1. Radiation Force Calculations on Apertured Piston Fields Pierre Gélat, Mark Hodnett and Bajram Zeqiri 3 April 2003

2. Background  The effective radiating area A ER 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 A ER was developed so that radiation force balances can be used to determine A ER 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

3. Schematic Representation of Aperture Technique Using an Absorbing Target

4. Schematic Representation of Aperture Technique Transducer ApertureAbsorbing target x y

5. Theory of Acoustic Radiation Force and Radiation Power on an Absorbing Target  Acoustic radiation stress tensor: Where:  ij is the Kronecker delta  Acoustic radiation force vector: is the time-averaged acoustic pressure i and j assume values of 1,2 and 3

6. Acoustic Radiation Force and Power on the Target  Acoustic power on the target resulting from normal acoustic intensity  In axisymmetric case, axial component of F is: Where b is the target radius and where (^) denotes the complex amplitude V is the potential energy density T x is the kinetic energy density due to the axial particle velocity T R is the kinetic energy density due to the radial particle velocity

7. Un-Apertured Case  Consider un-apertured case to validate Finite Element approach  Use velocity potential  to compute near-field pressure and axial particle velocity: Where: A 1 is the piston surface area is the maximum piston velocity r 1 is the position vector of a point on the piston r is the position vector of a point in the sound field  Acoustic pressure:  Axial component of particle velocity:

8. Analytical expression for ratio Fc/P  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)

9. Apertured Field (Aperture Diameter = 0 mm)

10. Apertured Field (Aperture Diameter = 4 mm)

11. Apertured Field (Aperture Diameter = 6 mm)

12. Apertured Field (Aperture Diameter = 9 mm)

13. Apertured Field (Aperture Diameter = 12 mm)

14. Apertured Field (Aperture Diameter = 16 mm)

15. Apertured Field (Aperture Diameter = 19 mm)

16. Apertured Field (Aperture Diameter = 22 mm)

17. Apertured Field (Aperture Diameter = 24 mm)

18. Apertured Field (Aperture Diameter = 30 mm)

19. Apertured Field (Aperture Diameter =  )

20. Fc/P Comparissons ka Fc/P (Analytical, Beissner) Fc/P (Rayleigh Integral) Fc/P (FE) 21 0.96730.98660.9857 24.5 0.97010.98940.9886 42 0.98680.99390.9928 55.5 0.98810.99530.9944

21. 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

22. 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