Radiation Force Calculations on Apertured Piston Fields

Pierre Gélat, Mark Hodnett and Bajram Zeqiri

3 April 2003

Background

The effective radiating area AER is the area at or close to the face of the treatment

head through which the majority of the ultrasonic power passes (IEC 61689)

The NPL aperture method for determining AER was developed so that radiation

force balances can be used to determine AER for physiotherapy treatment heads

Original implementation of method used a reflecting target radiation force balance; new implementation uses an absorbing target

In both cases, diffraction provides a source of systematic measurement uncertainty

There is a requirement to model and understand the way in which a circular absorbing aperture modifies the acoustic field – Use the Finite Element method

treatm en thead

w a te r su rface

m ask

rad ia tion forceba lance targe t

Schematic Representation of Aperture Technique Using an Absorbing Target

Schematic Representation of Aperture Technique

Transducer

Aperture Absorbing target

x

y

Theory of Acoustic Radiation Force and Radiation Power on an Absorbing Target

Acoustic radiation stress tensor:

jiijij uupS

A.FdS

Where:

ij is the Kronecker delta

Acoustic radiation force vector:

p is the time-averaged acoustic pressure

i and j assume values of 1,2 and 3

Acoustic Radiation Force and Power on the Target

Acoustic power on the target resulting from normal acoustic intensity

b

x RdRRxIP0

),(2

In axisymmetric case, axial component of F is:

b

Rxx RdRRxTRxTRxVdASF011 ,),(),(2

Where

b is the target radius and where (^) denotes the complex amplitude

V is the potential energy density

Tx is the kinetic energy density due to the axial particle velocity

TR is the kinetic energy density due to the radial particle velocity

Un-Apertured Case

Consider un-apertured case to validate Finite Element approach

Use velocity potential to compute near-field pressure and axial particle velocity:

1A

jktj

n Aut1

d-

eeˆ

2π1

),(1r-r

1rrr

Where:

A1 is the piston surface area

nu is the maximum piston velocityr1 is the position vector of a point on the piston

r is the position vector of a point in the sound field

Acoustic pressure: ),r(),r(),r( tjtt

tp

Axial component of particle velocity:x

u x

Analytical expression for ratio Fc/P

kakakaka

P

Fc2J

1

JJ1

1

21

20

Serves as an additional check for Rayleigh integral and Finite Element computations in un-apertured case (Beissner, Acoustic radiation pressure in the near field. JASA 1984; 93(4): 537-548)

Apertured Field (Aperture Diameter = 0 mm)

Apertured Field (Aperture Diameter = 4 mm)

Apertured Field (Aperture Diameter = 6 mm)

Apertured Field (Aperture Diameter = 9 mm)

Apertured Field (Aperture Diameter = 12 mm)

Apertured Field (Aperture Diameter = 16 mm)

Apertured Field (Aperture Diameter = 19 mm)

Apertured Field (Aperture Diameter = 22 mm)

Apertured Field (Aperture Diameter = 24 mm)

Apertured Field (Aperture Diameter = 30 mm)

Apertured Field (Aperture Diameter = )

Fc/P Comparissons 

ka Fc/P (Analytica

l, Beissner)

Fc/P (Rayleigh Integral)

Fc/P(FE)

21 0.9673 0.9866 0.9857

24.5 0.9701 0.9894 0.9886

42 0.9868 0.9939 0.9928

55.5 0.9881 0.9953 0.9944

 

Radiation Force on Target, Aperture Front Face and Rear Face, for ka=21, vs. Aperture Diameter Normalised to Radiation Force on Target in Absence of Aperture

Conclusions

Prediction of apertured transducer pressure field

Prediction of radiation force and radiation power on absorbing target for apertured transducer field using the Finite Element method

Comparison of FE derived Fc/P in absence of aperture with analytical expression and Rayleigh integral