Physics Letters A. Existence of traveling wave solutions of a high-order nonlinear acoustic wave equation
|
|
- Lesley Mason
- 5 years ago
- Views:
Transcription
1 Physics Letters A ) Contents lists aailable at ScienceDirect Physics Letters A 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 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. 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) /$ see front matter 009 Published by Elseier B.V. doi:0.06/j.physleta
2 038 M. Chen et al. / Physics Letters A ) 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)
3 M. Chen et al. / Physics Letters A ) 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 ) ) 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. )
4 040 M. Chen et al. / Physics Letters A ) 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 ) ] γ γ + ε)w w + ) ε. Hence Re Δ = ε ) ] γ + 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 = ± γ 5) ) γ + = ± γ 5) ) 4 = ± γ ) + 8 which proes the theorem.
5 M. Chen et al. / Physics Letters A ) 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 ) 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 ) ε ) w + w w + w) ) 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 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
6 04 M. Chen et al. / Physics Letters A ) 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)
7 M. Chen et al. / Physics Letters A ) 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 The authors would like to thank the anonymous referee for helpful comments. References ] N. Sugimoto J. Fluid Mech ) 55. ] N. Sugimoto M. Masuda J. Ohno D. Motoi J. Acoust. Soc. Am ) 97. 3] N. Sugimoto M. Masuda J. Ohno D. Motoi Phys. Re. Lett ) ] P.M. Jordan Phys. Lett. A ) 77. 5] P.M. Jordan A. Puri Phys. Lett. A ) 50. 6] P.M. Jordan A. Puri Phys. Lett. A ) 59. 7] P.M. Jordan Phys. Lett. A ) 6. 8] A. Rasmussen M. Sorensen Y. Gaididei P. Christainsen arxi: physics.flu-dyn]. 9] M. Hamilton D. Blackstock Eds.) Nonlinear Acoustics Academic Press ] L.H. Söderholm On the Kuznetso equation and higher order nonlinear acoustics equations in: ISNA 5 5th International Symposium AIP Conference Proceedings ol pp ] L.H. Söderholm Acta Acustica-ACUSTICA 87 00) 9. ] V.P. Kuznetso So. Phys. Acoust. 6 97) ] J. Wojcik J. Acoust. Soc. Am ) ] S. Makaro M. Ochmann Acustica 83 ) 997) 97. 5] G.M. Murphy Ordinary Differential Equations and Their Solutions D. Van Nostrand Company Inc ] W. Boyce R. DiPrima Elementary Differential Equations Wiley ] R.T. Beyer Nonlinear Acoustics Department of the Nay Sea Systems Command ] 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 ] A.D. Pierce Acoustics McGraw Hill New York 98.
Min Chen Department of Mathematics Purdue University 150 N. University Street
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
More informationResidual migration in VTI media using anisotropy continuation
Stanford Exploration Project, Report SERGEY, Noember 9, 2000, pages 671?? Residual migration in VTI media using anisotropy continuation Tariq Alkhalifah Sergey Fomel 1 ABSTRACT We introduce anisotropy
More informationLecture #8-6 Waves and Sound 1. Mechanical Waves We have already considered simple harmonic motion, which is an example of periodic motion in time.
