## 1 Introduction  
Understanding the interaction of elastic waves with surface defects like cracks, corrosion pits, topographic irregularities, etc., is critical for the potential development of ultrasonic nondestructive evaluation (NDE). However, the wave scattering problem remains a challenging area of research due to the complexity of the scattering phenomenon. For this reason, most of the works in the literature are performed with the use of numerical methods, such as the finite element (FE) method [1], the boundary element (BE) method [2], or the finite element boundary integral (FE-BI) approach [3]. For a three-dimensional (3D) scattering problem, investigation with numerical methods is highly computationally expensive. Computational times remain of the order of several hours for one particular incident wavefield, even on powerful machines. Analytical or semi-analytical solutions to such problems are therefore desirable. They provide independent validation of numerical methods and enable rapid simulations of scattering from large populations of different scatterers. They also enable the relative contributions of different modes (e.g., propagating or non-propagating) to be examined to provide insight into the physics of the scattering process. For the present paper, we studied the general three-dimensional problem of surface wave scattering by a cavity of arbitrary shape on the surface of a half-space. Two approximate methods using concentrated loading and distributed loading for the theoretical analysis of the scattered field are introduced. Examples of calculations include the scattering of surface waves by a spherical, a circular cylindrical (coin-shaped), and a square cylindrical cavity. Experimental work was carried out to verify the analytical solutions and it is followed by a discussion on the results of comparison. The investigation is the very first theoretical approach for obtaining the solutions of the scattering of surface waves by a three-dimensional cavity of arbitrary shape. Closed-form expressions that represent the scattered wave field due to the interaction with the cavity are derived. For these explicit solutions, computation of the scattered amplitudes is extremely simple. The approximation models can offer a much lower computational cost and time with a reasonable accuracy in comparison with the existing numerical methods. They are also able to provide insight into the physics of the scattering process that is important to solving inverse scattering problems. With respect to earlier work, much of the work in the literature has been concerned with scattering by two-dimensional surface-breaking cracks. Typical examples of analytical work are References [4; 5; 6]. Semi-analytical solutions were obtained, for example, in References [7; 8]. Numerical work was carried out using the finite element method [1] and the boundary element method (see References [9; 10]). The scattering of surface waves by a two-dimensional cavity was theoretically investigated in our earlier works [11; 12; 13]. Defect detection in rails using surface waves was also studied by [PERSON] and [PERSON] [14]. In a related category are papers on scattering by strips and grooves [15; 16; 17; 18; 19]. A review of the scattering of elastic waves by defects was presented in [PERSON] [20] (pp. 141-143). The related problem of the scattering of Lamb waves by a surface defect in a layer has received considerable attention (see, for example, References [2; 21]). In three dimensions, the scattering of elastic waves by canyons of arbitrary shape was studied using an indirect boundary integral equation method [22]. The 3D scattering of guided waves by a through-thickness cavity with an irregular shape in an isotropic plate is investigated in [PERSON] et al. [23] and scattering signal analysis of a cylindrical structure from a torsional wave is presented by [PERSON] [24]. An analytical solution for the scattering of ultrasonic guided waves by flat-bottomed cavities with arbitrary shape in a plate was introduced in [PERSON] et al. [25].