We investigate impact of a sphere onto a floating elastic sheet and the resulting formation and evolution of wrinkles in the sheet. Following impact, we observe a radially propagating wave, beyond which the sheet remains approximately planar but is decorated by a series of radial wrinkles whose wavelength grows in time. We develop a mathematical model to describe these phenomena by exploiting the asymptotic limit in which the bending stiffness is small compared to stresses in the sheet. The results of this analysis show that, at a time $t$ after impact, the transverse wave is located at a radial distance $r\sim t^{1/2}$ from the impactor, in contrast to the classic $r\sim t^{2/3}$ scaling observed for capillary–inertia ripples produced by dropping a stone into a pond. We describe the shape of this wave, starting from the simplest case of a point impactor, but subsequently addressing a finite-radius spherical impactor, contrasting this case with the classic Wagner theory of impact. We show also that the coarsening of wrinkles in the flat portion of the sheet is controlled by the inertia of the underlying liquid: short-wavelength, small-amplitude wrinkles form at early times since they accommodate the geometrically imposed compression without significantly displacing the underlying liquid. As time progresses, the liquid accelerates and the wrinkles grow larger and coarsen. We explain this coarsening quantitatively using numerical simulations and scaling arguments, and we compare our predictions with experimental data.

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对漂浮的弹性薄板的影响：数学模型

我们研究了球体对浮动弹性片的影响以及在片中产生的皱纹的形成和演变。撞击后，我们观察到径向传播的波，在该波之外，片材保持近似平面，但由一系列随时间增长的径向褶皱装饰。我们开发了一个数学模型，通过利用渐近极限来描述这些现象，在渐近极限中，弯曲刚度比板材中的应力小。分析结果表明，一次$Ť$ 撞击后，横波位于径向距离处 $\mathrm{[R}〜{Ť}^{\mathrm{1个}/2}$ 来自撞击器，与经典 $\mathrm{[R}〜{Ť}^{2/3}$观察到因将石头掉入池塘而产生的毛细惯性波纹的水垢。我们从点撞击器的最简单情况开始，然后介绍有限半径球形撞击器，然后将此情况与经典的瓦格纳撞击理论进行对比，描述了该波的形状。我们还显示出，在薄板的平坦部分中，皱纹的粗化程度受底层液体的惯性控制：在早期形成短波长，小振幅的皱纹，因为它们可以承受几何上施加的压缩力而不会显着移位底层液体。随着时间的流逝，液体加速，皱纹变大和变粗。我们使用数值模拟和缩放参数定量地解释了这种粗化，并且将我们的预测与实验数据进行了比较。