Lecture #8-6 Waes and Sound 1. Mechanical Waes We hae already considered simple harmonic motion, which is an example of periodic motion in time. The position of the body is changing with time as a sinusoidal
More informationRELATIVISTIC DOPPLER EFFECT AND VELOCITY TRANSFORMATIONS
Fundamental Journal of Modern Physics ISSN: 49-9768 Vol. 11, Issue 1, 018, Pages 1-1 This paper is aailable online at http://www.frdint.com/ Published online December 11, 017 RELATIVISTIC DOPPLER EFFECT
More informationE : Ground-penetrating radar (GPR)
Geophysics 3 March 009 E : Ground-penetrating radar (GPR) The EM methods in section D use low frequency signals that trael in the Earth by diffusion. These methods can image resistiity of the Earth on
More informationBlow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations
Mathematics Statistics 6: 9-9, 04 DOI: 0.389/ms.04.00604 http://www.hrpub.org Blow up of Solutions for a System of Nonlinear Higher-order Kirchhoff-type Equations Erhan Pişkin Dicle Uniersity, Department
More informationACUSTICA DE MEDIOS DEBILMENTE LINEALES
ACUSTICA DE MEDIOS DEBILMENTE LINEALES PACS: 43.25.+y Ramos Sobrados, Juan Ignacio Escuela de las Ingenierías, Universidad de Málaga. 29071 Málaga (España) E-mail: jirs@lcc.uma.es ABSTRACT We consider
More informationAnalytical and numerical modeling of front propagation and interaction of fronts in nonlinear thermoviscous
Downloaded from orbit.dtu.dk on: Dec 1, 18 Analytical and numerical modeling of front propagation and interaction of fronts in nonlinear thermoviscous Rasmussen, Anders Rønne; Sørensen, Mads Peter; Gaididei,
More informationTransmission lines using a distributed equivalent circuit
Cambridge Uniersity Press 978-1-107-02600-1 - Transmission Lines Equialent Circuits, Electromagnetic Theory, and Photons Part 1 Transmission lines using a distributed equialent circuit in this web serice
More information2008 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies
FINITE DIFFERENCE TIME DOMAIN MODELING OF INFRASOUND PROPAGATION: APPLICATION TO SHADOW ZONE ARRIVALS AND REGIONAL PROPAGATION Catherine de Groot-Hedlin Uniersity of California at San Diego Sponsored by
More informationSection 6: PRISMATIC BEAMS. Beam Theory
Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam
More informationSingle soliton solution to the extended KdV equation over uneven depth
Eur. Phys. J. E 7) : DOI./epje/i7-59-7 Regular Article THE EUROPEAN PHYSICAL JOURNAL E Single soliton solution to the etended KdV equation oer uneen depth George Rowlands, Piotr Rozmej,a, Eryk Infeld 3,
More informationDynamics ( 동역학 ) Ch.2 Motion of Translating Bodies (2.1 & 2.2)
Dynamics ( 동역학 ) Ch. Motion of Translating Bodies (. &.) Motion of Translating Bodies This chapter is usually referred to as Kinematics of Particles. Particles: In dynamics, a particle is a body without
More informationAlgebraic Derivation of the Oscillation Condition of High Q Quartz Crystal Oscillators
Algebraic Deriation of the Oscillation Condition of High Q Quartz Crystal Oscillators NICOLAS RATIER Institut FEMTO ST, CNRS UMR 67 Département LPMO a. de l Obseratoire, 50 Besançon FRANCE nicolas.ratier@femto-st.fr
More informationSUPPLEMENTARY MATERIAL. Authors: Alan A. Stocker (1) and Eero P. Simoncelli (2)
SUPPLEMENTARY MATERIAL Authors: Alan A. Stocker () and Eero P. Simoncelli () Affiliations: () Dept. of Psychology, Uniersity of Pennsylania 34 Walnut Street 33C Philadelphia, PA 94-68 U.S.A. () Howard
More informationMagnetic Fields Part 3: Electromagnetic Induction
Magnetic Fields Part 3: Electromagnetic Induction Last modified: 15/12/2017 Contents Links Electromagnetic Induction Induced EMF Induced Current Induction & Magnetic Flux Magnetic Flux Change in Flux Faraday
More informationv v Downloaded 01/11/16 to Redistribution subject to SEG license or copyright; see Terms of Use at
The pseudo-analytical method: application of pseudo-laplacians to acoustic and acoustic anisotropic wae propagation John T. Etgen* and Serre Brandsberg-Dahl Summary We generalize the pseudo-spectral method
More informationEvaluation of pressure and bulk modulus for alkali halides under high pressure and temperature using different EOS
Journal of the Association of Arab Uniersities for Basic and Applied Sciences (3) 4, 38 45 Uniersity of Bahrain Journal of the Association of Arab Uniersities for Basic and Applied Sciences www.elseier.com/locate/jaaubas
More informationFLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE
FLUID DYNAMICS FLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE S. A. ODEJIDE 1, Y. A. S. AREGBESOLA 1, O. D. MAKINDE 1 Obafemi Awolowo Uniersity, Department of Mathematics, Faculty of Science, Ile-Ife,
More informationSPACE-TIME HOLOMORPHIC TIME-PERIODIC SOLUTIONS OF NAVIER-STOKES EQUATIONS. 1. Introduction We study Navier-Stokes equations in Lagrangean coordinates
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 218, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SPACE-TIME HOLOMORPHIC
More informationOnline Companion to Pricing Services Subject to Congestion: Charge Per-Use Fees or Sell Subscriptions?
