A Paradigm for Time-periodic Sound Wave Propagation in the Compressible Euler Equations (original) (raw)
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Acta Mathematica Scientia, 2009
In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by entropy jumps. Conditions on the states and the periods are derived which allow for the existence of solutions in the Fourier 1-mode. In it is shown that these are the simplest linearized waves such that, for almost every period, they are isolated in the kernel of the linearized operator that imposes periodicity, and such that they perturb to nearby nonlinear solutions of the compressible Euler equations that balance compression and rarefaction along characteristics in the formal sense described in . Their fundamental nature thus makes them of interest in their own right.
arXiv (Cornell University), 2023
We prove the existence of "pure tone" nonlinear sound waves of all frequencies. These are smooth, space and time periodic, oscillatory solutions of the 3 × 3 compressible Euler equations in one space dimension. Being perturbations of solutions of a linear wave equation, they provide a rigorous justification for the centuries old theory of Acoustics. In particular, Riemann's celebrated 1860 proof that compressions always form shocks holds for isentropic and barotropic flows, but for generic entropy profiles, shock-free periodic solutions containing nontrivial compressions and rarefactions exist for every wavenumber k.
SIAM Journal on Mathematical Analysis, 2011
It has been unknown since the time of Euler whether or not time-periodic sound wave propagation is physically possible in the compressible Euler equations, due mainly to the ubiquitous formation of shock waves. The existence of such waves would confirm the possibility of dissipation free long distance signaling. Following our work in [27], we derive exact linearized solutions that exhibit the simplest possible periodic wave structure that can balance compression and rarefaction along characteristics in the nonlinear Euler problem. These linearized waves exhibit interesting phase and group velocities analogous to linear dispersive waves. Moreover, when the spacial period is incommensurate with the time period, the sound speed is incommensurate with the period, and a new periodic wave pattern is observed in which the sound waves move in a quasi-periodic trajectory though a periodic configuration of states. This establishes a new way in which nonlinear solutions that exist arbitrarily close to these linearized solutions can balance compression and rarefaction along characteristics in a quasi-periodic sense. We then rigorously establish the spectral properties of the linearized operators associated with these linearized solutions. In particular we show that the linearized operators are invertible on the complement of a one dimensional kernel containing the periodic solutions only in the case when the wave speeds are incommensurate with the periods, but these invertible operators have small divisors, analogous to KAM theory. Almost everywhere algebraic decay rates for the small divisors are proven. In particular this provides a nice starting framework for the problem of perturbing these linearized solutions to exact nonlinear periodic solutions of the full compressible Euler equations.
The large time stability of sound waves
Communications in Mathematical Physics, 1996
We demonstrate the existence of solutions to the full 3 x 3 system of compressible Euler equations in one space dimension, up to an arbitrary time T > 0, in the case when the initial data has arbitrarily large total variation, and sufficiently small supnorm. The result applies to periodic solutions of the Euler equations, a nonlinear model for sound wave propagation in gas dynamics. Our analysis establishes a growth rate for the total variation that depends on a new length scale d that we identify in the problem. This length scale plays no role in 2 x 2 systems, (or any system possessing a full set of Riemann coordinates), nor in the small total variation problem for n x n systems, the cases originally addressed by Glimm in 1965. Recent work by a number of authors has demonstrated that when the total variation is sufficiently large, solutions of 3 x 3 systems of conservation laws can in general blow up in finite time, (independent of the supnorm), due to amplifying instabilities created by the non-trivial Lie algebra of the vector fields that define the elementary waves. For the large total variation problem, there is an interaction between large scale effects that amplify and small scale effects that are stable, and we show that the length scale on which this interaction occurs is d. In the limit d ~ cx~, we recover Glimm's theorem, and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp.
A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations
Methods and Applications of Analysis, 2010
Following the authors' earlier work in 10], we show that the nonlinear eigenvalue problem introduced in [10] can be recast in the language of bifurcation theory as a perturbation of a linearized eigenvalue problem. Solutions of this nonlinear eigenvalue problem correspond to time periodic solutions of the compressible Euler equations that exhibit the simplest possible periodic structure identified in . By a Liapunov-Schmidt reduction we establish and refine the statement of a new infinite dimensional KAM type small divisor problem in bifurcation theory, whose solution will imply the existence of exact time-periodic solutions of the compressible Euler equations. We then show that solutions exist to within an arbitrarily high Fourier mode cutoff. The results introduce a new small divisor problem of quasilinear type, and lend further strong support for the claim that the time-periodic wave pattern described at the linearized level in [10], is physically realized in nearby exact solutions of the fully nonlinear compressible Euler equations.
