. Let w be the solution of 16) which approaches to w 1 at or ). We compute c = a 1 w 1 a w 1 a 3 w 3 1
6 TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION and hence 18) becomes where w =a 1 w w 1 ) + a w w 1) + a 3 w 3 w 3 1) =w w 1 )[a 1 + a w + w 1 ) + a 3 w + ww 1 + w 1)] =w w 1 )gw) 1) gw) = a 3 w + a 3 w 1 + a )w + a 1 + a w 1 + a 3 w 1. If the discriminant of this quadratic form is positie where if follows that := a 3 w 1 + a ) 4a 3 a 1 + a w 1 + a 3 w 1) = a 3w 1 + a a 3 w 1 4a 3 a 1 + a w 1 + a 3 w 1) + a = a 3 [4a 1 + a w 1 + 3a 3 w 1] + a ) w = a 3 w w 1 )w w )w w 3 ) = pw) where w w 3 are the real roots of 1). Thus the cubic polynomial pw) can hae three or two different real roots. In the first case there exist two different bounded solutions of 18) and in the second case w = w 1 or w 3 = w 1 meaning that w 1 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 1 < w 3. Since the constant functions w 1 w and w 3 sole the equation w = pw) the theory of ODE [] implies that there exist two different bounded solutions w and w of 18). Since a 3 < 0 we hae w = pw) < 0 if w < w < w 1 and hence lim wξ) = w 1 and ξ Also w = pw) > 0 if w 1 < w < w 3 which yields lim wξ) = w 1 and ξ lim wξ) = w. ξ + lim wξ) = w 3. ξ + In the second case by relabeling the roots if necessary we can write w < w 1. If w = a 3 w w 1 ) w w ) then w < 0 for w < w < w 1 and the only bounded solution w satisfies lim ξ wξ) = w 1 and lim ξ + wξ) = w. If w = a 3 w w 1 )w w ) then w > 0 for w < w < w 1 and w satisfies lim ξ wξ) = w and lim ξ + wξ) = w 1. If = 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 18) while in the second case we hae again only one bounded solution w. We now proceed to characterize the condition > 0. Since [ Re = ε ) 41 ) + ε)w 1 w ε ] ) 1 + ε ) 6 4
TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION 7 we hae 6 [ ] Re = ε ) 41 ) + ε)w 1 ε ) w1 + 3 ε ) [ ] 3 = ε ) ) + 41 ) + ε)w 1 w1ε ) [ = ε ) 4 5 + 3 ] γ + ε)w 1 w1ε ). Hence 3) 1 Re = ε )[8 5 + 3γ + ε)w 1 ε )w1] = ε ) [ 3γ 5) + ε)w 1 + 8 ε )w1 ] = ε )h γε w 1 ). From 3) it follows that > 0 if that is h γε w 1 ) = 3γ 5) + ε)w 1 + 8 ε )w 1 > 0; 4) ε w 1 εw 1 If we define θ = εw 1 then > 0 is true if where θ 1 are the roots of 8 + 3γ 5) θ 1 < θ = εw 1 < θ < 0. θ θ 8 + 3γ 5) = 0. Using that γ > 1 and writing θ 1 explicitly we obtain θ 1 = ± 4 + 48+3γ 5) ) γ+1 = ± + 8 + 3γ 5) ). 4 = ± γ 1) + 8 which proes the theorem. From the proof of Theorem.1 we see that it is also possible to prescribe two roots of the polynomial pw) say w 1 and w and sole for the alues w 1 w ε). One can then test the existence of non-triial traeling wae solutions with Theorem.1. Let w = Φ x t) be a solution of 16) obtained in this way with 5) lim ξ wξ) = w 1 and lim wξ) = w. ξ
8 TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION Thus from 16) and 5) we obtain Re1 ) εre) w 1 w1 + ε Re) w1 3 = 6) 6 Re1 ) εre) w w + ε Re) w 3 6. This expression can be simplified as follows 7) 1 ε) w 1 w ) w1 w ) + ε ) w1 3 w 3 6 ) = 0. Since w 1 w w 1 = w implies constant solution) we can factor the term w 1 w out of the preious expression. Upon doing this and multiplying through by we arrie at 8) 1 ε) ) + w 1 + w ) ε 6 )w 1 + w 1 w + w) = 0. Multiplying through by 1 we obtain 9) ε) w 1 + w ) + ε 6 )w 1 + w 1 w + w) 1 = 0. Therefore we can compute from the equation ε) = w 1 + w ) 4 30) ± 1 ε ) w 1 + w ) 4 3 ε )w1 + w 