Enhancing Ultrasonic Crack Sizing Accuracy in Rails: The Role of Effective Velocity and Hilbert Envelope Extraction

Ultrasonic testing is a prevalent method for non-destructive evaluation of railway rails; however, conventional Time-of-Flight (<i>ToF</i>) approaches applied in practical dry-coupled inspections often rely on simplified assumptions regarding wave propagation velocity and neglect complex waveform characteristics. This paper presents a robust depth estimation framework for surface-breaking cracks that enhances sizing accuracy through effective velocity calibration and Hilbert envelope extraction. Unlike standard methods that assume the free-space speed of sound in air (343 m/s) for wave propagation within the air-filled gap of a surface-breaking crack, we propose an effective velocity model derived from in situ calibration to account for the boundary layer viscosity and thermal conduction effects within narrow crack geometries. The signal processing chain incorporates spectral analysis, band-pass filtering, and Hilbert Transform-based envelope detection to mitigate noise and resolve phase ambiguities. Experimental validation on steel specimens with controlled defects (0.2-10.0 mm) demonstrates that the proposed method achieves an exceptional linear correlation (<i>R</i>² ≈ 0.9976). The calibrated effective velocity was determined to be 289.3 m/s, approximately 15.6% lower than the speed of sound in air, confirming the significant influence of confinement effects. Furthermore, excitation parameters were optimized, identifying that high-voltage excitation (≥110 V) and a tuned pulse width (≈150 ns) are critical for maximizing the signal-to-noise ratio. The results confirm that combining physical model calibration with advanced signal analysis significantly reduces systematic errors, paving the way for portable, high-precision rail inspection systems.