*Intermediate Physics for Medicine and Biology*, Russ Hobbie and I discuss the medical uses of ultrasound. One important problem we analyze is the pressure distribution produced by a piezoelectric transducer.

There are some important features of the radiation pattern from a transducer which we review next. Consider a circular transducer, the surface of which is oscillating back and forth in a fluid… Each small element of the vibrating fluid creates a wave that travels radially outward, the points of constant phase being expanding hemispheres. The amplitude of each spherical wave decreases as 1/r, the intensity falling as 1/r^{2}. We want the pressure at a point z on the axis of the transducer. It is obtained by summing up the effect of all the spherical waves emanating from the face of the transducer….

The [average intensity] is plotted in Fig. 13.13 for a fairly typical but small transducer (a = 0.5 cm, f = 2 MHz)... Close to the transducer there are large oscillations in intensity along the axis: there are corresponding oscillations perpendicular to the axis, as shown in Fig. 13.14. The maxima and minima form circular rings. This is called the near field or Fresnel zone… The depth of the Fresnel zone is approximately a^{2}/λ [where a is the radius of the transducer, λ is the wavelength, and f is the frequency].

Fig, 13.13. |

*Intermediate Physics for Medicine and Biology*we calculate why this happens mathematically, but it is illuminating to describe what is happening physically. Basically, this is a result of wave interference. Our statement that “each small element of the vibrating fluid creates a wave that travels radially outward” is often called Huygens principle. Each point on the face of the transducer produces such a wavelet. To understand the pressure distribution, we must examine the phase relationship among these various wavelets. Very near the face of the transducer, the waves that contribute significantly to the pressure are in phase; they all interfere constructively and you get a maximum (evaluate Eq. 13.39 at z = 0 and you get a nonzero constant). However, as you move away, more distant points on the transducer face contribute to the pressure on the axis, and these points may be out of phase with the pressure produced by the point at the center. For some value of z the in-phase and out-of-phase wavelets interfere destructively, resulting in zero intensity. Increase z a little more, and not only do the in-phase points at the center and the out-of-phase points just away from the center contribute to the pressure, but so do some in-phase points even farther from the center. When you add it all up, you get a net constructive interference and a non-zero intensity. And so it goes, as you move out farther and farther along the z axis.

Fig. 13.14. |

*Optics*(4th edition), by Eugene Hecht. (My bookshelf contains the first edition, by Hecht and Zajac, that I used in my undergraduate optics class at the University of Kansas).

If you want to be clever, you could make the ultrasound transducer vibrate only at those radii that result in constructive interference along the axis, and have it remain stationary at radii that cause destructive interference. (Of course, this would mean you would have to design your transducer face cleverly so concentric rings vibrate, separated by rings that do not, which might make constructing the transducer more difficult.) Using such a trick eliminates the dark spots along the z axis, increasing the intensity there. This method is commonly used to focus light waves, and is called a zone plate. It has been used occasionally with ultrasound.