Abstract: A spatial and temporal multiscale asymptotichomogenization method simulating the wave propagationproblem in periodic multiphase materials is systematically studied.Generalized function field governing equations of wave propagation are expressedin a unified form with both inertia and velocity items. Amplified spatialand reduced temporal scales are, respectively, introduced to account forspatial and temporal fluctuations and nonlocal effect of the homogenizedsolution due to material heterogeneity on different time scales. The model isderived from the higher-order homogenization theory with multiple spatialand temporal scales. By combining various orders of homogenized functionfield equations, the reduced time dependence is eliminated and then thefourth-order differential equations are derived. To avoid the necessity ofC1-continuity in finite element implementation, the C0-continuousmixed finite element approximation of the resulting nonlocal equations offunction field is put forward. Non-Fourier heat conduction and thermaldynamic problem are computed to demonstrate the efficiency and validity ofthe theories and models developed and indicate the disadvantages of theclassical spatial homogenization.