On dispersion relations for acoustic flows in finite. alternating domains

Transcription

1 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 On dispersion relations for acoustic flows in finite alternating domains Mikhail Deryabin and Morten Willatzen Mads Clausen Institute for Product Innovation University of Southern Denmark Grundtvigs Alle 50, DK-6400 Sønderborg, DENMARK Phone: Fax: Abstract A one-dimensional treatment of solutions to the flow-acoustic equations for a composed medium consisting of a finite number of alternating materials layers A and B for example water and air is presented. Special emphasis is given to mathematical properties of the solutions more precisely, the stability of the trivial zero-solution and the possible wavenumber values. The discussion is intimately related to the discussion of bandgap formation in the dispersion relations as the number of A B layers n increases towards infinity. Estimates are given for the distribution of wavenumber values β as a function of n, in cases when the trivial solution is either stable or unstable. We emphasize that the present analytical analysis allows determination of distributed in space domain as well as surface solutions in one calculation.

2 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 Introduction the distance between successive solutions is in the order of order /n. In these areas, regions may also exist where this distance can grow to O/ n. In the unstable areas, where TrA >, the distance between successive solutions is in the order of. Both the stable and the unstable regions do not depend on n. An immediate consequence is that for the stable regions the distance between successive solutions is not uniformly distributed. In this paper we study the acoustic problem for periodically arranged materials materials A and B in the -dimensional case, i.e., periodicity being extended only in one direction. We assume that a fluid flows in one or both of the constituent media. We study an acoustic band gap structure for the case, when the domain consists of a finite yet big number of layers n. As the periodicity is -dimensional, we can use simple analytical techniques to completely describe the dispersion relations. We first show that the acoustic equations in question are linear Hamiltonian equations with periodic coefficients. Thus, the question of whether the solution for acoustic equations is bounded in the infinite domain reduces to the well-known problem of stability of the zero solution of linear Hamiltonian equations: if the zero solution is stable, then there cannot be unbounded solutions. The zero solution is stable, if the absolute value of the trace of the monodromy matrix A is less than, and we calculate this matrix for our system. Suppose that the number of layers n is big enough. Then the asymptotics of the dispersion structure is the following. In the stable areas, where TrA <, Equations of flow acoustics Under the monofrequency condition, the linearized equations for the acoustic flow are iω c p + v 0 p c x + ρ vx 0 x + v y y + v z =0, z v x iωv x + v 0 x + v v 0 y y + v v 0 z z = p ρ 0 x, v y iωv y + v 0 x = p ρ 0 y, v z iωv z + v 0 x = p ρ 0 z, see Ref. [] for details. Here v 0 =v 0 y, z, 0, 0 is the background flow, ρ 0 y, z is the pressure of the background flow, p is the acoustic pressure, and v = v x,v y,v z is the acoustic flow velocity. We are looking for the solution of the following form: p x = iβp, v = iβv. x

3 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 Substituting relations into equations, we get the following expression for the pressure p, cf. Ref. []: p ρ 0 ω + βv 0 y ρ 0 p ρ 0 ω + βv 0 y y β v 0 p ρ 0 ω + βv 0 3 y y + p ρ 0 ω + βv 0 z 3 ρ 0 p ρ 0 ω + βv 0 z z β v 0 p ρ 0 ω + βv 0 3 z z + β + c ρ 0 ρ 0 ω + βv 0 p =0. We consider now the -dimensional case, when the functions ρ 0 and v 0 do not depend on y, and we assume that the pressure p does not depend on y either. where = / z, and the Hamiltonian equals Hp, q = ρ 0 ω + βv 0 q + c ρ 0 β ρ 0 ω + βv 0 p. The Hamiltonian system 5 is integrable: this is a consequence of the Lyapunov-Floquet theorem that states that there exists a linear transformation p, q P, Q, which is π-periodic in z, that sends the linear system 5 with π-periodic in z coefficients to the linear system with the constant coefficients. However, for arbitrary functions ρ 0 and v 0, one cannot find this variable transformation explicitly. We have also assumed periodicity in z, i.e., both v 0 and ρ 0 are π-periodic functions of z. Then Equation 3 can be written as z p ρ 0 ω + βv 0 z β ρ 0 ω + βv 0 + c ρ 0 p =0. 4 This is the Lagrange equation, which describes the linearized pendulum motion with the vibrating suspension point. We can put Equation 4 into the Hamiltonian form by introducing the canonical conjugate variable q = Equation 4 now becomes q = H p, ρ 0 ω + βv 0 p z. p = H q, 5 3 The monodromy matrix and the wavevector To determine the value of the wavevector x- component β, one has to solve Equation 4 with the following boundary conditions: for z = z L and z = z R, the z-component of the pressure gradient should be zero: p, e z = p/ z =0. Let p = pz,pz L,p z L be the solution to Equation 4. Then the equation for β becomes p z R,, 0 = 0. 6

