A time-dependent and fully nonlinear numerical model is employed to solve the Navier-Stokes equations in two spatial dimensions and to describe the propagation of a Gaussian gravity wave packet generated in the troposphere. A Fourier spectral analysis is used to analyze the frequency power spectra of the wave packet, which propagates through and dwells within several thermal ducting regions. The frequency power spectra of the wave packet are derived at several discrete altitudes, which allow us to determine the evolution of the packet. This spectral analysis also clearly reveals the existence of a stratospheric duct, a mesospheric and lower thermospheric duct, and a duct lying between the tropopause and the lower thermosphere. In addition, we determine the spatially localized wave kinetic energy density and the horizontally averaged, time-resolved, normalized vertical velocity. Examination of these diagnostic variables allows us to better understand the process of wave ducting and the vertical transport of wave energy among multiple thermal ducts. The spectral analysis allows us to unambiguously identify the ducted wave modes. These results compare favorably with those derived from a full-wave model.