Online Companion to Pricing Serices Subject to Congestion: Charge Per-Use Fees or Sell Subscriptions? Gérard P. Cachon Pnina Feldman Operations and Information Management, The Wharton School, Uniersity
More informationRelation between compressibility, thermal expansion, atom volume and atomic heat of the metals
E. Grüneisen Relation between compressibility, thermal expansion, atom olume and atomic heat of the metals Annalen der Physik 6, 394-40, 1908 Translation: Falk H. Koenemann, 007 1 Richards (1907) and his
More informationAnalysis of cylindrical heat pipes incorporating the e ects of liquid±vapor coupling and non-darcian transportða closed form solution
International Journal of Heat and Mass Transfer 42 (1999) 3405±341 Analysis of cylindrical heat pipes incorporating the e ects of liquid±apor coupling and non-darcian transportða closed form solution N.
More informationOn computing Gaussian curvature of some well known distribution
Theoretical Mathematics & Applications, ol.3, no.4, 03, 85-04 ISSN: 79-9687 (print), 79-9709 (online) Scienpress Ltd, 03 On computing Gaussian curature of some well known distribution William W.S. Chen
More informationTo string together six theorems of physics by Pythagoras theorem
To string together six theorems of physics by Pythagoras theorem H. Y. Cui Department of Applied Physics Beijing Uniersity of Aeronautics and Astronautics Beijing, 00083, China ( May, 8, 2002 ) Abstract
More informationElements of diffraction theory part II
Elements of diffraction theory part II Setosla S. Iano Department of Physics, St. Kliment Ohridski Uniersity of Sofia, 5 James Bourchier Bld, 1164 Sofia, Bulgaria (Dated: February 7, 2016 We make a brief
More informationReal Gas Thermodynamics. and the isentropic behavior of substances. P. Nederstigt
Real Gas Thermodynamics and the isentropic behaior of substances. Nederstigt ii Real Gas Thermodynamics and the isentropic behaior of substances by. Nederstigt in partial fulfillment of the requirements
More informationDynamic Vehicle Routing with Moving Demands Part II: High speed demands or low arrival rates
ACC 9, Submitted St. Louis, MO Dynamic Vehicle Routing with Moing Demands Part II: High speed demands or low arrial rates Stephen L. Smith Shaunak D. Bopardikar Francesco Bullo João P. Hespanha Abstract
More informationUNDERSTAND MOTION IN ONE AND TWO DIMENSIONS
SUBAREA I. COMPETENCY 1.0 UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS MECHANICS Skill 1.1 Calculating displacement, aerage elocity, instantaneous elocity, and acceleration in a gien frame of reference
More informationApplication of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate
Physics Letters A 37 007) 33 38 www.elsevier.com/locate/pla Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate M. Esmaeilpour, D.D. Ganji
More information0 a 3 a 2 a 3 0 a 1 a 2 a 1 0
Chapter Flow kinematics Vector and tensor formulae This introductory section presents a brief account of different definitions of ector and tensor analysis that will be used in the following chapters.
