In this paper, a general solution to the field equations of a generalized thermoelastic medium with double porosity has been obtained. To investigate the problem, we use the Lord–Shulman theory in the thermoelasticity. The half-space of an isotropic homogeneous thermoelastic material is considered. Using the normal mode analysis and the numerical inversion technique, the analytic expressions of the physical quantities are obtained. Numerically, computed results for these quantities and its depicted graphically lead to study the effect of porosity. Comparisons in the presence and absence of double porosity, in two different times, are obtained. Although the problem has been solved theoretically, it is possible for researchers to benefit from their results in many different sciences, for example, in the field of geophysics, earthquake engineering, along with seismologist working in the field of mining tremors and drilling into the crust of the earth.