In order to determine the steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body, it is typically necessary to impose a radiation condition that allows for capillary waves upstream, but disallows those corresponding to gravity. However, this radiation condition is not known a priori and consequently, the computation of accurate numerical solutions to the steady-state problem remains problematic. Although the physical model might be modified (e.g. with viscosity), recovery of the original problem is not always possible. In this work, we discuss the above radiation problem, and show how, in the low-speed limit, the steady gravity-capillary waves can be resolved using a Sommerfeld-type boundary condition applied to an asymptotically reduced set of water-wave equations. These results allow us to validate the specialized classes of low-speed waves theoretically predicted by Trinh & Chapman (2013) using methods in exponential asymptotics [J. Fluid Mech. 724, pp. 392--424]. The issues of numerically solving the full set of gravity-capillary equations for a potential flow are discussed, and the sensitivity to errors in the boundary conditions is clearly demonstrated.

中文翻译：

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波浪-结构相互作用简化模型中的重力-毛细管波

为了确定由通过造波体的势流产生的稳态亚临界重力毛细波，通常需要施加辐射条件，允许上游毛细波，但不允许对应于重力的毛细波。然而，这种辐射条件不是先验的，因此，稳态问题的精确数值解的计算仍然存在问题。尽管可能会修改物理模型（例如，使用粘度），但并非总是可以恢复原始问题。在这项工作中，我们讨论了上述辐射问题，并展示了如何在低速极限下使用适用于渐近简化的水波方程组的 Sommerfeld 型边界条件来解析稳定的重力毛细波。这些结果使我们能够使用指数渐近学中的方法验证 Trinh & Chapman (2013) 理论预测的低速波的特殊类别 [J. 流体机械 724，第 392--424 页]。讨论了对势流的全套重力毛细管方程进行数值求解的问题，并清楚地证明了对边界条件误差的敏感性。