On finite amplitude elastic waves propagating in compressible solids
Physical Review E Statistical Physics Plasmas Fluids and Related Interdisciplinary Topics, 2005
The paper studies the interaction of a longitudinal wave with transverse waves in general isotropic and unconstrained hyperelastic materials, including the possibility of dissipation. The dissipative term chosen is similar to the classical stress tensor describing a Stokesian fluid and is commonly used in nonlinear acoustics. The aim of this research is to derive the corresponding general equations of motion, valid for any possible form of the strain energy function and to investigate the possibility of obtaining some general and exact solutions to these equations by reducing them to a set of ordinary differential equations. Then the reductions can lead to some exact closed-form solutions for special classes of materials (here the examples of the Hadamard, Blatz-Ko, and power-law strain energy densities are considered, as well as fourth-order elasticity). The solutions derived are in a time/space separable form and may be interpreted as generalized oscillatory shearing motions and generalized sinusoidal standing waves. By means of standard methods of dynamical systems theory, some peculiar properties of waves propagating in compressible materials are uncovered, such as for example, the emergence of destabilizing effects. These latter features exist for highly nonlinear strain energy functions such as the relatively simple power-law strain energy, but they cannot exist in the framework of fourth-order elasticity.
Nonlinear wave propagation in binary mixtures of Euler fluids
Continuum Mechanics and Thermodynamics, 2004
In the present paper, we study the propagation of acceleration and shock waves in a binary mixture of ideal Euler fluids, assuming that the difference between the atomic masses of the constituents is negligible. We evaluate the characteristic speeds, proving that they can be separated into two groups: one is related to the case of a single Euler fluid, provided that an average ratio of specific heats is introduced; the other is new and related to the propagation speed due to diffusion. We evaluate the critical time for sound acceleration waves and compare its value to that of a single fluid. We then study shock waves, showing that three types of shock waves appear: sonic and contact shocks, which have counterparts in the single fluid case, and the diffusive shock, which is peculiar to the mixture. We discuss the admissibility of the shock waves using the Lax-Liu conditions and the entropy growth criterion. It is proved that the sonic and the characteristic shock obey the same properties as in the single fluid case, while for the diffusive shock there exists a locally exceptional case that is determined by a particular value of the concentration of the constituents, for which the genuine nonlinearity is lost and no shocks are admissible. For other values of the unperturbed concentration, the diffusive shock is stable in a bounded interval of admissibility.
Finite amplitude elastic waves propagating in compressible solids
Physical Review E, 2005
The paper studies the interaction of a longitudinal wave with transverse waves in general isotropic and unconstrained hyperelastic materials, including the possibility of dissipation. The dissipative term chosen is similar to the classical stress tensor describing a Stokesian fluid and is commonly used in nonlinear acoustics. The aim of this research is to derive the corresponding general equations of motion, valid for any possible form of the strain energy function and to investigate the possibility of obtaining some general and exact solutions to these equations by reducing them to a set of ordinary differential equations. Then the reductions can lead to some exact closed-form solutions for special classes of materials ͑here the examples of the Hadamard, Blatz-Ko, and power-law strain energy densities are considered, as well as fourth-order elasticity͒. The solutions derived are in a time-space separable form and may be interpreted as generalized oscillatory shearing motions and generalized sinusoidal standing waves. By means of standard methods of dynamical systems theory, some peculiar properties of waves propagating in compressible materials are uncovered, such as for example, the emergence of destabilizing effects. These latter features exist for highly nonlinear strain energy functions such as the relatively simple power-law strain energy, but they cannot exist in the framework of fourth-order elasticity.
A Nash–Moser Framework for Finding Periodic Solutions of the Compressible Euler Equations
Journal of Scientific Computing, 2014
In this paper, which summarized a talk given by the second author in Waterloo, reports on recent progress the authors have made in a long term program to prove the existence of time-periodic shockfree solutions of the compressible Euler equations. We briefly recall our previous results, describe our recent change of direction, and discuss the estimates that must be obtained to get the Nash-Moser method to converge. Assuming these estimates, we present a new convergence theorem for the Nash-Moser method. Our approach reduces the problem to that of obtaining small divisor estimates for finite dimensional projections of linearized operators. These operators can be described in detail, and in principle, this reduces the proof of periodic solutions to the calculation of the smallest singular value of an N × N matrix.