_applied sciences_  
_Article_  
**Edge and Notch Detection in a Plate Using Time Reversal Process of Leaky Lamb Waves**  
**[PERSON] 1,2,*, [PERSON] 2, [PERSON] 1, [PERSON] 1 and [PERSON] 2**  
Footnote 1: CEA, DES, IRESNE, DTN, Center of Cadarache, F-13108 Saint-Paul-Lez-Durance, France; [EMAIL_ADDRESS] (FB. ); [EMAIL_ADDRESS] (M.C.) Footnote 2: Aix Marseille Univ, CNRS, Centrale Marseille, LMA 7031, F-13453 Marseille, France; [EMAIL_ADDRESS] (M.-A.P. ); [PERSON]-F.C.) Footnote 3: email: {e-mail: [EMAIL_ADDRESS]}  
Footnote 4: email: {e-mail: [EMAIL_ADDRESS]}  
Footnote 5: email: {e-mail: luc  
The selection of the mode can be done by imposing an incidence to the generated acoustic beam through a wedge or a delay law if a linear array transducer is used [10]. Several imaging methods involving Lamb waves are proven effective. For example, a classical B-scan may provide an image of a defect assuming the knowledge of the propagating mode and its velocity in a single plate or in the second plate of the layered structure [2]. Other linear methods using delay-and-sum can also be applied, such as the Synthetic Aperture Focusing Technique (SAFT) [11; 12; 13], the Total Focusing Method (TFM) [14; 15] and their adaptation to Lamb waves, Lamb-SAFT [16] to take into account the dispersive behavior of the Lamb waves, and the sparse-TFM [17] to carry out short range inspection. However, those techniques are focused on the time travel of a single mode and do not use information of the mode conversion that may occur at the reflection on a defect, such as a crack [18; 19]. Later on, time-reversal techniques were introduced [20; 21; 22; 23; 24] to optimize the resolution of a reconstructed image of an inspected medium. For example, the DORT (Decomposition of the Time-Reversal Operator) method [21] is a selective detection of scatterers using the eigenvectors of the time-reversal operator, extracted from the full-matrix capture of a linear array of transducers. Each significant eigenvector is associated with a scatterer in the inspected medium, and the associated signal can be retropropagated independently in a numerical medium to locate the scatterer. This operation is made possible because of the completeness of the fundamental Lamb modes as proven by [PERSON] [3]. Meanwhile, the topological energy method [22; 24; 25; 26; 27; 28] requires the numerical solution of only two problems: the direct problem where the experimental source is generated in the healthy corresponding structure and the adjoint problem where the source is the time-reversed difference between the ultrasonic field measured on the transducers of the inspected medium and the reference medium. This time-reversed source can be interpreted as the time-reversed signature of the defects and could provide data of very good quality. This method has already been applied on Lamb waves in the literature. For example, [PERSON] et al. applied it to the inspection of a free anisotropic plate by a monomodal inspection. The transducers are located on the edge of the plate, and several defects are detected [29]. Further works of [PERSON] et al. showed the efficiency of the topological imaging method towards other methods, notably the TFM [27]. Thus, this method is applicable and efficient in cases of free single plates. But no work was found concerning the leaky Lamb waves in a set of immersed plates. The topological energy method was preferred in this work because the adjoint field and the time reversal phenomenon allow maintaining a good sensibility and a great localization. Moreover, it involves fewer simulations that can become very time consuming. The topological energy method is also more versatile regarding the complex geometry of two parallel and immersed plates, in particular considering the multiple reflections between the two plates.