Planar shock waves in single-crystal copper were simulated using nonequilibrium molecular dynamics with a realistic embedded atom potential. The simulation results are in good agreement with new experimental data presented here, for the Hugoniot of single-crystal copper along ⟨100⟩. Simulations were performed for Hugoniot pressures in the range 2 GPa – 800 GPa, up to well above the shock induced melting transition. Large anisotropies are found for shock propagation along ⟨100⟩,⟨110⟩, and ⟨111⟩, with quantitative differences from pair potentials results. Plastic deformation starts at Up≳0.75km∕s, and melting occurs between 200 and 220 GPa, in agreement with the experimental melting pressure of polycrystalline copper. The Voigt and Reuss averages of our simulated Hugoniot do not compare well below melting with the experimental Hugoniot of polycrystalline copper. This is possibly due to experimental targets with preferential texturing and/or a much lower Hugoniot elastic limit.

Abstract. Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrödinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength ε. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrödinger equations subject to highly oscillatory initial data of the form Ae iΦ/ε . Through a careful estimate of an oscillatory integral operator, we prove that the k-th order Gaussian beam superposition converges to the original wave field at a rate proportional to ε k/2 in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, ε-scaled, energy norm and for the Schrödinger equation in the L 2 norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in R 2 to analyze the sharpness of the theoretical results.

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