This study considers the propagation of attenuated waves in a double porosity solid half‐space permeated by two different pore regimes. Pores in both the regimes are saturated with the same viscous fluid. The compressional effect of the Rayleigh wave transforms into different fluid‐pressure fields in two pore regimes. The resulting pressure gradient in porous aggregate induces a local flow of fluid through the connected pores. In return, this local wave‐induced fluid flow affects the propagation and polarization of Rayleigh waves in a double porosity solid half‐space. The pores exposed at the surface control the fluid discharge and characterize the boundary of the porous solid to be pervious or impervious. For either boundary, the propagation and attenuation of Rayleigh waves are represented by a complex irrational dispersion equation, which is solved numerically after rationalizing into an algebraic equation. Complex solution of this dispersion equation is resolved to define the phase velocity and attenuation coefficient for the propagation of Rayleigh waves as an inhomogeneous wave in dissipative double porosity solid. A numerical example is solved to illustrate the propagation, attenuation, and polarization of the Rayleigh wave. Numerical results suggest the existence of an additional (second) Rayleigh wave in the double porosity solid saturated with low‐viscosity pore fluid.