In this paper, we initiate the study of wave propagation in a recently proposed mathematical model for stretch-limited elastic strings. We consider the longitudinal motion of a simple class of uniform, semi-infinite, stretch-limited strings under no external force with finite end held fixed and prescribed tension at the infinite end. We study a class of motions such that the string has one inextensible segment, where the local stretch is maximized, and one extensible segment. The equations of motion reduce to a simple and novel shock front problem in one spatial variable for which we prove existence and uniqueness of local-in-time solutions for appropriate initial data. We then prove the orbital asymptotic stability of an explicit two-parameter family of piece-wise constant stretched motions. If the prescribed tension at the infinite end is increasing in time, our results provide an open set of initial data launching motions resulting in the string becoming fully inextensible and tension blowing up in finite time.

在本文中，我们开始研究最近提出的拉伸受限弹性弦数学模型中的波传播。我们考虑一类简单的均匀的、半无限的、受拉伸限制的弦在没有外力的情况下的纵向运动，有限端保持固定，无限端有规定的张力。我们研究了一类运动，使得弦有一个不可伸展的部分，其中局部拉伸最大化，还有一个可伸展的部分。运动方程简化为一个空间变量中的一个简单而新颖的激波前沿问题，我们证明了适当初始数据的局部时间解的存在性和唯一性。然后，我们证明了分段恒定拉伸运动的显式双参数族的轨道渐近稳定性。