Whilst the dynamics of tapered structures have been extensively studied numerically and analytically, very few experimental results have been presented to date. The main aim of this paper is to derive and demonstrate an experimental method enabling both axial and flexural wavenumbers in exponentially tapered bars to be estimated. Our particular interest in this type of tapering is motivated by its occurrence in naturally grown structures such as tree roots, with an outlook towards remote root mapping. Decomposing a dynamic response into a sum of contributing waves, we propose a method in which two independent wavenumbers can be calculated from five equispaced measurements. The approach was demonstrated in an experiment on a freely suspended wooden specimen supported by theoretical modelling. For axial waves we used the well-established elementary rod theory, whereas for flexural waves we build a piecewise uniform model based on the Timoshenko beam theory. The estimates calculated from the experimental data were compared with the analytical and numerical results and showed good agreement. The limitations of the method include an appropriate choice of sensor spacing, the effect of sensor misalignments and the assumption of small wavenumber variation for flexural waves.