Scattering of an elastic wave by a rigid sphere in a semi-bounded domain

The problem of scattering of plane elastic waves by a rigid sphere near a rigid boundary is considered. This leads to the appearance of multiply re-reflected dilatation and shear waves, which generate strong oscillations of the wave field. The problem for a vector operator of the shear waves is reduced to the definition of scalar functions as a consequence of symmetry. Approximate formulas for the far field and the long-wave Rayleigh approximation are presented. The construction of multiply re-reflected waves by the image method is presented and analyzed. Calculations of the scattered wave fields are plotted in the form of scattering diagrams .


Scattering of an elastic wave by a rigid sphere in a semi-bounded domain
in multiply connected domains is presented in (Selezov et al., 2018) [2]. In (Jackson, 1962) [3] a useful presentation of the image method is given. The construction of multiple scattered fields is presented in (Selezov et al., 2018) [2]. The basic relations in the spherical coordinate system are considered in (Kratzer & Franz, 1963) [4], for cylindrical functions in (Watson, 1945) [5]. The addition theorem for spherical functions is given in (Friedman & Russek, 1954) [6], for cylindrical functions in (Watson, 1945) [5]. Diffraction of plane waves by a rigid sphere in infinite domain was considered in (Knopoff, 1959) [7], where previous studies in the field of seismology are noted, and in particular in (Ying & Truell, 1956) [8]. Diffraction of elastic waves by an elastic sphere was considered in (Jain & Kanwal, 1980) [9]. This article presents the statement of the problem of diffraction of elastic waves by a rigid spherical inclusion located near a flat rigid boundary. The transformation of a vector field to the definition of scalar functions is considered. The far-fields and approximate solutions in the long-wave Rayleigh approximation are constructed. It is shown how to construct a primary and secondary fields from geometric considerations using the image method. Calculations of the re-reflected wave fields are carried out and their strong oscillations are shown.

Statement of the problem
We consider a spherical coordinate system (radial, zenith and azimuthal coordinates) corresponding to a rectangular Cartesian coordinate system (Fig. 1). The axis Oy is perpendicular to the flat boundary with the origin of coordinates in the center of an rigid spherical inclusion (scatterer) and is directed from infinity to the flat boundary. When plane waves propagate from infinity (plane waves propagate along the Oy axis), a diffracted field of multiply re-reflected waves occurs in the system.
The motion of an elastic medium is described by the equations and the displacement vector is defined by the formula The boundary conditions on the sphere and on the flat boundary are 0, 0 Conditions (4) mean that the displacement vector on the surface of the sphere r a  is zero, and on the surface of the flat boundary y h   only a normal component of the displacement takes place while the shear stress are zero (slip).The required functions must also satisfy the conditions of Sommerfeld radiation. Dimensionless quantities are realized everywhere. The characteristic values are: length [m] is the radius of a sphere a , time [s] is 1 /  , kilogram-mass [kg] is Young's modulus E . Thus, the dimensionless distance from the center of the sphere to the flat boundary is / h a . In the far field approximation, the quantity / h a is assumed large 1 h a  , and in the Rayleigh long-wave approximation, the quantities pa and qa are The expression for the incident wave (1) corresponds to the dilatation waves, so from (1) and (3) we get The potential  is determined to an arbitrary function, which can be taken 0 f  and written in accordance with (2) the equation for the incident wave Hence, the function  in (6) defines the potential of the incident wave.
  In the case of axial symmetry   0 we obtain from (7)   The third term in expression (8) is also zero, since 0 a   . The components a  are the projections of the vector a  to the coordinate line  and are zero in the case of axial symmetry. This can be established from physical considerations: shear deformations are possible only in planes normal to lines  . For a dilatation field (expansion -compression waves), the problem is solved simply (Seismic diffraction, 2016) [11]. In the case of the equivoluminal field characterizing the tangential stresses, the problem is more complicated. When a flat incident wave is incident on a spherical scatterer in the direction of the Oy axis, the elastic displacements due to axial symmetry, i.e. shear stresses and corresponding deformations will be only in directions orthogonal to the coordinate line e   . As a result, we obtain that the construction of an equation for  , one can introduce a scalar function   , r   with normalization 0 u depending on two scalar arguments r and  , i.e. we obtain the scalar wave equation for the expression 2 a   in (8) also depends on two coordinates r and  . By analogy with the scalar function from which, after the separation of variables, the Legendre equation and the Bessel equation for spherical functions follow.To implement the image method in the semi-bounded region using the above formulas, the well-known solution of the diffraction of elastic waves on a sphere in the infinite region is applied (Knopoff, 1959) [7].

Approximate solution
Solutions in the infinite domain for functions  and  are written as The pa values correspond to the ratios of the size of the scatterer а to the wavelengh l. The smaller the pa, the better the series converge, and for large pa the solution in the series is inapplicable and it is necessary to pass to high-frequency asymptotics. In the work (Knopoff, 1959) [7], scatterplots of waves with different values of pa are given, which leads in some cases to incorrect results that do not satisfy the Rayleigh approximation 1 pa  , and the series do not converge well. In the Rayleigh approximation, the inequalities 1 pa  1 qa  are valid. In this case, from (10) and (11) it can be established that the coefficients 1 a and 1 b are dominant Relations (12) where the total displacement components for the scattered multiplicity field of the multiplicity k are The difference in distances from the real and imaginary obstacles to a certain point , r  and the difference in time between the arrival of P  and S  waves in the first approximation are taken into account by the formulas Formulas (15) follow from the geometrical relations obtained below for main and mirror obstacles. According to formulas (12) -(15), after a series of transformations for a single scattered field,, we find     In the right-hand sides of (16), (17), the multiplier   Thus, a single diffracted field is described by formulas (16), (17). The first approximation includes the field from the incident wave (before reflection from the flat boundary in the coordinate system with origin in 0 z  ) and the field from the reflected waves from the flat boundary to scattering (this field is described in the coordinate system with origin in * 0 z  ), which includes only normal components in connection with slippage (the fourth condition in (2)). Waves scattered on a sphere, reach the border, these scattered waves are reflected from the border and again scattered on the sphere. It is all the same that waves in image coordinates ( * * * , , r   ) are scattered, but they must be converted into coordinates , , r   by the addition theorem. This completes the construction of solutions of the primary field.
4. Geometric relations and the second approximation Note that the real and mirror fields are defined in different coordinate systems. In some cases, the mirror field can be expressed by means of geometric transformations and the use of addition theorems in the variables of the main field. We present the basic formulas for an obstacle characterized by functions     * * exp n Z pr in (see Fig. 2). In the case of axial symmetry (diffraction of plane waves), it is sufficient to consider the change in the plane with the meridional line  , but using the addition theorems for spherical or cylindrical functions.  also to estimate the modules of the coefficients 1 a , 1 b . In the case of a two-term approximation, the data given above can be applied.

Conclusion
In this article, we have investigated using the image method the problem of plane elastic wave scattering by a rigid sphere located at some distance from a flat rigid boundary. On the sphere surface the conditions of zero displacements are satisfied, and on the flat boundary, the conditions for slippage are satisfied, i.e. a zero tangent stresses. Exact solutions in spherical functions have written in an infinite region, from which approximate solutions are obtained in the case of the far field (a large distance of the sphere from the flat boundary) and the long-wave Rayleigh approximation. As a result, solutions describing a multiply re-reflected wave field for the primary and secondary fields were obtained. Formulas from geometric relations were derived and using them the single and secondary fields were constructed. Calculations have been carried out and scattering diagrams plotted showing a strongly oscillating wave field. literatura