Abstract The assessment of dispersive media may become a challenge due to the high attenuation ratios, anisotropy and geometrical particularities of the specimen under study. Besides, non-contact methods that preserve the integrity of samples are needed when some degree of purity is required, e.g., when water immersion is not allowed, and the direct contact is not feasible. In this work, we study composite plates, in which acoustic dispersion is present. We use a non-contact hybrid optoacoustic system to implement the experimental setup that makes it possible the detection of the non-exposed damages of the samples. Different relevant aspects, such as cost, maximum frequency achieved, bandwidth, sensitivity, signal-to-noise ratio and spatial resolution were considered before setting the experimental configuration. Though dispersion curves are reconstructed, we emphasize the use of the instantaneous information of signals to detect the faults. Moreover, the instantaneous phase front of the acoustic field becomes a relevant complementary tool for ultrasonic inspection and characterization of dispersive media. In particular, it highlights hidden characteristics which are not visible with other commonly-used representations, namely, the instantaneous amplitude, the time-frequency distributions or the dispersion curves widely employed in non-destructive testing. This work provides an alternative approach to study highly attenuating media where the signal-to-noise ratio is a challenge, and the available scanning path is limited, e.g., layered and poroelastic media such as biological tissues.