## 1 Introduction
Analytical solutions corresponding to Lamb waves scattering in the presence of delamination do not exist for composite laminates due to their multilayer characteristics [16]. Within the framework of a plane strain assumption, the reflection of the fundamental symmetric (S\({}_{0}\)) Lamb wave from a delamination in unidirectional and cross-ply laminates showed that S\({}_{0}\) Lamb wave cannot be used to detect the delamination at locations with zero shear stress [17]. On the other hand, the use of a 2D method to study the reflection characteristics of S\({}_{0}\) and A\({}_{0}\) Lamb waves from a delamination showed that the A\({}_{0}\) Lamb wave is sensitive to delamination at all through-thickness locations [18]. However, it should be noticed that the wave scattering mechanism depends on the involved guided mode as well as on the depth and size of the damage zone. In the case of a delamination, various investigations have shown that A\({}_{0}\) and S\({}_{0}\) modes propagate individually in the two regions separated by delamination and then interact with each other after exiting the delamination area [19; 20]. However, the scattering of guided waves in the case of an impact damage is different from the one corresponding to a delamination created along the full width of the composite laminates. Indeed, [PERSON] et al. [21] found that in thin composite specimens, the damage is initiated in the bottom layers, whereas in thick specimens, the damage is initiated in the top layers. As a result of the stresses induced by the hemispherical impactor, fiber breakage is expected in the top plies. The impact damage in the bottom layers is mainly dominated by delamination due to bending stresses, which does not cover the whole thickness of the composite [22; 23]. For a given impact energy, the delamination area is expected to be proportional to the bending stiffness mismatching coefficient between the matrix and reinforcements. Therefore, high mismatching coefficients are to be expected when the difference between young's moduli in the fiber and matrix directions is important [24]. Scatter amplitudes and scatter directivity distributions depend on the delamination size to wavelength ratio and the through-thickness location of the delamination damage. This effect is often studied by bonding masses to the surface of composite laminates in order to simulate delamination damage [16]. Different techniquesand approaches have been used in order to model the propagation of ultrasonic guided waves within impacted composite plates. [PERSON] et al. proposed a modelling that combines low-velocity impact damage with the analysis of guided wave propagation, where delamination and matrix cracking are both directly meshed in the guided-wave model [25]. However, different types of damage could be created when composites are submitted to a low velocity impact. Indeed, impact damages in CFRP structures include delamination, shear cracks, transverse cracks, etc., which makes the precise modelling of all these micro-damages difficult [26; 27; 28]. The complexity of the multiple scattering phenomena, local resonances, etc., makes the use of a physical approach important and more realistic. Indeed, the effective medium approximation offers a good opportunity to simplify the modeling of complex media, where the inhomogeneous medium can be considered as homogeneous with effective properties. An application of the homogenization approach in the case of a strongly scattering medium can be found in the following references [29; 30; 31; 32]. This work investigates the directivity pattern of the scattered field of incident A\({}_{0}\) guided wave by the impact damage and provide much more material and a more in-depth study in connection with new experimental results to our previous contribution [33]. Unlike most of the studies in the literature, the damage zone considered in this paper is not across the full width of the laminates.