## 1 Introduction  
The wing-in-ground (WIG) craft [1] is a kind of aircraft that makes full use of ground effect at low altitudes to improve flight efficiency. It is mainly used for high-speed water transportation. Compared with ships, it has the advantages of fast speed and low resistance; compared with aircrafts, it has the advantages of a high lift-to-drag ratio and a high energy utilization efficiency. However, the WIG craft has unique aerodynamic characteristics when flying near to the ground which are significantly different from those of conventional aircraft. Therefore, many scholars have carried out research on the unique aerodynamic characteristics of the WIG craft [2; 3; 4; 5; 6; 7; 8]. The WIG craft is designed to fly at a low altitude, but there are various obstacles such as buildings, trees, hills, and other obstacles on the land. The vast ocean is flat, which provides a good application scenario for the WIG craft flying at a low altitude. Therefore, most WIG crafts are used for water transportation. Up to present, most of the researchers focusing on the ground effect assume that the ground is flat. However, there are often waves on the water surface. Therefore, it is necessary to study the ground effect under wavy ground conditions. [PERSON] [3] studied the aerodynamic characteristics of the airfoil in ground effect by solving the steady Reynolds Averaged Navier-Stokes equations (RANS). He found that ground clearance had a significant impact on the lift coefficient, and when the ground clearance was less than a specific value, the lift would be lost. [PERSON] [8] applied the numerical method to study the aerodynamic characteristics of the airfoil with a deflection flap in ground effect. [PERSON] [9] carried out numerical simulations on WIG craft with a power-augmented ram (PAR) engine. [PERSON] [4] numerically investigated the aerodynamic characteristics of compound wings with the ground effect. He concluded that, compared with the rectangular wing, the compound wing had greater lift, less drag, and better aerodynamic characteristics, especially in the case of small ground clearance. [PERSON] [10] investigated the aerodynamics and flow physics of a NACA4412 airfoil with ground effect for a wide range of angles of attack; his research showed that the angle of attack (AOA) versus ride height plane could be divided into three regions based on the sign of the lift increment value. Besides this, [PERSON] [11] also studied the aerodynamic properties of the dynamic ground effect (DGE), supposing that the distance between an aircraft and the ground keeps constantly changing during takeoff and landing. These studies all assume that the ground is a rigid flat ground, and preliminary studies of the ground effect are carried out on this basis. Different from the ground covered with obstacles, such as trees, buildings, hills, and so on, the flatness of the sea surface provides a good flying environment for WIG crafts, therefore this kind of aircraft is mainly used for high-speed transportation along the water surface. However, the actual sea surface is a free surface. To get closer to reality, some scholars begin to employ free surface instead of rigid ground to study the ground effect. [PERSON] [12] studied the water surface deformation when the wing moved over a free surface at a small ride height; her results showed that any deformation occurring on the water surface was likely to be caused by the wingtip vortices rather than an increased pressure distribution beneath the wing. With the increase in velocity, the deformation of the water surface became less significant. [PERSON] [6] numerically investigated the effect of an endplate on the 3-D WIG problem using the assumptions of steady incompressible viscous flow. He found that endplate could effectively increase lift and reduce drag with the ground effect, and the wingtip vortex will cause a small deformation of the water surface.