## 1 Introduction  
For high-speed moving objects, the current methods to reduce the flow-noise mainly include changing the shape of the object [1, 2] and controlling the flow on the surface of the object [1, 3]. Wherein the first one is the passive noise reduction method to change the actual shape of the object, such as the serration shape [4, 5], the wavy shape [6, 7] and bionic shape [8, 9, 10]. The other one is the active noise reduction method to control the boundary layer flow near the surface of the object, such as blowing [11, 12], suction [13], air curtain [3], plasma excitation [14, 15] and porous structure [16, 17]. In essence, the second one is to generate a kind of virtual shape of the object via energy input or absorption. There will be a consumption of energy for these flow control methods to change the local flow field. In contrast, the first one is a method of zero energy consumption. Therefore, the passive method will become a more promising and engineering-favorable noise reduction method due to no consumption advantage. Adopting the wavy shape is an important passive method to reduce flow-noise. Early research shows that the wavy shape was used to reduce drag [18, 19, 20]. A lot of research on the drag reduction using wavy shape has been conducted via numerical simulation [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]including the large eddy simulation method and three-dimensional POD method. Meanwhile, many experiments for drag reduction also have been performed [31; 32; 33; 34; 35; 36] to obtain the surface pressure distributions and flow visualization results. Furthermore, the numerical simulation and experimental studies were coupled [6; 37; 38; 39] to obtain more detailed analysis including a refined flow field structure. The significant differences in the wake vortex patterns and fluctuating pressure between the surface wavy shapes and baseline shape were revealed and verified. Along with the development of some related research, it is found that the wavy shape can not only reduce the drag, but also decrease the flow-noise [40; 41; 42] of the leading edge or trailing edge. According to the published literature analysis, the research of noise reduction using a wavy cylinder is mainly related to numerical simulations [43; 44; 45] but experiments in aeroacoustic wind tunnels are relatively less [6]. The large eddy simulation (LES) method [46; 47; 48] and direct numerical simulation (DNS) method [49] are commonly used as the computational tool for noise reduction studies, as they are more reliable and accurate than the Reynolds Averaged Navier-Stokes (RANS) method. Furthermore, the idea of noise reduction using wavy objects is not just limited to cylinders, but also extended to airfoils. In addition, there is also a considerable noise reduction effect for airfoils [50; 51; 52; 53]. In fact, the wavy shapes for noise reduction include the square wave (Figure 1a), sawtooth wave (Figure 1b), the cosine wave (Figure 1c) and the sine shape (Figure 1d). In essence, there are common features of periodic variation shapes with both peaks and troughs. However, the curves connecting the peaks and troughs are quite different. The main parameters controlling these shapes are the wavelength of a period, the amplitude of a period wave and the aspect ratio (wavelength divided by amplitude). The most common representative curve is the cosine or sine function of the wave shape. For simplification, the role of this periodic cosine function of the wave shape on the noise reduction and its mechanism is studied in the present study. Moreover, some reference paper [26; 35] mentioned that, a large spanwise wavelength of wavy shapes can change the wake of a sine wavy cylinder for drag reduction. Generally, the wake is closely related to the far-field noise [54]. For further consideration, maybe it can reduce the flow-noise if the large spanwise wavelength and large aspect ratio are coupled on the cosine wavy shape.