Fiber Bragg Gratings (FBGs) are increasingly used for ultrasonic wave measurements in the field of Structural Health Monitoring. The interrogation method generally employed for that purpose is the so-called edge filtering method, which measures the dynamic strain at the location of the FBG. Beneficially, wavelength multiplexing can be used to provide several simultaneous measurement points on a single fiber. In this framework, we demonstrate that Fabry-Perot (FP) cavity-related signals can be observed between untuned FBGs (i.e. FBGs inscribed at different wavelengths) or between an FBG and the fiber tip. This signal is typically seen as a crosstalk effect, as it corresponds to dynamic strains measured at another point alongside the optical fiber, preventing the unbiased analysis of the ultrasonic waves. We suggest to use a blind source separation algorithm to separate the contribution of the FP cavity from the contribution of the interrogated FBG. Hence, the Independent Component Analysis (ICA) methodology is applied on signals from two different experimental setups with fibers bonded to an aluminum plate or embedded into a resin block. The quality of the separation is quantitatively assessed using the correlation coefficient between the signal separated by ICA and the signal from a reference FBG placed near that fiber tip, yielding a median value of 82 % between the reference and the FP-related signal over the various acquired signals.