1w + w ) + 4. It is interesting to note that the solutions for are independent of the acoustic Reynolds number Re. Since the HOAWE is a generalization of Kusnetso s equation we should expect the traeling wae elocity to reduce to the one obtained in [5] in the limit of small ɛ. Indeed using that 1 + x 1 + 1 x for small x in the preious formula for and choosing the positie sign we obtain: ε) 31) 1 + w 1 + w ) + Oε ) 4 which shows that when ε 0 our wae speed w 1 w ) agrees with the wae speed w 1 w ) of Kusnetso s equation gien in [5] equation 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: 3) δɛ) := aɛ + 4 where 33) ) a = w 1 + w ) 4 3 )w 1 + w 1 w + w) [ ) =w 1 + w ) ) w 1 + w 1 w + w ) )] 4 3 w 1 + w ). When a > 0 there is always at least one positie solution for any alue of ɛ. When a < 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 TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION 9 34) ε c = a. We note that if γ < 5 3 and w 1w < 0 then a < 0 because w 1 +w 1w +w w 1 +w > 1 and a < [ ] ) w 1 + w ) ) γ+1) 4 3 < 0. In specific if a < 0 which is the case when γ < 5 3 and w 1 w < 0 there is not solution when ε > ε c. 3. Traeling wae solutions of Kuznetso s equation We also seek traeling wae solutions to equation 9) which take the form 35) ΦX T ) = ΦX T ) = Φξ) and proceed as in to obtain the equation 36) w b 1 w b w c = 0 where c is the constant of integration as in 14) and 15) wξ) = Φ ξ) and 37) 38) If w is a solution of 36) by defining b 1 := Re 1 b := εre[1 + β 1)]. 39) w 1 := lim wξ) and w := lim wξ) ξ ξ + and assuming lim ξ ± w = 0 then from 36) it follows that 40) c = b 1 w 1 b w 1 = b 1 w b w. Therefore 36) becomes 41) w b 1 w b w + b 1 w 1 + b w 1 = 0. For solutions of 41) of the form we hae the following theorem. wξ) = A + B tanhλξ) Theorem 3.1. If the following equations hold 4) w 1 + w = b 1 b λ = b 43) w 1 w ) for some constants w 1 w λ b 1 and b then 44) wξ) = b 1 b λ b tanhλξ) satisfies equation 41) and 45) w 1 := lim ξ wξ) and w := lim ξ wξ)
10 TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION Proof. We look for solutions of the form w = A + B tanhλξ) and thus 46) w = Bλ[1 tanh λξ)] 47) w = A + AB tanhλξ) + B tanh λξ). If we substitute 46) and 47) into 41) and use the notation y = tanhλξ) we obtain Bλ Bλy b 1 A + By) b A + ABy + B y ) + b 1 w 1 + b w1 = 0. From this and 40) we get the system of equations: 48) Bλ b 1 A b A + b 1 w 1 + b w1 = 0 49) b 1 w 1 + b w1 = b 1 w + b w 50) b 1 B b AB = 0 51) Bλ + b B = 0 and we add to the system the two equations 5) 53) A = 1 w 1 + w ) B = 1 w 1 w ). Since we want B λ 0 then we must hae w 1 w. Using 5) and 53) equations 50) and 51) simplify to: 54) w 1 + w = b 1 b λ = b 55) w 1 w ). We note that the equation 49) reduces to exactly equation 54) and thus it does not add any information. Thus since λ = b B A = b 1 b and B = b 1 b w 1 we only need to show that equation 48) is satisfied: Bλ b 1 A b A + b 1 w 1 + b w1 = b B b 1 A b A + b 1 w 1 + b w1 [ = b w1 + b ] 1 w 1 + b 1 b 4b which implies our desired result. b 1 b 1 + b 1 b b 4b + b 1 w 1 + b w1 = b w1 b 1 w 1 b 1 + b 1 b 1 + b 1 w 1 + b w1 4b b 4b = 0 Remark 3.1. From the conditions 4) and 43) we conclude that 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 1 and w. ii) If we look for a solution with prescribed behaior at or + ); that is if we prescribe w 1 or w ) then there is a solution for any ε γ).
TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER ACOUSTIC WAVE EQUATION 11 Remark 3.. If we look for a solution with prescribed behaior both at and + ; that is if we prescribe both w 1 and w then the possible alues of and ε for which there is solution are restricted. This situation was analyzed in [5] 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 paper 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 [5]. This makes sense since the high-order wae equation is a generalization of the Kuznetso equation considered in [5]. Acknowledgments. Monica Torres s research was supported in part by the National Science Foundation under grant DMS-0540869. References [1] R. T. Beyer. Nonlinear Acoustics. Department of the Nay Sea Systems Command 1974. [] W. Boyce and R. DiPrima. Elementary Differential Equations. Wiley 005. [3] B. O. Enflo and C. M. Hedberg. Theory of Nonlinear Acoustics in Fluids. Kluwer Academic Publishers 00. [4] M. F. Hamilton and Eds. D. T. Blackstock. Nonlinear Acoustics. Academic Press 1998. [5] P. M. Jordan. An analytical study of Kuznetso s equation: Diffusie solitons shock formation and solution bifurcation. Physics Letters A 36:77 84 004. [6] P. M. Jordan and A. Puri. A note on traeling wae solutions for a class of nonlinear iscoelastic media. Physics Letters A 335:150 156 005. [7] P. M. Jordan and A. Puri. Addendum to: A note on traeling wae solutions for a class of nonlinear iscoelastic media. Physics Letters A 361:59 533 007. [8] V. P. Kuznetso. Equations of nonlinear acoustics. So. Phys. Acoust. 16:467 470 1971. [9] S. Makaro and M. Ochmann. Nonlinear and thermoiscous phenomena in acoustics part ii. Acustica 83):197 1997. [10] G. M. Murphy. Ordinary Differential Equations and Their Solutions. D. Van Nostrand Company Inc. 1960. [11] K. Naugolnykh and L. Ostrosky. Nonlinear Wae Processes in Acoustics. Cambridge Uniersity Press 1998. [1] Allan D. Pierce. Acoustics. McGraw-Hill New York 1981. [13] L. H. Soderholm. On the Kuznetso equation and higher order nonlinear acoustics equations. Proc. 15th International Symposium on Nonlinear Acoustics Gngen 541):133 136 000. [14] N. Sugimoto. Propagation of nonlinear acoustic waes in a tunnel with an array of helmholtz resonators. J. Fluid Mech 44:55 7 199. [15] N. Sugimoto M. Masuda J. Ohno and D. Motoi. Acoustic solitary waes in a tunnel with an array of helmholtz resonators. J. Acoust. Soc. Am. 99:1971 1976 1996. [16] N. Sugimoto M. Masuda J. Ohno and D. Motoi. Experimental demonstration of generation and propagation of acoustic solitary waes in an air-filled tube. Physical Reiew Letters 83:4053 4056 1999. [17] J. Wojcik. Conseration of energy and absorption in acoustic fields for linear and nonlinear propagation. J. Acoust. Soc. Am. 104:654 663 1998.