More informationVelocity, Acceleration and Equations of Motion in the Elliptical Coordinate System
Aailable online at www.scholarsresearchlibrary.com Archies of Physics Research, 018, 9 (): 10-16 (http://scholarsresearchlibrary.com/archie.html) ISSN 0976-0970 CODEN (USA): APRRC7 Velocity, Acceleration
More informationTHE FIFTH DIMENSION EQUATIONS
JP Journal of Mathematical Sciences Volume 7 Issues 1 & 013 Pages 41-46 013 Ishaan Publishing House This paper is aailable online at http://www.iphsci.com THE FIFTH DIMENSION EQUATIONS Niittytie 1B16 03100
More informationRelativistic Energy Derivation
Relatiistic Energy Deriation Flamenco Chuck Keyser //4 ass Deriation (The ass Creation Equation ρ, ρ as the initial condition, C the mass creation rate, T the time, ρ a density. Let V be a second mass
More information10. Yes. Any function of (x - vt) will represent wave motion because it will satisfy the wave equation, Eq
CHAPER 5: Wae Motion Responses to Questions 5. he speed of sound in air obeys the equation B. If the bulk modulus is approximately constant and the density of air decreases with temperature, then the speed
More informationModule 1. Energy Methods in Structural Analysis
Module 1 Energy Methods in Structural Analysis esson 5 Virtual Work Instructional Objecties After studying this lesson, the student will be able to: 1. Define Virtual Work.. Differentiate between external
More informationReversal in time order of interactive events: Collision of inclined rods
Reersal in time order of interactie eents: Collision of inclined rods Published in The European Journal of Physics Eur. J. Phys. 27 819-824 http://www.iop.org/ej/abstract/0143-0807/27/4/013 Chandru Iyer
More informationReaction/Diffusion at Electrode/Solution Interfaces: The EC 2 Reaction
Int. J. Electrochem. Sci., 4(9) 1116-117 International Journal of ELECTROCHEMICAL SCIENCE www.electrochemsci.org Reaction/Diffusion at Electrode/Solution Interfaces: The EC Reaction Michael E G Lyons *
More informationStationary states and energy cascades in inelastic gases
Los Alamos National Laboratory From the SelectedWorks of Eli Ben-Naim May, 005 Stationary states and energy cascades in inelastic gases E. Ben-Naim, Los Alamos National Laboratory J. Machta, Uniersity
More informationTransmission Line Transients
8 5 Transmission Line Transients CHAPTER OBJECTIES After reading this chapter, you should be able to: Proide an analysis of traelling waes on transmission lines Derie a wae equation Understand the effect
More informationKinematics on oblique axes
Bolina 1 Kinematics on oblique axes arxi:physics/01111951 [physics.ed-ph] 27 No 2001 Oscar Bolina Departamento de Física-Matemática Uniersidade de São Paulo Caixa Postal 66318 São Paulo 05315-970 Brasil
More informationTHE CAUCHY PROBLEM FOR ONE-DIMENSIONAL FLOW OF A COMPRESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION
GLASNIK MATEMATIČKI Vol. 4666, 5 3 THE CAUCHY POBLEM FO ONE-DIMENSIONAL FLOW OF A COMPESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION Nermina Mujakoić and Ian Dražić Uniersity of ijeka, Croatia Abstract.