4 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 Indeed, this equation is linear in pz L, thus this value canbesetto. F B = c B ρb 0 F B = ρ B 0 ω + βv B 0, β ρ B 0 ω + βvb 0 Let n denote the number of the layers of materials A and B. For big values of n, we may expect, that the number of solutions β on a bounded interval may depend on this interval. In the other words, the density of solutions on an interval β min,β max may vary. We denote by ρ A 0, v0 A and c A the values of the background density, fluid velocity and the velocity of When we glue the solutions in the materials A and B together, we demand that the canonical coordinates q, p should be continuous, rather than p, p cf., e.g., []. This is important for the derivative p z ρ 0 ω + βv 0 z sound in the material A, andbyρ B 0, v B 0, c a the corresponding values in the material B: ρ 0 = ρ A 0, v 0 = v A 0, c = c A for z [0,z, ρ 0 = ρ B 0, v 0 = v B 0, c = c a for z [z, π Both ρ A 0, v A 0, c A, ρ B 0, v B 0 and c B are the functions of z, and we assume that they are smooth. However, the functions ρ 0, v 0 and c need not even to be continuous. Equation 4 reads p = F A q, q = F A p, z [0,z, p = F B q, q = F B p, z [z, π, 7 where the functions F A,B, equal F A = c A ρa 0 F A = ρ A 0 ω + βv A 0, β ρ A 0 ω +, βva 0 8 to be defined in the generalized sense. To this extend, our model closely resembles the Ben Daniel- Duke model. Remark. One can see that the Hamiltonian formalism is very useful for this kind of systems. First, the canonical variables are continuous, while the derivative of the pressure pressure p is not. Second, the Hamilton equations are regular when we tend both β and ω to zero, which is important if we want to determine the speed of sound in the composite media. Our boundary conditions are now q =0forz = z L, z = z R,andp =forz = z L. Let z L,z R 0,z mod π. A mapping p, q z=0 p, q z=π is linear and thus can 3

5 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 be written as pπ qπ =A p0 q0, ing form: cos ψ sin ψ B T AB = sin ψ cos ψ where B is the orthogonal matrix. The n-th power where A is a constant matrix i.e., A/ z = 0. This A n becomes now: matrix is called the monodromy matrix, and we write it as a product of two matrices A L and A R, where the matrix A L defines the mapping p, q z=zl p, q z=π, and the matrix A R defines the mapping p, q z=0 A n =BB T AB n B T =B cos nψ sin nψ sin nψ cos nψ B T p, q z=zr. The equation for β now reads Let us rewrite Equation 9 as 0, A R A n A L 0 = 0 9 βcosnψβ+ βsinnψβ = 0 0 Suppose that + 0 for all β this is a general case. Then Equation 0 can be written as sin nψ + arccos = Wavenumber distribution for the stable case: TrA < We now assume that the trace TrA < for any Suppose that functions ψ,, are smooth. The structure of solutions to Equation 6 is described by the following results. In the neighbourhood of a regular point, i.e., such β that β 0, values of the parameters. Then in the appropriate the distance between the solutions is at most of order coordinates the matrix A can be put into the follow- /n. More precisely, the following Theorem is true: 4

6 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 Theorem. Consider an interval L, such that there exists a constant c>0, such that for any β L β c Then there is a constant C>0, that does not depend on n, such that on any interval β,β + C/n L there is at least one solution to Equation 6. The proof rests on the following Proposition: Proposition. Let β 0 be such that β β 0 0 Then there is a constant C >0, that does not depend on n, such that for any n there exists a solution β to Equation 6 that satisfies β β 0 C n. where β 0 [β 0,β 0 + n /m β ]. Notice that this is an exact equation. One can see that the expression nψβ 0 + β β 0 β + arccos β πk + changes sign on the interval β [π/c, π/c], if n is sufficiently big, where c = / ββ 0. Thus, on this interval there is a solution to our equation, and the interval itself does not depend on n. If we drop the condition / β c, then the distance between the solutions may be of order / n instead of /n: this is the case for a typical degenerate point, i.e., such β that β =0, ψ β 0. Theorem. Let β 0 be such that β β 0=0, ψ β β 0 0 Proof. If + β 0 = 0, then β = β 0 and the Then there is an interval L, mesl >c/ n for Proposition is proved. Let now + β 0 0. Let k Z be such that nψβ 0 + arccos β + 0 πk = α<π Let β = β 0 + n β. Substituting this relation to Equation, we get nψβ 0 + β β 0 β + + arccos β πk =0 + some c>0 and the distance between L and β 0 being at most of order /n, such that there are no solutions to Equation 6 in L. The point β 0 needs not to belong the interval L : there may be some exceptional values of n, such that there are solutions in an O/n of the degenerate point β 0. Proof. Take β = β 0 + n β and substitute into Equa- 5

