Cross-waves are standing waves with crests perpendicular to a wave-maker; they are subharmonic waves excited by parametric instability. The modulational and chaotic behaviours of nonlinear cross-waves have been studied widely since the 1970s. Most of the previous work has focused on gravity waves where surface tension can be neglected. In this work we study cross-waves that are highly dependent on surface tension as well as gravity. By oscillating a planar wave-maker either vertically or horizontally with frequencies of 25 Hz through 40 Hz at one end of a rectangular basin, two-dimensional multi-component surface patterns are realized. Using the free-surface synthetic Schlieren technique to measure the surface elevations, multi-dimensional Fourier transforms are utilized to track the evolutionary spectrum of the water surface in both the temporal and spatial domains. Wavelet transforms are implemented to show the development of the various frequency components. Three-wave resonances with and without first subharmonics are observed for small nonlinearity. Three-dimensional oblique propagating cross-waves are generated at higher nonlinearity; unlike most previous cross-wave experiments, this staggered pattern propagates far downstream. Experimental evidence shows that two oblique propagating waves form a two-dimensional short-crested pattern, and that the lateral component of the waves develops into parametric sloshing modes corresponding to the width of the tank. Two regimes of nonlinear wave patterns, resonant triads and oblique propagating cross-waves, are delineated.

中文翻译：

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由单频造波器振荡和随后的共振产生的重力-毛细管多分量波型

横波是波峰垂直于造波器的驻波；它们是由参数不稳定性激发的次谐波。自 1970 年代以来，非线性交叉波的调制和混沌行为得到了广泛的研究。以前的大部分工作都集中在可以忽略表面张力的重力波上。在这项工作中，我们研究高度依赖于表面张力和重力的横波。通过在矩形盆的一端以 25 Hz 至 40 Hz 的频率垂直或水平振荡平面波发生器，实现了二维多分量表面图案。使用自由表面合成纹影技术测量表面高程，多维傅里叶变换用于跟踪时空域中水面的演化谱。实现小波变换以显示各种频率分量的发展。观察到具有和不具有第一次谐波的三波谐振，因为非线性很小。非线性度较高时产生三维斜向传播交叉波；与大多数以前的交叉波实验不同，这种交错模式传播到下游很远。实验证据表明，两个斜向传播的波形成一个二维短波峰，并且波的横向分量发展为与水箱宽度相对应的参数晃动模式。两种非线性波形模式，共振三重奏和倾斜传播的交叉波，