This thesis is part of the research at Labtau (Laboratory of therapeutic applications of ultrasound) and ISTerre (Institut des sciences de la terre - earth science institute) at the interface of medical imaging and seismology, two research disciplines that are based on the propagation of elastic waves. It investigates the nature of elastic wave propagation in complex conditions by ultrafast ultrasound imaging, also known as transient elastography or shear wave imaging. This medical imaging technique allows for retrieval of the dynamic shear wave field inside a soft elastic material and is commonly applied in hospitals for elasticity mapping in, e.g., the liver and breast. In the present manuscript, two research questions of interest for bio- and geophysics are tackled. The first part treats elastic wave propagation in porous materials. The dispersion of the shear and secondary compression wave in lung-mimicking materials is analyzed experimentally and compared to Biot's theory of poro-elasticity. The results show a good agreement for the shear wave and qualitative agreement for the secondary compression wave. This has direct implications for elasticity imaging: the properties of the viscous fluid govern the shear wave dispersion in highly porous soft elastic materials. The thesis thus contributes to the emerging branch of lung elasticity imaging. The results could have clinical implications for other organs as well. The liver and spleen contain a high percentage of blood, a non-Newtonian fluid which exposes a highly varying viscosity. The conclusions drawn from the comparison of the experimental results and poro-elastic theory imply, that the role of the pore-filling fluid should be investigated in liver elastography: The clinically observed dispersion of shear waves in the liver remains partly unexplained by purely visco-elastic models. Furthermore, the experimental proof of the secondary compression wave is of general interest for poroelasticity. Originally, this wave has been the object of geophysical studies and has scarcely been shown experimentally. In the second part, the ultrafast ultrasound shear wave imaging technique is applied to a geophysical research question. What does the elastic wavefield, which is emitted by a frictional instability, reveal about the nature of dynamic rupture propagation? How does rupture, the process behind earthquakes, nucleate? By mapping the shear wave-field during rupture of a granular asperity at the source point and in the medium, unique insights into rupture nucleation are gained. The experimental setup, which relies on soft elastic phantoms, is shown to reproduce many characteristics of sliding friction that have been show for real rocks in the earth and the laboratory. These include supershear and sub-Rayleigh rupture propagation, a nucleation phase and stick-slip friction. Neither a singular-force nor a double-couple source mechanism explain the entirety of observed rupture modes. Finally, in order to statistically analyze the complex spatio-temporal evolution of the presented experiment, a semi-automated data analysis workflow, taking advantage of image segmentation and computer vision, is suggested.