We investigate three and four-wave resonances of capillary–gravity water waves arising as the free surface of water flows exhibiting piecewise constant vorticity. More precisely, our type of flow has a jump in the vorticity distribution that separates a rotational layer at the top (commonly generated by wind-shear) from another rotational layer adjacent to the sea-bed, region that accommodates strong sheared currents. Instrumental in deriving our findings is the dispersion relation for such flows having a jump in the distribution of the vorticity. In general, for rotational flows of non-constant vorticity, the dispersion relation is very intricate. However, we show that a disentanglement occurs in the case of capillary–gravity water waves for which the ratio “thickness of the near-surface vortical layer/the wavelength of the surface wave” is sufficiently large. More precisely, we find explicitly two solutions, ${\lambda }_0$ and ${\lambda }_1$ that represent the (relative) surface wave speed. We then confirm analytically that ${\lambda }_1$ gives rise to three-wave resonances for capillary–gravity water waves with wavelengths not exceeding 2 cm. However, for significant wavelength range, we establish that ${\lambda }_1$ does not lead to four-wave resonances. In contrast to the previous conclusion ${\lambda }_0$ does not bring about three-wave resonances, but is able to generate four-wave resonances.

我们研究毛细重力水波的三波和四波共振，这是由于水流的自由表面呈现分段恒定涡旋而产生的。更准确地说，我们的流动类型在涡度分布中具有跳跃性，将顶部的旋转层（通常由风切变产生）与邻近海床的另一个旋转层（容纳强剪切流）分开。有助于得出我们的发现的是这种流动在涡度分布中具有跳跃性的色散关系。通常，对于非恒定涡旋的旋转流，色散关系非常复杂。然而，我们表明，在毛细管重力水波的情况下，“近表面涡旋层的厚度/表面波的波长”之比足够大，就会发生纠缠。更确切地说，我们明确找到两种解决方案，${\lambda }_0$ 和 ${\lambda }_{\mathrm{1个}}$代表（相对）表面波速度。然后我们通过分析确认${\lambda }_{\mathrm{1个}}$引起波长不超过2 cm的毛细管重力水波的三波共振。但是，对于较大的波长范围，我们确定${\lambda }_{\mathrm{1个}}$不会导致四波谐振。与先前的结论相反${\lambda }_0$ 不会引起三波谐振，但是能够产生四波谐振。