More informationJournal of Computational and Applied Mathematics. New matrix iterative methods for constraint solutions of the matrix
Journal of Computational and Applied Mathematics 35 (010 76 735 Contents lists aailable at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elseier.com/locate/cam New
More informationHasna BenSaid Faculté des sciences de Gafsa Campus Universitaire Sidi Ahmed Zarroug 2112 Gafsa, Tunisie
On the Ealuation of Linear and Non-Linear Models Using Data of Turbulent Channel Flows Hasna BenSaid bs_hasna@yahoo.fr Faculté des sciences de Gafsa Campus Uniersitaire Sidi Ahmed Zarroug Gafsa, Tunisie
More informationBessel s Equation and Bessel Functions: Solution to Schrödinger Equation in a Neumann and Hankel Functions
International ournal of Noel Research in Physics Chemistry & Mathematics Vol. 3, Issue 1, pp: (43-48), Month: anuary-april 2016, Aailable at: www.noeltyjournals.com Bessel s Equation and Bessel Functions:
More informationNonlinear electromagnetic wave propagation in isotropic and anisotropic antiferromagnetic media
Nonlinear electromagnetic wae propagation in isotropic anisotropic antiferromagnetic media V. Veerakumar Department of Physics, Manipal Institute of Technology, MAHE Deemed Uniersity, Manipal - 576 4 India.
More informationPatterns of Non-Simple Continued Fractions
Patterns of Non-Simple Continued Fractions Jesse Schmieg A final report written for the Uniersity of Minnesota Undergraduate Research Opportunities Program Adisor: Professor John Greene March 01 Contents
More informationMotion in Two and Three Dimensions
PH 1-A Fall 014 Motion in Two and Three Dimensions Lectures 4,5 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
More informationDerivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations H. A. Erbay Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794,
More informationDoppler shifts in astronomy
7.4 Doppler shift 253 Diide the transformation (3.4) by as follows: = g 1 bck. (Lorentz transformation) (7.43) Eliminate in the right-hand term with (41) and then inoke (42) to yield = g (1 b cos u). (7.44)
More informationMotion in Two and Three Dimensions
PH 1-1D Spring 013 Motion in Two and Three Dimensions Lectures 5,6,7 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
More informationS 1 S 2 A B C. 7/25/2006 Superposition ( F.Robilliard) 1
P S S S 0 x S A B C 7/5/006 Superposition ( F.Robilliard) Superposition of Waes: As we hae seen preiously, the defining property of a wae is that it can be described by a wae function of the form - y F(x
More informationMOTION OF FALLING OBJECTS WITH RESISTANCE
DOING PHYSICS WIH MALAB MECHANICS MOION OF FALLING OBJECS WIH RESISANCE Ian Cooper School of Physics, Uniersity of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECORY FOR MALAB SCRIPS mec_fr_mg_b.m Computation
More informationOn the influence of horizontal temperature stratification of seawater on the range of underwater sound signals. H. Lichte, Kiel, Germany
On the influence of horizontal temperature stratification of seawater on the range of underwater sound signals Original title: Über den Einfluß horizontaler Temperaturschichtung des Seewassers auf die
More informationLect-19. In this lecture...
19 1 In this lecture... Helmholtz and Gibb s functions Legendre transformations Thermodynamic potentials The Maxwell relations The ideal gas equation of state Compressibility factor Other equations of
More informationWeakly Nonlinear Harmonic Acoustic Waves in Classical Thermoviscous Fluids: A Perturbation Analysis
Weakly Nonlinear Harmonic Acoustic Waves in Classical Thermoviscous Fluids: A Perturbation Analysis P. M. Jordan Code 78, Naval Research Laboratory Stennis Space Center, Mississippi 39529 5004, USA Email:
More information[Abdallah, 5(1): January 2018] ISSN DOI /zenodo Impact Factor
GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES THE PHYSICAL INTERPRETATION OF LAGRARGIAN AND KINETIC ENERGY ON THE BASIS OF WORK Mubarak Dirar Abdallah *, Ahmed Abdalla Husain, Mohammed Habeeb A.Elkansi
More informationThe Kinetic Theory of Gases
978-1-107-1788-3 Classical and Quantum Thermal Physics The Kinetic Theory of Gases CHAPTER 1 1.0 Kinetic Theory, Classical and Quantum Thermodynamics Two important components of the unierse are: the matter
More information4-vectors. Chapter Definition of 4-vectors
Chapter 12 4-ectors Copyright 2004 by Daid Morin, morin@physics.harard.edu We now come to a ery powerful concept in relatiity, namely that of 4-ectors. Although it is possible to derie eerything in special
More informationThe perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component
Aailable online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016, 5382 5397 Research Article The perturbed Riemann problem for the chromatography system of Langmuir isotherm with one inert component Pengpeng
More informationARE THERE BETTER WAYS TO UNDERSTAND THE SECOND LAW OF THERMODYNAMICS AND THE CARNOT EFFICIENCY OF HEAT ENGINES?
