Fresnel Number Concept and Revision of some Characteristics in the Linear Theory of Focused Acoustic Beams

TL;DR

This paper advocates for using the Fresnel number to characterize diffraction and focusing in ultrasonic beams, providing analytical formulas for pressure shift and gain dependence, improving accuracy over traditional methods.

Contribution

It introduces the Fresnel number as a more accurate measure for ultrasonic beam focusing, with analytical formulas for pressure shift and gain dependence.

Findings

Fresnel number better characterizes ultrasonic beam focusing.

Analytical formulas accurately describe axial pressure shift.

Real gain dependence on Fresnel number is established.

Abstract

The advisability of the use of the Fresnel number as the measure (characteristic) of the ratio of diffraction and focusing properties for ultrasonic transducers and its radiated beams is proposed and demonstrated. Althought this characteristic is more habitual in optics, in acoustics the equivalent (mathematically although not fully in its physical meaning) parameter of linear gain is used as a rule. However, the preference and the more accuracy of the Fresnel number use is demonstrated here on the basis that the usual determination of the linear gain parameter ceases to correspond to the real value of the gain for low Fresnel number acoustic beams. It connects with the linear effect of axial maximum pressure shift from the geometrical focus towards the transducer. This effect is known for a long time, but here the analytical formulas describing this shift with a high accuracy for arbitrary Fresnel numbers are presented. As a consequence, also the analytical dependence of the real gain on the Fresnel number is obtained.