Model for thickness measurements of steel plates using half-wave resonances

Previously reported measurement have shown how air-coupled ultrasound transmission measurements can be used to measure the thickness of steel plates, resolving thickness variations down to 0.2 mm. Simple plane wave theory predicts that at normal incidence, the steel plate is excited into compressional resonances when the plate thickness is an integer number of halfwavelengths. No shear waves are excited at normal incidence. The plate investigated previously had thickness 10.15 mm, with half-wave resonances expected at 288.18 kHz and 576.35 kHz. Transmission measurements on this plate showed resonance peaks at 267.33 kHz, 464.63 kHz and 571.98 kHz. The former and latter values deviate by 7% and 0.7%, respectively, from the values predicted by the plane wave model. The peak at 464.6 kHz was assumed to be due to the third harmonic resonance of the shear wave, but this was not confirmed. The aim of this work is to investigate whether this deviation can be explained by using a more realistic model for the plate, including shear waves and the finite extent of the source and receiver. A frequency domain model was developed using the wavenumber integration method. The transducer is modeled as a plane piston in an infinite planar baffle and the sound field is decomposed into plane waves over a range of angles. The stainless steel plate is modeled as an elastic layer, including compressional and shear waves, immersed in a fluid. The impulse response of this system was found by multiplying the plane wave decomposition of the sound field from the transducer with the transmission coefficient for the stainless steel plate and integrate over all angles. The transmitted sound pulse was then found by convolving this impulse response and the transmitted pulse, a chirp covering the frequency range from 200 kHz to 800 kHz Compared to the previously reported experimental results, the developed model was able to explain the deviation of the lowest pressure wave resonance peak from the p- ane wave theory, and predicted its position within 2%. The existence of the shear wave resonance was confirmed, and the position of the observed peak explained within the accuracy of the shear wave velocity of the steel plate.