ARE THERE BETTER WAYS TO UNDERSTAND THE SECOND LAW OF THERMODYNAMICS AND THE CARNOT EFFICIENCY OF HEAT ENGINES? W. John Dartnall 979 reised Abstract The Second Law of Thermodynamics imposes the Carnot
More informationOn Some Distance-Based Indices of Trees With a Given Matching Number
Applied Mathematical Sciences, Vol. 8, 204, no. 22, 6093-602 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ams.204.48656 On Some Distance-Based Indices of Trees With a Gien Matching Number Shu
More informationA NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (1+2)-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION
A NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (+2-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION ALI FILIZ ABDULLAH SONMEZOGLU MEHMET EKICI and DURGUN DURAN Communicated by Horia Cornean In this
More informationSLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS
3th AIAA Thermophysics Conference 3- June 3, Orlando, Florida AIAA 3-5 SLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS Matthew J. McNenly* Department of Aerospace Engineering Uniersity of Michigan, Ann
More informationСollisionless damping of electron waves in non-maxwellian plasma 1
http:/arxi.org/physics/78.748 Сollisionless damping of electron waes in non-mawellian plasma V. N. Soshnio Plasma Physics Dept., All-Russian Institute of Scientific and Technical Information of the Russian
More informationChapter 6: Operational Amplifiers
Chapter 6: Operational Amplifiers Circuit symbol and nomenclature: An op amp is a circuit element that behaes as a VCVS: The controlling oltage is in = and the controlled oltage is such that 5 5 A where
More informationChem 4521 Kinetic Theory of Gases PhET Simulation
Chem 451 Kinetic Theory of Gases PhET Simulation http://phet.colorado.edu/get_phet/simlauncher.php The discussion in the first lectures centered on the ideal gas equation of state and the modifications
More informationSound, Decibels, Doppler Effect
Phys101 Lectures 31, 32 Sound, Decibels, Doppler Effect Key points: Intensity of Sound: Decibels Doppler Effect Ref: 12-1,2,7. Page 1 Characteristics of Sound Sound can trael through any kind of matter,
More informationCases of integrability corresponding to the motion of a pendulum on the two-dimensional plane
Cases of integrability corresponding to the motion of a pendulum on the two-dimensional plane MAXIM V. SHAMOLIN Lomonoso Moscow State Uniersity Institute of Mechanics Michurinskii Ae.,, 99 Moscow RUSSIAN
More informationy (m)
4 Spring 99 Problem Set Optional Problems Physics February, 999 Handout Sinusoidal Waes. sinusoidal waes traeling on a string are described by wae Two Waelength is waelength of wae?ofwae? In terms of amplitude
More informationChapter 1. The Postulates of the Special Theory of Relativity
Chapter 1 The Postulates of the Special Theory of Relatiity Imagine a railroad station with six tracks (Fig. 1.1): On track 1a a train has stopped, the train on track 1b is going to the east at a elocity
More informationTarget Trajectory Estimation within a Sensor Network
Target Trajectory Estimation within a Sensor Network Adrien Ickowicz IRISA/CNRS, 354, Rennes, J-Pierre Le Cadre, IRISA/CNRS,354, Rennes,France Abstract This paper deals with the estimation of the trajectory
More informationIntroduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles
Introduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles by James Doane, PhD, PE Contents 1.0 Course Oeriew... 4.0 Basic Concepts of Thermodynamics... 4.1 Temperature
More informationStatus: Unit 2, Chapter 3
1 Status: Unit, Chapter 3 Vectors and Scalars Addition of Vectors Graphical Methods Subtraction of Vectors, and Multiplication by a Scalar Adding Vectors by Components Unit Vectors Vector Kinematics Projectile
More informationSELECTION, SIZING, AND OPERATION OF CONTROL VALVES FOR GASES AND LIQUIDS Class # 6110
SELECTION, SIZIN, AND OERATION OF CONTROL VALVES FOR ASES AND LIUIDS Class # 6110 Ross Turbiille Sales Engineer Fisher Controls International Inc. 301 S. First Aenue Marshalltown, Iowa USA Introduction
More informationNumerical Methods Applied to Chemical Engineering Homework #3. Nonlinear algebraic equations and matrix eigenvalue problems SOLUTION
0.34. Numerical Methods pplied to Chemical Engineering Homework #3. Nonlinear algebraic equations and matrix eigenalue problems SOLUTION Problem. Modeling chemical reaction with diffusion in a catalyst
More informationPHYS1169: Tutorial 8 Solutions
PHY69: Tutorial 8 olutions Wae Motion ) Let us consier a point P on the wae with a phase φ, so y cosϕ cos( x ± ωt) At t0, this point has position x0, so ϕ x0 ± ωt0 Now, at some later time t, the position
More informationNonequilibrium thermodynamics of driven amorphous materials. I. Internal degrees of freedom and volume deformation
Nonequilibrium thermodynamics of drien amorphous materials. I. Internal degrees of freedom and olume deformation Eran Bouchbinder Racah Institute of Physics, Hebrew Uniersity of Jerusalem, Jerusalem 91904,
More informationADVANCEMENT OF SMALL-SCALE THERMOACOUSTIC ENGINE
ADVANCEMENT OF SMALL-SCALE THERMOACOUSTIC ENGINE By SUNGMIN JUNG Masters in Mechanical Engineering WASHINGTON STATE UNIVERSITY School of Mechanical and Material Engineering MAY 2009 To the Faculty of Washington
More informationPropagation of Electromagnetic Field From a Pulsed Electric Dipole in a Dielectric Medium
CHINESE JOURNAL OF PHYSICS VOL. 39, NO. 2 APRIL 2001 Propagation of Electromagnetic Field From a Pulsed Electric Dipole in a Dielectric Medium Osama M. Abo-Seida 1 and Samira T. Bishay 2 1 Department of
More informationAn Alternative Characterization of Hidden Regular Variation in Joint Tail Modeling
An Alternatie Characterization of Hidden Regular Variation in Joint Tail Modeling Grant B. Weller Daniel S. Cooley Department of Statistics, Colorado State Uniersity, Fort Collins, CO USA May 22, 212 Abstract
More informationInvestigation on Ring Valve Motion and Impact Stress in Reciprocating Compressors
Purdue Uniersity Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 2010 Inestigation on Ring Vale Motion and Impact Stress in Reciprocating Compressors Yu Wang
More informationChapter 11. Perfectly Matched Layer Introduction
Chapter 11 Perfectly Matched Layer 11.1 Introduction The perfectly matched layer (PML) is generally considered the state-of-the-art for the termination of FDTD grids. There are some situations where specially
More informationPrediction of anode arc root position in a DC arc plasma torch
Prediction of anode arc root position in a DC arc plasma torch He-Ping Li 1, E. Pfender 1, Xi Chen 1 Department of Mechanical Engineering, Uniersity of Minnesota, Minneapolis, MN 55455, USA Department
More informationDomenico Solution Is It Valid?
