This paper presents experimental results for wave propagation in an anisotropic multilayered structure with linearly varying cross section. Knowing the dispersion and wave propagation properties in such a structure is of great importance for non-destructive material testing and structural health monitoring applications for accurate damage detection and localization. In the proposed study, the wavefield is generated by a circular piezoelectric wafer active sensor and measured by a scanning laser-Doppler-vibrometer. The measurements are compared with a theoretical group delay estimation and a signal prediction for the antisymmetric wave motion along the non-uniform propagation path. The required dispersion curves are derived from the well-known global matrix method for segments of constant thickness. A multidimensional frequency-wavenumber analysis of linescan data and the full wavefield provides further insight of the adiabatic wave motion because the wavenumber changes along the tapered geometry of the waveguide. In addition, it is demonstrated that a terahertz time-domain system can be used in glass-fiber reinforced plastic structures as a tool to estimate the thickness profile of thin structures by means of time-of-flight measurements. This information is particularly important for guided wave-based diagnostics of structures with unknown thickness.