Laboratoire Ondes et Acoustique, Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, INSERM ERL U979, 10 rue Vauquelin, 75231 Paris Cedex 05, France.
J Acoust Soc Am. 2010 Feb;127(2):683-91. doi: 10.1121/1.3277202.
For fluids, the theoretical investigation of shock wave reflection has a good agreement with experiments when the incident shock Mach number is large. But when it is small, theory predicts that Mach reflections are physically unrealistic, which contradicts experimental evidence. This von Neumann paradox is investigated for shear shock waves in soft elastic solids with theory and simulations. The nonlinear elastic wave equation is approximated by a paraxial wave equation with a cubic nonlinear term. This equation is solved numerically with finite differences and the Godunov scheme. Three reflection regimes are observed. Theory is developed for shock propagation by applying the Rankine-Hugoniot relations and entropic constraints. A characteristic parameter relating diffraction and non-linearity is introduced and its theoretical values are shown to match numerical observations. The numerical solution is then applied to von Neumann reflection, where curved reflected and Mach shocks are observed. Finally, the case of weak von Neumann reflection, where there is no reflected shock, is examined. The smooth but non-monotonic transition between these three reflection regimes, from linear Snell-Descartes to perfect grazing case, provides a solution to the acoustical von Neumann paradox for the shear wave equation. This transition is similar to the quadratic non-linearity in fluids.
对于流体，当入射激波马赫数较大时，激波反射的理论研究与实验吻合较好。但当马赫数较小时，理论预测马赫反射在物理上是不现实的，这与实验证据相矛盾。本文通过理论和模拟研究了软弹性固体中的切向激波的冯·诺依曼佯谬。通过带有立方非线性项的傍轴波动方程对非线性弹性波方程进行了近似。利用有限差分和 Godunov 格式对该方程进行了数值求解。观察到了三种反射区域。通过应用瑞利-霍尼格顿关系和熵约束来研究激波的传播。引入了一个与衍射和非线性相关的特征参数，并显示其理论值与数值观察值相匹配。然后将数值解应用于冯·诺依曼反射，观察到弯曲的反射激波和马赫激波。最后，研究了弱冯·诺依曼反射的情况，其中没有反射激波。这三种反射区域之间的平滑但非单调的转变，从线性斯涅尔-笛卡尔到完美掠射情况，为剪切波方程的声学冯·诺依曼佯谬提供了一个解决方案。这种转变类似于流体中的二次非线性。