Domenico Solution Is It Valid? by V. Sriniasan 1,T.P.Clement, and K.K. Lee 3 Abstract The Domenico solution is widely used in seeral analytical models for simulating ground water contaminant transport
More informationPlasma electron hole kinematics. I. Momentum conservation
Plasma electron hole kinematics. I. Momentum conseration The MIT Faculty has made this article openly aailable. Please share how this access benefits you. Your story matters. Citation As Published Publisher
More informationEINSTEIN S KINEMATICS COMPLETED
EINSTEIN S KINEMATICS COMPLETED S. D. Agashe Adjunct Professor Department of Electrical Engineering Indian Institute of Technology Mumbai India - 400076 email: eesdaia@ee.iitb.ac.in Abstract Einstein,
More informationUnit 11: Vectors in the Plane
135 Unit 11: Vectors in the Plane Vectors in the Plane The term ector is used to indicate a quantity (such as force or elocity) that has both length and direction. For instance, suppose a particle moes
More informationGalerkin method for the numerical solution of the RLW equation using quintic B-splines
Journal of Computational and Applied Mathematics 19 (26) 532 547 www.elsevier.com/locate/cam Galerkin method for the numerical solution of the RLW equation using quintic B-splines İdris Dağ a,, Bülent
More informationSolution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 307-315 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7418 Solution of the Hirota Equation Using Lattice-Boltzmann and the
More informationHORIZONTAL MOTION WITH RESISTANCE
DOING PHYSICS WITH MATLAB MECHANICS HORIZONTAL MOTION WITH RESISTANCE Ian Cooper School of Physics, Uniersity of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS ec_fr_b. This script
More informationarxiv: v1 [cond-mat.mes-hall] 28 Jun 2017
Loss of adiabaticity with increasing tunneling gap in non-integrable multistate Landau-Zener models Rajesh K. Malla and M. E. Raikh Department of Physics and Astronomy, Uniersity of Utah, Salt Lake City,
More informationAuto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any order
Physics Letters A 305 (00) 377 38 www.elsevier.com/locate/pla Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any
More informationGeneral Lorentz Boost Transformations, Acting on Some Important Physical Quantities
General Lorentz Boost Transformations, Acting on Some Important Physical Quantities We are interested in transforming measurements made in a reference frame O into measurements of the same quantities as
More informationThe optimal pebbling number of the complete m-ary tree
Discrete Mathematics 222 (2000) 89 00 www.elseier.com/locate/disc The optimal pebbling number of the complete m-ary tree Hung-Lin Fu, Chin-Lin Shiue Department of Applied Mathematics, National Chiao Tung
More informationPhysics 2A Chapter 3 - Motion in Two Dimensions Fall 2017
These notes are seen pages. A quick summary: Projectile motion is simply horizontal motion at constant elocity with ertical motion at constant acceleration. An object moing in a circular path experiences
More informationStability of negative ionization fronts: Regularization by electric screening?
PHYSICAL REVIEW E 69, 36214 24 Stability of negatie ionization fronts: Regularization by electric screening? Manuel Arrayás 1,2 and Ute Ebert 2,3 1 Uniersidad Rey Juan Carlos, Departmento de Física, Tulipán
More informationSolution to 1-D consolidation of non-homogeneous soft clay *
Xie et al. / J Zhejiang Uni SCI 25 6A(Suppl. I):29-34 29 Journal of Zhejiang Uniersity SCIENCE ISSN 19-395 http://www.zju.edu.cn/jzus E-mail: jzus@zju.edu.cn Solution to 1-D consolidation of non-homogeneous
More informationResearch Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
International Differential Equations Volume 2010, Article ID 764738, 8 pages doi:10.1155/2010/764738 Research Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
More informationProceedings of Meetings on Acoustics
Proceedings of Meetings on Acoustics Volume 19, 213 http://acousticalsociety.org/ ICA 213 Montreal Montreal, Canada 2-7 June 213 Noise Session 4pNSb: Noise Control 4pNSb7. Nonlinear effects of Helmholtz
More information