We investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

中文翻译：

---

深水横向调制有限振幅毛细管波的线性稳定性

我们研究了在整个波幅范围内在无限深度的水上以永久形式进行的纯毛细管波的周期性运动的三维线性稳定性。在针对二维稳定毛细管波引入基于共形图的坐标变换之后，我们对变换空间中的有限振幅毛细管波进行了线性稳定性分析。为了解决小幅度扰动的线性化方程，假设横向扰动的波长比传播方向的波长长得多，因此，扰动是弱三维的。这个假设以及解的周期性使得我们可以将线性化方程写成矩阵形式的特征值问题。遵循矩阵的摄动理论，我们通过测量弱三维的小参数扩展了该特征值问题的解，并通过数值获得了近似的特征值。对于弱三维超谐波扰动，数值结果表明，对于几乎所有波幅，纯毛细管波都是二维稳定的，但是三维不稳定的。另一方面，对于已知为二维不稳定的次谐波扰动，发现横向上的长波扰动会降低临界振幅附近的二维增长率，超过该临界振幅纯毛细管波就不稳定了。 。并在数值上获得近似特征值。对于弱三维超谐波扰动，数值结果表明，对于几乎所有波幅，纯毛细管波都是二维稳定的，但是三维不稳定的。另一方面，对于已知为二维不稳定的次谐波扰动，发现横向上的长波扰动会降低临界振幅附近的二维增长率，超过该临界振幅纯毛细管波就不稳定了。 。并在数值上获得近似特征值。对于弱三维超谐波扰动，数值结果表明，对于几乎所有波幅，纯毛细管波都是二维稳定的，但是三维不稳定的。另一方面，对于已知为二维不稳定的次谐波扰动，发现横向上的长波扰动会降低临界振幅附近的二维增长率，超过该临界振幅纯毛细管波就不稳定了。 。