Acoustic radiation pressure in laterally unconfined plane wave beams

Abstract

The acoustic radiation pressure in laterally unconfined, plane wave beams in inviscid fluids is derived via the direct application of finite deformation theory for which an analytical accounting is made ab initio that the radiation pressure is established under static, laterally unconstrained conditions, while the acoustic wave that generates the radiation pressure propagates under dynamic (sinusoidal), laterally constrained conditions. The derivation reveals that the acoustic radiation pressure for laterally unconfined, plane waves along the propagation direction is equal to (3/4)〈2K〉, where á ñ K is the mean kinetic energy density of the wave, and zero in directions normal to the propagation direction. The results hold for both Lagrangian and Eulerian coordinates. The value á ñ ( ) / K 3 4 2 differs from the value á ñ K 2 ,traditionally used in the assessment of acoustic radiation pressure, obtained from the Langevin theory or from the momentum flux density in the Brillouin stress tensor. Errors in traditional derivations leading to the Brillouin stress tensor and the Langevin radiation pressure are pointed out and a long-standing misunderstanding of the relationship between Lagrangian and Eulerian quantities is corrected. The present theory predicts a power output from the transducer that is 4/3 times larger than that predicted from the Langevin theory. Tentative evidence for the validity of the present theory is provided from measurements previously reported in the literature, revealing the need for more accurate and precise measurements for experimental confirmation of the present theory.wave. This result differs considerably from the value á ñ K 2 for the radiation pressure obtained from the Langevin theory [29] or from the Brillouin stress tensor [26,27]. In the directions normal to the propagation direction, the radiation pressure is zero for both Lagrangian and Eulerian coordinates.The difference between the present assessment of radiation pressure and that obtained from the Langevin theory or from the Brillouin stress tensor is significant, since the assessment is used to link the radiation force on a target with the power generated by acoustic sources. As pointed out by Beissner [21], the 'measured radiation force must be converted to the ultrasonic power value and this is carried out with the help of theory.' It is generally assumed that for laterally unconfined, plane wave beams the relationship between the acoustic radiation pressure and the energy density for plane waves in the direction of wave propagation is that obtained from the Langevin theory [29] or from the Brillouin stress tensor [26,27]. The present model predicts a transducer output power 4/3 times larger than that predicted from the Langevin theory or from the Brillouin stress tensor. This has considerable implications regarding safety issues for medical transducers, calibrated using radiation pressure.Issenmann et al [52] point out that 'despite the long-lasting theoretical controversies K the Langevin radiation pressure K has been the...