Objective - Shear wave elastography has enriched ultrasound medical imaging with quantitative measurements of tissue stiffness. However, this method still suffers from some limitations due to viscoelasticity, guiding geometry or static deformations. Approach - To explore these limitations, a nearly-incompressible soft elastomer strip is chosen to mimic the mechanical behavior of an elongated tissue. A supersonic shear wave scanner measures the propagation of shear waves within the strip. By repeating the experiment on the same sample for different orientations and static strains, the scanner estimates the shear wave velocity in a wide range from 2 to 6 m/s. Main results - The effect of waveguiding is highlighted and the spatio-temporal Fourier transform of the raw data provides dispersion diagrams. We provide a theoretical model that accounts for the static deformation and allows the extraction of the mechanical parameters of the sample, including its rheology and hyperelastic behavior. Significance - To overcome some limitations of current elastography, we propose a method that would allow the simultaneous characterization of the viscoelastic and hyperelastic properties of soft tissues, paving the way for robust quantitative elastography of elongated tissues.