## 1 Introduction  
Periodic structures or locally resonant systems, also known as metamaterials, have revolutionized the control of waves these last years [1; 2]. The main physical properties of these structured media, as for example the opening of band gaps [3; 4], the slow wave frequency band [5; 6], or the spatial filtering [7], among others, can be directly derived from the complex dispersion relation [8; 9; 10; 11; 12]. In addition to dispersive effects, wave attenuation can be caused by factors such as geometric attenuation or intrinsic material loss (e.g., heat or viscous dissipation). In all cases, wave attenuation can be interpreted in terms of complex wavenumbers, which are functions of the angular frequency \(\omega\) via a complex dispersion relation \(k(\omega)\). The real part of the wavenumber translates the propagative (dispersive) properties of the system while its imaginary part is related to the attenuation. This rich but sometimes hardly to interpret information is of crucial relevance for the design of metamaterial-based devices. Therefore, the recovery of the complex dispersion relation, i.e., the measurement or calculation of both the dispersion and the attenuation of waves in a finite system is of utmost importance in wave physics and material science, since it provides insightful information on the tested system also allowing the characterization of the effective properties of the material. Several methods have been proposed during these last years to calculate the complex dispersion relation of waves in discrete or continuous systems. The extended plane wave expansion [9; 10] or methods based on Multiple Scattering Theory (MST) [13; 14] are efficient tools to evaluate the complex dispersion relation in non-resonant structures made of scatterers of simple geometry. Numerical approaches are more relevant [15] when the scatterer geometry is more complicated or the scatterers are resonant. Recently, the SLaTCoW (Spatial LAplace Transform for COmplex Wavenumber) [11; 12] method has been proposed as a general procedure for the recovery of the complex dispersion relation. It has been successfully applied to several systems of different kind, such as Lamb-like modes in lossy materials in the low kHz regime, elastic waves in a Duralumin plate in the MHz regime at the Zero-Group Velocity (ZGV) point, surface acoustic waves in a two-dimensional microscale granular crystal adhered to a substrate near 100 MHz [11] and also spoof surface acoustic waves at a lossy resonant metasurface [12]. The method is based on the spatial Laplace transform (LT) instead of the usual spatial Fourier transform (FT) of the measured wavefield in the frequency domain. The LT, providing information on both the real and imaginary parts of the wavenumber, is analyzed by minimizing a correctly chosen cost function. The complex wavenumbers (as well as the complex amplitudes) of the modes are reconstructed even when they are interacting one with another or overlapping. Moreover, the SLaTCoW method is particularly suitable for the experimental characterization of real periodic systems of finite-size (resonant or non-resonant), which, while exhibiting similar properties as systems of infinite extent such as band gaps [16; 17; 18] or other dispersive effects [19; 20], can not be experimentally characterized using methods considering infinite structures, such as EPWE, MST or solving an eigenvalue problem using the Finite Element Method (FEM). In this work, we present an advanced version of the SLaTCoW method to characterize the complex dispersion relation of two-dimensional systems. We apply this methodology to both periodic media as Sonic Crystals (SC) and locally resonant acoustic materials. The article is organized as follows. The methodology and the extension of the method is presented Section 2. Section 3 depicts the different two-dimensional configurations the methodology is applied to. Finally, results are provided and validated in Section 4.