Spherical wave attenuation under multiple energy source in viscous fluid-saturated elastic porous media

In viscous fluid-saturated elastic porous media, the spherical dilatational wave displacement equations are developed on Biot's theory and the spherical shear wave displacement equations are developed based on a modified wave frame. Analytical forms of the solutions are introduced by considering the wave magnitude dispersion due to the viscosity of Newtonian fluids. The attenuation of dilatational wave and shear wave are separately studied. Due to the high attenuation of the shear wave, the superposition characteristic is only considered for dilatational waves under the multiple energy source models. Many advantages of the multiple sources over the single source have been theoretically or numerically demonstrated, such as multiple energy sources can produce a larger relative displacement than single source, the wave direction is controllable and the magnitude of the relative displacement can be adjusted as required under the multiple source.