## 2 Statement of the Problem  
Consider a homogeneous, isotropic, linearly elastic half-space \(z\geq 0\) in the Cartesian coordinate system \((x,y,z)\). The half-space contains a cavity on the surface as shown in Figure 1a. A two-dimensional plane surface wave propagating in the \(x\)-direction is incident on the cavity. The interaction of the surface wave with the cavity generates a scattered wavefield \(\mathbf{u}_{sc}\) (Figure 1c), which is of interest and will be studied theoretically in this paper. Using the decomposition principle, which is demonstrated in Figure 1, the total wavefield \(\mathbf{u}_{to}\) (Figure 1a) is equal to the summation of the incident wavefield \(\mathbf{u}_{in}\) (Figure 1b) and the scattered field \(\mathbf{u}_{sc}\). It should be noted that the incident field considered here is a two-dimensional (2D) surface wave, while the scattered field is a 3D surface wave. The far-field displacements of 3D surface waves radiated from a cavity were determined analytically and experimentally and are presented in the next sections.  
With the linear decomposition given in Figure 1, the scattered field is equivalent to the field generated by the application of a distribution of the tractions applied on the surface of the cavity. These traction components are equal in magnitude but opposite in sign to the corresponding tractions due to the incident wave on a virtual cavity in the half-space without the cavity. Therefore, they can be computed from the stress components of the incident Rayleigh wave and the outward normal vectors of the cavity. It should be noted that for the scattered problem given in Figure 1c, the calculated tractions will generate both body waves and three-dimensional surface waves. It was discussed in the literature (see page 55 of [PERSON] and [PERSON] (1967) and page 321 of [PERSON] (1967) for details) that for the three-dimensional case, the surface waves attenuate with a distance from the point of excitation according to \(r^{-1/2}\) while the body waves decay according to \(r^{-1}\), where \(r\) is the distance from the point of application. Therefore, the surface waves dominate at a sufficiently large distance from the excitation point. The displacements of the scattered surface wave field are of interest and are determined in this article.  
Two analytical approximation approaches will be presented. When the depth and the width of cavities are both much smaller than the incident wavelength, the tractions acting on the cavity surface can be replaced by three resultant loads applied at the origin of the coordinate system.  
Figure 1: Linear decomposition technique: (**a**) the interaction of the incident surface wave with the three-dimensional cavity, (**b**) the propagation of the incident wave in the half-space without the cavity, and (**c**) the scattered field produced by the incident wave interaction with the cavity.  
This approximation is simple in the calculation but is supposed to be less accurate when the size of the cavity (depth and width) is comparable to the incident wavelength. As an improvement, a distributed approach is considered in which the application of the distributed tractions acting on the cavity surface is shifted to the surface of the half-space. The traction components are first computed for every point of the cavity surface. The reciprocity theorem is then applied to the equivalent loads that are applied on the surface of the half-space to obtain the displacement amplitudes of the scattered field. The total displacement amplitude of the scattered field is a superposition of the amplitudes generated by the loads at every point on the surface.