## 1 Introduction
### Laser Generated Finite Amplitude Rayleigh Waves
The evolution of the \(v_{1}\) Rayleigh waveform from a sinusoid to sawtooth-like is described theoretically by two different models. The first model represents the wavefield as a Fourier series and uses Hamilton's principle to determine the amplitudes of the harmonics [23; 24; 25; 26; 27; 28]. The second model makes no a-priori assumption about the displacement profiles of the harmonics and uses the slowly varying wave profile method (described by [PERSON] and [PERSON] [29]) to handle the secular terms associated with shock front formation [21; 22]. Both models have been shown to predict shock wave formation in good agreement with experimental results; i.e.,  
1. for materials having a 'positive nonlinearity' the shock front in the \(v_{1}\) waveform corresponds with a negative spike-like pulse in the \(v_{3}\) waveform that narrows with increasing amplitude with propagation,
2. for materials having a 'negative nonlinearity' shock fronts in the \(v_{1}\) waveform form at the leading and trailing edges, which correspond to two positive peaks in the \(v_{3}\) waveform that are separating as the valley between them expands with propagation. The workhorse aerospace alloys Ti-6 Al-4V and IN-718 are of interest in the current work. These polycrystalline metals have a positive nonlinearity and are therefore expected to display evolution towards a single, spike-like pulse in the \(v_{3}\) waveform and towards shock formation in the \(v_{1}\) waveform as in case (1) above. We consider the application of Rayleigh waveform evolution for process monitoring during additive manufacturing, therefore the use of an absorbent fluid on the surface is impractical, making shock formation unlikely. The prior experiments used the dual-probe-beam deflection method to detect the surface slope and compute the \(v_{3}\) particle velocity component, which is impractical in the additive manufacturing environment. Instead, an adaptive laser interferometer is used to measure the \(u_{3}\) particle displacement component. Time differentiation of the time series \(u_{3}\) data enables computation of \(v_{3}\), and the Hilbert transform can be used to compute the in-plane components \(u_{1}\) and \(v_{1}\). To show what these four components look like for a metal (i.e., a polycrystalline aluminum alloy) having positive nonlinearity, we have digitized the \(v_{3}\) waveform from [PERSON] and Schuessler [14] and then computed the associated waveforms. The Kolomenskii and Schuessler [14] results are plotted in Figure 2 at the propagation distances of 13.9 and 26.7 mm. Figure 1: Nonlinear Rayleigh particle velocity waveform sketches based on [14]. The waveform evolution predicted by the [PERSON] et al. (2018) model for an IN-718 half-space using the material parameters (\(\rho=8170\) kg/m\({}^{3}\), \(\lambda=121\) GPa, \(\mu=80\) GPa, \(A=-484\) GPa, \(B=-382\) GPa, \(C=-174\) GPa) determined by [PERSON] and [PERSON] (2018) is shown in Figure 3. The waveform evolution is from a sinusoid to a cusped sawtooth as the propagation distance increases. A shock front occurs in the \(v_{1}\) waveform due to steepening, while the \(v_{3}\) initial cosine-shape evolves into a negative spike with increasing amplitude. The displacement component waveform evolutions are much different, with \(u_{1}\) steepening and its peak sharpening and \(u_{3}\) exhibiting some sharpening, but retaining its overall character. The next section introduces process monitoring for additive manufacturing. Figure 2: Particle velocity and displacement waveforms from aluminum substrate at (**a**) 13.9 mm and (**b**) 26.7 mm. \(v_{3}\) was detected by [PERSON] and [PERSON] (2018) and digitized by us, then \(v_{1}\) was computed using the Hilbert transform, and finally \(u_{3}\) and \(u_{1}\) were computed by numerical integration. All waveforms are at the surface \(x_{3}=0\).