An instantaneous sub-surface disturbance in a two-dimensional elastic half-space is considered. The disturbance propagates through the elastic material until it reaches the free surface, after which it propagates out along the surface. In conventional theory, the free-surface conditions on the unknown surface are projected onto the flat plane \(y = 0\), so that a linear model may be used. Here, however, we present a formulation that takes explicit account of the fact that the location of the free surface is unknown a priori, and we show how to solve this more difficult problem numerically. This reveals that, while conventional linearized theory gives an accurate account of the decaying waves that travel outwards along the surface, it can under-estimate the strength of the elastic rebound above the location of the disturbance. In some circumstances, the non-linear solution fails in finite time, due to the formation of a “peakon” at the free surface. We suggest that brittle failure of the elastic material might in practice be initiated at those times and locations.

考虑了二维弹性半空间中的瞬时地下扰动。扰动通过弹性材料传播，直到到达自由表面，然后沿着表面传播出去。在传统理论中，未知表面上的自由表面条件被投影到平面\(y = 0\) 上，因此可以使用线性模型。然而，在这里，我们提出了一个公式，该公式明确考虑了自由表面的位置是先验未知的事实，我们展示了如何用数值解决这个更困难的问题。这表明，虽然传统的线性化理论准确说明了沿表面向外传播的衰减波，但它可能低估了扰动位置上方的弹性回弹强度。在某些情况下，由于在自由表面形成“峰子”，非线性求解在有限时间内失败。我们建议弹性材料的脆性破坏实际上可能在这些时间和位置开始。