Ultrasonic fields in fluids: theoretical prediction using difference equations and three dimensional measurement using optical techniques

Abstract

A technique for calculating bulk ultrasonic fields which uses implicit difference equations to evaluate the parabolic approximation to the Helmholtz equation is described. The parabolic approximation assumes that the field varies much faster in the transverse directions than in the direction of propagation and results in a partial differential equation which is formulated as a pure i initial value problem . Finite difference equation descriptions are derived for one-dimensional, cylindrically symmetric problems and for more general two-dimensional problems. A Fourier stability analysis is performed on the simpler numerical scheme to determine acceptable values for parameters such as the grid spacing and increment step sizes. Several example calculations for each geometry are presented to demonstrate the usefulness of this parabolic-difference equation technique in transducer modeling. Included in these examples are gaussian, rectangular, circular, and concentric ring amplitude distributions, as well as uniform and focused phase distributions. In many cases, qualitative agreement between the numerical results and analytic diffraction theory predictions can be observed.

An experimental system is developed for the detection of acoustic fields in water using a dual beam differential interferometer which is scanned through a large tank. The output from this system is proportional to the average difference in refractive index sampled by the two beams and is effectively integrated along the path of the beams through the specimen. An algorithm is presented which reconstructs a radial profile of the measured field from the integrated data assuming cylindrical symmetry. Raw, averaged, and reconstructed data is given for scans taken in the far-field of a circular, uniform transducer.