Radiation force of an arbitrary acoustic beam on an elastic sphere in a fluid

Abstract

A theoretical approach is developed to calculate the radiation force of an arbitrary acoustic beam on an elastic sphere in a liquid or gas medium. First, the incident beam is described as a sum of plane waves by employing conventional angular spectrum decomposition. Then, the classical solution for the scattering of a plane wave from an elastic sphere is applied for each plane-wave component of the incident field. The net scattered field is expressed as a superposition of the scattered fields from all angular spectrum components of the incident beam. With this formulation, the incident and scattered waves are superposed in the far field to derive expressions for components of the radiation stress tensor. These expressions are then integrated over a spherical surface to analytically describe the radiation force on an elastic sphere. Limiting cases for particular types of incident beams are presented and are shown to agree with known results. Finally, the analytical expressions are used to calculate radiation forces associated with two specific focusing transducers.

Characterization of a multi-element clinical HIFU system using acoustic holography and nonlinear modeling

“…To determine “measured” acoustic powers for different settings, the power of the measured hologram at 259 ampvals (Fig. 2) was calculated using an angular spectrum approach [29] and then scaled based on relative pressure changes from the near-source measurement data. Figure 4 (bottom) shows the discrepancy between the measured acoustic powers and the nominal powers provided by Philips, expressing the difference as a percentage of the nominal power.…”

“…The angular spectrum is based on the idea that an arbitrary acoustic field can be decomposed into a superposition of plane waves propagating at different angles, where the angles are represented by different spatial frequencies. This method is computationally efficient for acoustic propagation between parallel planes [28] and was used here to evaluate the acoustic powers represented by measured holograms [29]. With this approach, true acoustic powers were calculated without assuming that the field comprised a plane wave.…”

The effect of shear waves in an elastic sphere on the radiation force from a quasi-Gaussian beam

“…In the case of an axisymmetric beam, the corresponding problem can be solved using the results obtained in Ref. 4. Axial symmetry leads to some simplifications in the radiation force calculation.…”

“…Because of the axisymmetry of quasi-Gaussian beam, the force has two components F = ( F z , F r ) It is convenient to perform numerical calculations by the method from Ref. 4, based on the on the representation of the incident acoustic beam in the form of plane waves: $$P_{\mathrm{italicinc}}\left(x,y,z\right)=\frac{1}{4{\pi }^2}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}d{k}_{x}d{k}_{y}S\left(k_x,k_y\right)e^{i{k}_{x}x+i{k}_{y}y+i\sqrt{k^2-k_x^2-k_y^2}z},$$ where S ( k x , k y ) is the angular spectrum of the incident wave. For the quasi-Gaussian beam, the angular spectrum is written as follows: $$S\left(k_x,k_y\right)=\frac{P_0\pi z_d}{{sinh}^2(k{z}_d)}\frac{sinh\left[z_d(k+\sqrt{k^2-k_r^2})\right]}{\sqrt{k^2-k_r^2}},$$ where $k_r=\sqrt{k_x^2+k_y^2}$.…”

“…The calculations were conducted for elastic spheres of two types using a previously developed method for the radiation force 4 and the exact solution of the Helmholtz equation in the form of a quasi-Gaussian beam 5 .…”