## 2 Analysis Methodology  
### Harmonic Generation and Pulse-Inversion Technique  
The higher harmonic waves are generated when two conditions are satisfied: first, modal excitability given the selected transducer, andsecond, phase matching between the primary and its higher harmonics [5; 20]. Once such conditions are satisfied, the amplitude of the nonlinear waves, \(U_{NL}\), increases for the second order interaction as  
\[U_{NL}\propto\beta\cdot U_{L}^{2}\cdot x \tag{1}\]  
where \(\beta\) is the ultrasonic nonlinearity, \(U_{L}\) is the amplitude of the primary wave, and \(x\) is the length of the damaged zone in which the wave travels. In principle, there are an infinite number of mode pairs that satisfy these conditions, and thus a careful modal analysis can be theoretically and experimentally performed. If the tested structure has a complex geometry (for example, involving an arbitrarily varying geometry such as multiple joints, anisotropic multi layers, etc. ), selecting the relevant mode pairs becomes quite complicated, and might render such a nonlinear guided wave technique impractical. Our proposed method generates multiple modes (both symmetric and antisymmetric modes) over a broad frequency range from small-area (length and width of the interrogated medium is much larger than lateral dimensions of transducer) single-element transducers with compressive excitation on the surface of the structure under test. Under such excitation, many guided wave modes with a wide range of frequencies, thus generated, are expected to strongly interact with micro-cracks (or, in general, microstructural defects) in the entire interrogated structure. To achieve broadband excitation, this study uses a chirp signal with linearly increasing frequency as an input source, written as  
\[u_{in}(t)=\pm A\sin[2\pi f_{\text{min}}t+\pi(f_{\text{max}}-f_{\text{min}})t^{ 2}/T_{d}] \tag{2}\]  
where \(f_{\text{min}}\) and \(f_{\text{max}}\) are the minimum and maximum frequencies, respectively, \(T_{d}\) is the duration of the chirp and \(A\) is the initial amplitude of the chirp. This linear chirp signal, thus injected perpendicular to the surface of the medium under test, diffusely propagates across the interrogated medium. The energy of the propagating multi-mode guided wave signal is transferred to the higher harmonic guided wave modes after interacting with micro-cracks. Such higher harmonics (so-called nonlinear guided waves) are used to assess the density of the micro-cracks in the entire interrogated zone. Since ultrasonic amplitudes at a specific site and strength of nonlinear interaction for a given microstructural defect cannot be accurately known in most situations, our proposed technique should be considered foremost as an initial, early-warning technique to determine the presence of early-stage microstructural defects such as micro-cracks in an inspection zone, which can be followed by a more systematic and quantitative examination for identifying defect types and obtaining detailed information on local distribution of defects. The amplitude of the nonlinear guided waves can be much smaller than that of the linear primary waves [21]. To measure such small amplitude signals in the presence of much larger linear signals, a pulse-inversion technique can be used to remove the linear primary response signal and accentuate the nonlinear response signal. First, a base measurement is conducted where a signal of amplitude \([A]\) is excited in a medium with a certain phase. It is followed by a second measurement having the same amplitude but opposite polarity \([-A]\) as compared to the first measurement. The base measurement \(x_{b}(t)\) and the inverted measurement \(x_{i}(t)\) are recorded in the time domain. Due to hardware artifacts, in practice the two measurements may not have identical amplitudes.