Browsing by Subject "Inverse Abel Transform"

The feasibility of quantitatively measuring ultrasound in air with a Schlieren arrangement has been demonstrated before, but previous work demonstrating calibration of the system combined with computation to yield the 3D pressure field does not exist. The present work demonstrates the feasibility of this both in theory and practice, and characterizes the setup used to gain the results. Elementary ray optical and Schlieren theory is exhibited to support the claims. Derivation of ray optical equations related to quantitative Schlieren measurements are shown step by step to help understand the basics. A numerical example based on the theoretical results is then displayed: Synthetic Schlieren images are computed for a theoretical ultrasonic field using direct numerical integration, then the ultrasonic field is recovered from the Synthetic Schlieren images using the inverse Abel transform. Accuracy of the inverse transform is evaluated in presence of synthetic noise. The Schlieren arrangement, including the optics, optomechanics, and electronics, to produce the results is explained along with the stroboscopic use of the light source to freeze ultrasound in the photographs. Postprocessing methods such as background subtraction and median and Gaussian filtering are used. The repeatability and uncertainty of the calibration is examined by performing repeated calibration while translating or rotating the calibration targets. The ultrasound fields emitted by three transducers (100 kHz, 175 kHz, and 300 kHz) when driven by 5 cycle sine bursts at 400 Vpp are measured at two different points in time. The measured 3D pressure fields measured for each transducer are shown along with a line profile near the acoustic axis. Pressure amplitudes range near 1 kPa, which is near the acoustic pressure, are seen. Nonlinearity is seen in the waveforms as expected for such high pressures. Noise estimates from the numerical example suggest that the pressure amplitudes have an uncertainty of 10% due to noise in the photographs. Calibration experiments suggest that additional uncertainty of about 2% per degree of freedom (Z, X, rotation) is to be expected unless especial care is taken. The worst-case uncertainty is estimated to be 18%. Limitations and advantages of the method are discussed. As Schlieren is a non-contacting method it is advantageous over microphone measurements, which may affect the field they are measuring. As every photograph measures the whole field, no scanning of the measurement device is required, such as with a microphone or with an LDV. Suggestions to improve the measurement setup are provided.