A simple model of piston-driven spherical and cylindrical shocks is suggested. The model is based on a consistent use of two factors: (a) an almost uniform pressure across the shocked layer and, (b) continuous geometrical stretching of the surface elements of the expanding piston. It turns out that for a uniform pre-shock medium the gas between the piston and the shock behaves essentially as an incompressible fluid. An algebraic equation for the shock vs piston position is obtained. Detailed evaluation of the accuracy of the proposed solution shows that its accuracy is a few percent for the adiabatic index $\gamma =5/3$ (as in ideal plasma). A closed-form solution describing enhancement of a weak ambient magnetic field by the shock is presented. The proposed model of piston-driven shocks goes beyond the classical self-similar solutions in that it: (1) naturally covers an early, non-asymptotic dynamics and its transition to asymptotic regime; (2) allows for smooth radial density variation of an ambient gas of the form of bumps, dimples or ramps between two constant values; and (3) allows for smooth temporal variation of piston velocity of the form of bumps, dimples, or ramps. This simple and versatile model provides some new insights into a classical hydrodynamical problem.

中文翻译：

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由球形或圆柱形活塞驱动的强冲击的简单模型

建议使用活塞驱动的球形和圆柱形冲击的简单模型。该模型基于对两个因素的一致使用：(a) 冲击层上几乎均匀的压力，以及 (b) 膨胀活塞表面元素的连续几何拉伸。事实证明，对于均匀的预激波介质，活塞和激波之间的气体基本上表现为不可压缩的流体。获得了冲击与活塞位置的代数方程。对所提出的解决方案的准确度的详细评估表明，其对于绝热指数的准确度仅为百分之几$\gamma =5/3$（如在理想等离子体中）。提出了一种封闭形式的解决方案，描述了冲击对弱环境磁场的增强。提出的活塞驱动冲击模型超越了经典的自相似解，因为它：（1）自然地涵盖了早期的非渐近动力学及其向渐近状态的过渡；(2) 允许在两个恒定值之间以隆起、凹坑或斜坡形式的环境气体的平滑径向密度变化；(3) 允许以隆起、凹坑或斜坡形式的活塞速度的平滑时间变化。这个简单而通用的模型为经典流体动力学问题提供了一些新的见解。