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Mathematics Colloquium


Date:  Tuesday, November 20, 2007

Title:  A Mechanism for the Propagation of Nonlinear Time-Periodic Sound Waves

Abstract:  The compressible Euler equations were derived by Leonard Euler in 1752 as an extension of Newton's laws of motion to the setting of continuous media. In that year, Euler derived the equations from physical principles, linearized the equations, and thereby confirmed D'Ambert's idea that sound waves are oscillations in the density that evolve according to the linear wave equation. Hence came the linear theory of sound, with modes of vibration, harmonics, and everything that is the starting point of the theory of music today. Since Euler's time, it has been unknown whether the fully nonlinear equations could support oscillatory time-periodic solutions analogous to the sinusoidal oscillations of sound waves in the linearized theory. For most of the last 250 years, experts have believed that time-periodic solutions of the compressible Euler equations which propagate like sound waves, were physically impossible due to the ubiquitous formation of shock waves. (Formation of shock waves from smooth solutions was first demonstrated by Riemann.) This intuition was supported in 1970 by a famous paper of Peter Lax and James Glimm, in which they proved that solutions of the compressible Euler equations starting from time-periodic initial data must form shock waves and decay away by shock wave dissipation at a rate $1/t$...but their theorem required the simplifying assumption that the entropy (or temperature) be constant, required so that the complexity of the system dropped from $3$ to $2$ coupled equations. In this talk I discuss joint work with Robin Young in which we derive the simplest nonlinear wave pattern that is consistent with periodic evolution when the entropy is not constant, and discuss progress in the authors' program to prove that such time-periodic solutions solve the compressible Euler equations exactly.

Speaker:  Blake Temple
Institution:  UC Davis


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