# Computational Vibro-Acoustic Time Reversal for Source and Novelty Localization
## 2 Statement of the Problem  
Time reversibility in a vibro-acoustic environment is a promising procedure that can be useful in several applications, e.g., scatterer (cavities or obstacles) identification and localization as well as crack or general defect and damage localization in a non destructive testing context [13]. The first problem we are interested in is source localization. Here, we try to infer the spatial location of the original source that created the signals measured from a collection of receivers that record the acoustic pressure responses via microphones as well as the kinematic responses, via accelerometers. The source might be a mechanical or acoustic excitation (e.g., a blast). It has been demonstrated that using kinematic responses together with acoustic recordings makes the procedure more stable and less sensitive to background noise, especially since microphones are prone to such noise [11]. On the other hand, the use of microphones offers the opportunity to speed-up the inspection by saving time in the sensor installation. Additionally, there are certain applications where it is difficult to place sensors onto the structure in order to acquire kinematic response or it is not proper to modify the dynamics of the system by added masses (i.e., lightweight structures). In such cases, the use of microphones is found to be beneficial [9]. The second problem we are interested in is the localization of novelties taking place within the coupled vibro-acoustic system. These novelties may appear in either the acoustic or the mechanical elastic part. It is more common for such novelties to happen in the elastic part, e.g., in the form of some occurred damage on the elastic body or on the support of it. In order to find which part of the structural assembly has suffered such modifications and to accurately define its spatial location, we assume that such a novelty will act as a secondary source. Using the time reversibility of the waves, it is possible to localize this secondary source similarly to the original excitation source. In this case, to focus on the field produced by the novelty, we subtract the wavefield produced by the original source when the novelty did not exist. By using this latter signal, which is the difference of the fields recorded before and after the appearance of the novelty, we are able to localize the novelty. Let us now describe the procedure to be numerically implemented using the descriptive sketch of Figure 1. For the source localization in a healthy medium, as shown in Figure 1a, a source acts as acoustic or mechanic excitation at time \(t_{0}\), and the response to this excitation is recorded on a finite number of recording stations as pressure (via microphones) or kinematic responses (using accelerometers or displacement/velocity sensors) for a duration time \(T\). We denote this wavefield \(u_{r}\), including both acoustic and elastic responses, while the acquisition process is called the forward step. In the current computational approach, all data are obtained numerically using simulations of virtual experiments. Time reversibility assures us that reverting in time the recorded signals and re-emitting them as excitation, in a procedure called backward step, an energy focus will occur at the time \(T-t_{0}\) on the spatial location of the original source. Note that actually, one needs to revert the whole wavefield and re-emit it, although it is well documented that the reduced (or incomplete) time reversal, where the signal is recorded at a finite number of sensors, still keeps the refocusing property. This is the source localization process, while for the novelty localization, a few extra datum are needed. In this case, we record the wavefield for a modified system configuration, which includes the novelty, e.g., some modification on one of the beams in the domain as depicted in Figure 1b; we call this wavefield \(u_{r}^{t}\).