The elevation of ocean waves is always modeled in linear theory as a superposition of the sinusoidal components with crests and troughs of identical heights. However, under some circumstances, the wave amplitude is outside the linear range and presents as a weakly nonlinear asymmetrical waveform with sharper crests and shallower troughs. We studied the impact of the weakly nonlinear effect of ocean waves in deep and intermediate waters on acoustic scattering from the surface of the ocean using two rough surface models with fractal geometry and power law spectral behavior in the equilibrium range. The classic Weierstrass–Mandelbrot function was used to model the linear waves and a new fractal function, the fractional Weierstrass function developed in studies of electromagnetism, was used to model the weakly nonlinear waves. We evaluated these two models using the Pierson–Moskowitz spectrum and the incident wavelength. The bistatic scattering strength was obtained via a numerical method based on the “exact” solution of the integral equation. The weakly nonlinear phenomenon led to a very small reduction in the narrow area around the specular reflection angle and a small increase in the remaining wide area, including the backpropagation area with a scattering angle < 90∘. The differences in backscattering strength between the two models were similar to the bistatic scattering strength in the backpropagation area and did not depend on the incident grazing angle.

中文翻译：

---

弱非线性波对海面声散射的影响

海浪的高程总是在线性理论中建模为具有相同高度的波峰和波谷的正弦分量的叠加。但在某些情况下，波幅超出线性范围，表现为波峰较尖、波谷较浅的弱非线性不对称波形。我们使用具有分形几何和平衡范围内的幂律谱行为的两个粗糙表面模型研究了深水和中水海浪的弱非线性效应对海洋表面声散射的影响。经典的 Weierstrass-Mandelbrot 函数用于模拟线性波，而新的分形函数，即在电磁学研究中开发的分数 Weierstrass 函数，用于模拟弱非线性波。我们使用 Pierson-Moskowitz 光谱和入射波长评估了这两个模型。双基地散射强度是通过基于积分方程“精确”解的数值方法获得的。弱非线性现象导致镜面反射角附近的狭窄区域减少非常小，而剩余的宽区域则略有增加，包括具有散射角的反向传播区域 < 90∘. 两种模型之间的反向散射强度差异与反向传播区域的双基地散射强度相似，并且不依赖于入射掠射角。