Passive control of 3D wave propagation with a functionally graded layer

An analytical formulation is presented for stress wave propagation in a half-space or a full-space, each constructed from a combination of homogeneous half-space(s) and finite thickness layers, which are either homogeneous or continuously inhomogeneous, totally subjected to arbitrary P-, SV- or SH-waves incoming from far field. The governing equations of motion in both homogeneous and continuously inhomogeneous domains are uncoupled with the use of potential functions, after which the variable separation method is employed to achieve modified Bessel ordinary differential equations in nonhomogeneous domain and scalar wave equations in homogeneous domain, both with the known solutions. Subsequently, employing the transfer matrix method, the solution is extended to the layered media. It is shown that if the half-space is coated by a Functionally Graded (FG) layer, the characteristics of FG-layer can be selected in such a way that it prevents traveling of the main energy of the wave to the surface, and decreases the displacement response at the zone adjacent to the free surface of the FG-layer significantly. This may have important applications in passive control of the wave induced load such as seismic load. In addition, it is shown that when two half-spaces with different characteristics are connected with the use of a middle specific FG-layer sandwiched in between the half-spaces, in the case of less stiffness for the second half-space, the wave propagation pattern is changed in such a way that the reflected wave to the first medium is approximately omitted. Moreover, in this case, the incoming waves with different incident angles will be guided to the directions nearly normal to the interface, which can increase the efficiency of absorbing boundaries, if used. As another application, FG-layer can be used for filtering of the incoming wave, and transmitting some specific waves.