^{1,a)}and Michael R. Bailey

^{2}

### 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.

This work was supported by the National Institutes of Health (DK43881 and DK92197), the National Space Biomedical Research Institute through NASA NCC 9-58, and the Russian Foundation for Basic Research (RFBR 11-02-01189, 12-02-00114). Funding and support also came from the UW Center for Commercialization, the Washington Research Foundation, the Coulter Foundation, and the Institute of Translational Health Science. Thanks to our colleagues at the Center for Industrial and Medical Ultrasound, Physics Faculty of Moscow State University, and the Consortium for Shock Waves in Medicine. In particular, we thank Wayne Kreider for help with the manuscript.

I. INTRODUCTION

II. BASIC EQUATIONS

A. On the radiation force calculation

B. Scattering and radiation force due to an axisymmetric field—case of a plane wave

C. Scattering of an arbitrary beam

D. Radiation force of an arbitrary beam

III. SPECIFIC CASES

A. Radiation force of a plane wave

B. Radiation force due to Bessel beams

C. Radiation force of an arbitrary beam on a small spherical scatterer ()

IV. RADIATION FORCE CREATED BY FOCUSED BEAMS OF TYPICAL ULTRASOUND SOURCES

A. Focused transducer in the form of a spherical cap with a circular central opening

B. Multi-element linear array

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Acoustic waves
- 27.0
- Acoustic scattering
- 22.0
- Transducers
- 15.0
- Elasticity
- 14.0
- Sound pressure
- 13.0

## Figures

Geometry of the problem when the acoustic source has the form of a spherical cap with a circular central opening.

Geometry of the problem when the acoustic source has the form of a spherical cap with a circular central opening.

(Color online) Spatial distributions in the plane of the amplitude of the incident pressure wave (top), the radiation force axial component (middle), and the lateral component (bottom) when the sphere is positioned on the axis for the source shown in Fig. 1 . The image box size is 20 × 10 mm. Note the sphere is included for scale and is not in fact fixed in only the shown position.

(Color online) Spatial distributions in the plane of the amplitude of the incident pressure wave (top), the radiation force axial component (middle), and the lateral component (bottom) when the sphere is positioned on the axis for the source shown in Fig. 1 . The image box size is 20 × 10 mm. Note the sphere is included for scale and is not in fact fixed in only the shown position.

(Color online) Spatial distributions in the plane of the full acoustic pressure at time for different positions of the 2-mm spherical stone. The lateral coordinate of the stone center is marked in the left-upper corner of each image, and the colors denote pressure magnitudes. An arrow shows the radiation force direction and relative magnitude. The image box size is 20 × 10 mm. The incident beam axis is directed to the right.

(Color online) Spatial distributions in the plane of the full acoustic pressure at time for different positions of the 2-mm spherical stone. The lateral coordinate of the stone center is marked in the left-upper corner of each image, and the colors denote pressure magnitudes. An arrow shows the radiation force direction and relative magnitude. The image box size is 20 × 10 mm. The incident beam axis is directed to the right.

Axial distribution of pressure amplitude (top) and normalized radiation force for different radii of the scatterer. Solid lines represent calculations based on the full model, while dotted lines correspond to the low-frequency approximation, Eq. (85) .

Axial distribution of pressure amplitude (top) and normalized radiation force for different radii of the scatterer. Solid lines represent calculations based on the full model, while dotted lines correspond to the low-frequency approximation, Eq. (85) .

Geometry of the problem in the case of a linear array source. The focal point can be steered within the imaging plane *y* = 0.

Geometry of the problem in the case of a linear array source. The focal point can be steered within the imaging plane *y* = 0.

(Color online) Spatial distributions in the plane (left column) and plane (right column) of the amplitude of the incident pressure wave (upper), the radiation force axial component (center), and lateral component (lower left) and (lower right) for the 5-MHz source shown in Fig. 5 . The target diameter 2 = 2 mm is shown by the dotted white circles in the top plots. The image box size is 6 × 4 mm.

(Color online) Spatial distributions in the plane (left column) and plane (right column) of the amplitude of the incident pressure wave (upper), the radiation force axial component (center), and lateral component (lower left) and (lower right) for the 5-MHz source shown in Fig. 5 . The target diameter 2 = 2 mm is shown by the dotted white circles in the top plots. The image box size is 6 × 4 mm.

(Color online) Spatial distribution in the plane of the radiation force for the same parameters used in Fig. 6 . The force magnitude is plotted by color. The force direction is shown by the white arrows.

(Color online) Spatial distribution in the plane of the radiation force for the same parameters used in Fig. 6 . The force magnitude is plotted by color. The force direction is shown by the white arrows.

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