Answer #1

The correct answer is E: 2 cm.

While current day transducers have multiple crystals, it is important to know the basics of a simple single crystal transducer, as the principles are similar.

The figure shows the beam emanating from a crystal of length D.

The beam actually has three dimensions - lateral (side to side), axial (also called radial, longitudinal or depth) and elevation (slice thickness). The beam is typically hourglass shaped with a minimal width at the natural focus equal to D/2.

The area before the natural focus is called the Fresnel zone; the area after is the Fraunhofer zone, where the beam diverges and becomes wider. At twice the natural focus the beam width is the same as the diameter of the crystal.

The distance to the natural focus is called the 'near zone length" and is calculated with the following equation:

Near zone length = D^2/4*wavelength

where D is the diameter of the crystal.
In tissue since c = 1540m/s this is also equivalent to
D^2*frequency/6
where diameter is mm and frequency is in MHz.

In this question if the near zone length is at 6 cm, then at 12 cm, it is at twice the near zone length and the beam width will be the same as the crystal diameter again, or 2 cm, so this is the correct answer. At the natural focus the beam width would be 1 cm.