**The radiation force formula for ultrasonic fields produced by rectangular transducers has been calculated in order to improve the reliability of the ultrasonic power measurement by means of a radiation force balance with particular respect to those field types.**

The temporal average of the emitted ultrasonic power is an important parameter of ultrasound devices. It is determined mostly by using a radiation force balance. The ultrasonic beam in water is directed towards a target (mostly an absorbing one) and the force exerted on the target is measured using an electronic micro or semi-micro balance. Knowledge of the theoretical relation between force and power is needed to be able to derive the ultrasonic power, and this relation depends on several influencing factors, but mainly on the field structure.

The radiation force formulae stated so far in the literature and in standards apply to fields with circular symmetry, i.e., to circular transducers. In practice, however, rectangular transducers are often used, particularly in ultrasonic medical diagnostics. The relation between power *P* and the forward component of the force *F* has, therefore, been calculated for the fields produced by rectangular transducers and by using numerical algorithms based on the radiation force theory. Three scenarios have been taken into account: constant vibration amplitude of the transducer (piston source), spatial distribution of the transducer's vibration amplitude (apodization), and focussing, which means spatial distribution of the transducer's vibration phase.

Examples of calculation results are shown in figure 1. The relevant ratio *c*·*F*/*P* is depicted as a function of (*k*·*h*)^{-1} in the range of practical interest, 100 = *k*·*h* = 15. Here, *c* and *k* are the speed of sound and the circular wave-number, respectively, in the sound-propagating fluid (water), and *h* is generally the geometric mean of the two half dimensions of the rectangular transducer which in this case, however, are assumed to be equal. The figure includes curves for three different values of the focus (half) angle *?* under the condition of constant vibration amplitude. The indication *?* = 0° stands for "unfocussed" and refers to a piston source in this case.

There are two reasons for the deviation from the plane-wave value *c*·*F*/*P* = 1, namely (a) diffraction at the beam edge, depending on the wave-length, which means on *k*·*h*, and (b) the geometric beam focussing which leads to the vertical shift between the curves and whose influence can be characterized particularly by the extrapolation value to (*k*·*h*)^{-1} ? 0. Whereas the entire curves can only be obtained numerically by sample calculations for certain parameter values, it was possible to derive a closed formula for the diffraction-free extrapolation value, a formula that may be used for any geometric parameter values.

Figure 1: Calculation result for the quantity of interest, the ratio *c*·*F*/*P* as a function of (*k*·*h*)^{-1} for three values of the focus (half) angle *?*. Assumptions in this case: constant transducer amplitude, equal sides of the rectangular transducer.

### Contact person:

Klaus Beissner, Senior Scientist 1.60, ultrasonics@ptb.de