## 1 Introduction
### Laser Generated Finite Amplitude Rayleigh Waves  
Pulses from a Q-switched Nd:YAG laser formed into an intense beam nominally 6-15 mm long and 0.05 mm wide can generate finite amplitude surface waves that can be approximated as planar for a reasonable range of propagation distances. In practice, the laser pulses usually have a wavelength of 1064 nm, a duration of -7 ns, and energies in the 100-200 mJ range. The transduction from light (electromagnetic wave) energy to sound (elastic wave) energy can be enhanced by a thin layer of light-absorbing fluid on the surface. If the solid media is isotropic, homogeneous, and well represented by a half-space having a traction-free surface, then Rayleigh waves are the dominant type of wave generated. Given the short pulse duration, the Rayleigh waves are broadband and have finite amplitudes associated with extremely high stresses that cause nonlinear evolution of the waveform with propagation distance. In this literature investigating the physics of nonlinear surface wave pulses [9; 10; 11; 12], and their application to surface cleaning [13], characterization [14; 15; 16], and fracture [17; 18; 19; 20], the dual-probe-beam deflection method with continuous wave lasers (-4 \(\upmu\)m spot size) was used to determine the surface slope at two points separated by about 16 mm. The surface slope is proportional to the out-of-plane particle velocity component, \(v_{3}\). In this article, without loss of generality, we take the Rayleigh waves to propagate in the \(x_{1}\) direction and the outward normal to the surface to be in the negative \(x_{3}\) direction. If the surface slope measurement is properly calibrated then \(v_{3}\) has engineering units (e.g., m/s). A smooth surface is required for this detection method because the small diameter beam is reflected at an oblique angle from the surface to a photodetector. The in-plane particle velocity \(v_{1}\) can be computed using the Hilbert transform and may have clearly defined shock fronts, which depend upon the algebraic sign of the nonlinearity parameters [11]. Experimental studies on fused quartz [9; 11; 18], fused silica [10; 14], aluminum [14], stainless steel [12; 15; 16], and single-crystal silicon [9; 13; 17; 18; 19; 20] are reported. Single-crystal silicon is anisotropic, therefore the waves are not Rayleigh waves, but rather are described simply as surface acoustic waves (SAWs). Waveform evolution occurs because the particle velocity depends on the wave amplitude, which varies with depth. Shock formation occurs in the \(v_{1}\) Rayleigh waveform in a similar fashion to shock formation in fluids, depending upon the'sign of the nonlinearity' as described below. Herein, we refer to the material parameter \(\varepsilon_{1}\) used in the [PERSON] et al model [21; 22] for the sign of the nonlinearity, as it has been shown to dominate the waveform evolution [14]. Materials with positive \(\varepsilon_{1}\) nonlinearity parameter tend to have a \(v_{1}\) waveform described by a large negative peak followed by a smaller positive peak as shown in Figure 1. The negative peak travels faster than the positive peak resulting in waveform compression, or steepening, leading to a higher peak frequency known as a frequency up-conversion. If the steepening is sufficiently high, it can cause shock formation. Materials with negative \(\varepsilon_{1}\) nonlinearity parameter tend to have a \(v_{1}\) waveform described as an inverted N-shape as sketched in Figure 1, where the leading negative peak travels faster than the trailing positive peak. When this occurs, the peaks separate, resulting in a lower peak frequency known as a frequency down-conversion. In this case, shock fronts can form at the leading and trailing edges. We will see that nonlinearity in the \(v_{3}\) Rayleigh waveformis evident by sharpening of the negative peak for material with positive nonlinearity and widening of the valley between the positive peaks for material with negative nonlinearity.