A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

中文翻译：

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深水宽带宽三维表面重力波

针对海洋表面波提出了一种新的非线性薛定谔方程（NLSE）。描述深水波演变的 NLSE 的早期推导仅限于窄带宽，其中波陡度中的二阶束缚波在前导近似中描述。这项工作概括了这些早期的工作，以允许具有大方向传播的宽带宽深水波。新的 NLSE 允许简单的数值实现，并且可以以直接的方式进行扩展，以便解释有限深度水面上的波浪。对于二阶波的描述，本文提出了一种半解析方法，可以提供准确且计算有效的预测。在新 NLSE 的领先阶近似下，研究了斯托克斯波的不稳定区域和能量增长率。与基于 McLean 的精确结果相比 (J.流体机械。, 卷。511, 1982, p. 135)，新 NLSE 的预测比 Trulsen 的预测更一致等。(物理。流体, 卷。12, 2000, pp. 2432–2437)。通过新 NLSE 的数值实现，通过检查定向扩展聚焦波群的演变来研究波方向性的影响。由于能量变化率的不对称性比均匀斯托克斯波更复杂，因此观察到光谱峰向下移动。观察到线性焦点附近组附近的快速倾斜能量转移，这可能是由于均匀斯托克斯波的不稳定性引起的，该波出现在受斜边带干扰的窄谱中。