Abstract Long waves in shallow water propagating over a background shear current towards a sloping beach are investigated, and exact solutions are found using a hodograph transform and separation of variables. Inspired by the work of Carrier and Greenspan on steady waves over a uniform beach profile in the irrotational setting, we study waves which propagate over a background shear current. The shallow-water equations are obtained from the nonlinear Benney equations, and exact solutions are found with help of the hodograph transformation in conjunction with several further changes of variables. The hodograph transformation is effected by finding the Riemann invariants after the equations are written in the standard form of barotropic gas dynamics. In the current work, the background flow features zero mass flux, as would be required by a real flow at a beach. Moreover, in contrast with previous work, the present approach allows separate study of the influence of the strength of the shear current and the slope of the bottom profile. This enables us to provide an estimate of the run-up as a function of the shear flow while keeping the bottom slope constant.

中文翻译：

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背景剪切流上长波的上升

摘要 研究了浅水中长波通过背景剪切流向倾斜海滩传播，并使用全息图变换和变量分离找到了精确解。受 Carrier 和 Greenspan 对非旋转环境中均匀海滩剖面上稳定波的研究的启发，我们研究了在背景剪切流上传播的波。浅水方程是从非线性 Benney 方程中得到的，通过全息图变换，结合几个变量的进一步变化，得到了精确解。以正压气体动力学的标准形式写出方程后，通过找到黎曼不变量来实现全息图变换。在目前的工作中，背景流具有零质量通量，就像海滩上真正的水流所需要的那样。此外，与以前的工作相比，本方法允许单独研究剪切流强度和底部剖面斜​​率的影响。这使我们能够在保持底部斜率不变的情况下提供作为剪切流函数的上升估计。