This paper describes linear stability analysis of the two-dimensional steady motion of periodic deep-water waves with symmetric non-overhanging profiles propagating on a linear shear current, namely a vertically sheared current with constant vorticity. In order to investigate numerically with high accuracy the stability of large-amplitude waves, we adopt a formulation using conformal mapping, in which the time-varying water surface is always mapped onto the real axis of a complex plane. This formulation allows us to apply numerical methods developed for large-amplitude irrotational waves without a shear current directly to the present problem, and reduces the linear stability problem to a generalized eigenvalue problem. Numerical solutions describe both super- and sub-harmonic instabilities of the periodic waves for a wide range of wave amplitudes and clarify how the behaviours of dominant eigenvalues change with the strength of the shear current. In particular, it is shown that, even in the presence of a linear shear current, the steady periodic waves lose stability due to superharmonic disturbances at the wave amplitude where the wave energy attains an extremum, similarly to the case of irrotational waves without a shear current. It is also found that re-stabilization with an increase in wave amplitude characterizes subharmonic instability for weak shear currents, but the re-stabilization disappears for strong shear currents.

中文翻译：

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基于非定常共形映射的线性剪切流上的深水波稳定性分析

本文描述了具有对称非悬垂剖面的周期性深水波在线性剪切流（即具有恒定涡度的垂直剪切流）上传播的二维稳态运动的线性稳定性分析。为了高精度地数值研究大振幅波的稳定性，我们采用了一种使用保角映射的公式，其中时变水面总是映射到复平面的实轴上。该公式使我们能够将针对无剪切流的大振幅无旋波开发的数值方法直接应用于当前问题，并将线性稳定性问题简化为广义特征值问题。数值解描述了周期波在宽范围波幅下的超谐波和次谐波不稳定性，并阐明了主导特征值的行为如何随剪切电流的强度而变化。特别是，它表明，即使在线性剪切电流的存在下，由于波幅处的超谐波扰动，在波能量达到极值的情况下，稳态周期波失去稳定性，类似于没有剪切的无旋波的情况当前的。还发现随着波幅增加的重新稳定表征弱剪切流的次谐波不稳定性，但重新稳定对于强剪切流消失。即使在存在线性剪切流的情况下，稳定的周期性波也会由于波幅处的超谐波扰动而失去稳定性，其中波能量达到极值，类似于没有剪切流的无旋波的情况。还发现随着波幅增加的重新稳定表征弱剪切流的次谐波不稳定性，但重新稳定对于强剪切流消失。即使在存在线性剪切流的情况下，稳定的周期性波也会由于波幅处的超谐波扰动而失去稳定性，其中波能量达到极值，类似于没有剪切流的无旋波的情况。还发现随着波幅增加的重新稳定表征弱剪切流的次谐波不稳定性，但重新稳定对于强剪切流消失。