7 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 tion. Then we get n ψβ 0 + ψ β β 0 β n arccos β πk =0 As ψ/ β β 0 0, one can choose β such that nψβ 0 + ψ β β 0 β n + + arccos β + 0 πk > In the set of such values of β, one can pick up an interval L, such that mesl >c/ n, which follows by construction. Proposition. Let β 0 be such that β β 0=0, ψ β β 0 0 Then there is a set of the parameters of positive Lebesgue measure, such that for any N > 0 there is not satisfied for some sequence of values n. Take such n, and consider a solution of the form β = β 0 + n β. As above, we have to solve the equation nψβ 0 + ψ n β β 0 β arccos β πk =0 + Neglecting the terms of order /n and higher, we get a quadratic equation for β, that always have two real solutions due to condition. The rest of the proof is similar to Proposition. Summarizing the results for the stable case, i.e., when TrA <, we see that most of wavenumbers β are /n-close to each other. There also can be gaps in wavenumbers, that are of order / n and are located in an / n-neighbourhood of zeroes of the derivative / β. is n>nsuch that there exists a pair of solutions β, to Equation 6 that both satisfy β, β 0 C n where the constant C does not depend on n. Proof. Let ψ/ β β 0 > 0. For an arbitrary small constant c>0, there is a set of parameters of positive measure, such that the Diofantine condition nψβ 0 + arccos β + 0 πk > c n 5 Distribution for wavenumbers for the unstable case: TrA > The situation is different in the unstable case: although the gaps have the similar position, they are now of order. Moreover, the distance between any two solutions is also of order. 6

8 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 The the matrix A can still be put into the eigenvectorform: B T AB = cosh ψ sinh ψ sinh ψ cosh ψ where B is the orthogonal matrix. The n-th power A n becomes now: cosh nψ A n =B sinh nψ sinh nψ cosh nψ As above, we rewrite Equation 9 as B T βcoshnψβ+ βsinhnψβ = 0 4 Suppose that 0 for all β this is a general case, and also that > 0. Then Equation 4 can be written as sinhnψ + φ =0, cosh φ = 5 For n, this equation has only solutions that are close to zeroes of the function ψβ. Proposition 3. Let β 0 be such that ψβ 0 =0, β β 0 0. Then there is an interval of order, containing β 0, such that on this interval there is precisely one solution to Equation 6. This solution is /n-close to β 0. Proof. We rewrite Equation 5 as ψβ+ φβ =0, n and use the implicit function theorem: for n this equation has locally only one solution β = β 0, as / ββ 0 0. For big, but finite values of n, the solution will be /n-close to β 0, and it can be found by the same method as in the previous section. 6 Bandgap structure Numerical experiments show that the stable and the unstable areas may be mixed with each other. an example, we consider the water-air mix, and we assume that in each of the media the background density ρ 0, the media background velocity v 0 and the velocity of sound c are constants. The background velocity in air is assumed to be zero. The band gap areas are the sets, where TrA >, see fig.. Along the horizontal line is the number of a solution, i.e., - st, -nd,..., 00-th, starting from the first one that is greater than zero. corresponding values of β. As On the vertical line are the The distribution of solutions when n is the fol- 7

9 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp Conclusions Figure : Bond gaps for air-water mix. The seemingly horizontal parts of the graph are the soltions that lie in the stability regions. Notice that these regions are very small compared with the unstable regions. An analytical study of solutions to the flow-acoustic problem for a material composed of alternating layers of materials A and B has been presented. Depending on the trace of the monodromy matrix, that solutions are stable trace less than or unstable trace larger than. Asymptotics of distances in wavenumber space have been given as the number of AB layers in the medium goes to infinity, including a discussion of bandgap formation well-known from solid-state physics applications. Finally, the present analysis gives information about possible pressure solutions including distributed and localized pressure eigenstates in space domain. lowing. In the stable areas, where TrA <, the solutions lie /n-close to each other, and in these areas there also may be regions, where the distance between two successive solutions may grow to O/ n. In the unstable areas, where TrA >, the distance between two successive solutions is of order. It is important to note that the stable and unstable regions themselves do not depend on n. The present analysis for phononic bandgap structures shows many similarities with the corresponding analysis of photonic bandgap structures. In actual fact under some assumptions, they both obey the Helmholtz wave equation for propagation in homogeneous steady media. Hence, many of the mathematical implications obtained in the present work can to some degree be carried over to photonics applications. 8

10 Proceedings of the 7th WSEAS International Conference on Acoustics & Music: Theory & Applications, Cavtat, Croatia, June 3-5, 006 pp3-3 References [] Willatzen M., Lew Yan Voon L.C. Flow acoustics in periodic structures. To appear in Ultrasonics. [] Arnold V.I. Mathematical methods in classical mechanics. Springer-Verlag. 9