Most ultrasonic transducers can be simply represented as a vibrating piston located in an infinitely large, rigid wall. The geometry of this is shown here:

We wish to know the sound pressure at a point A located at a distance r and an angle [theta] from the center of the piston. To do this, you divide the surface of the piston into a number of small elements, each of which is a simple source vibrating in phase with all the other elements. The pressure at A is the sum in magnitude and phase of the pressures from those elements. The summation looks like this:

(where:

u_{o} = rms velocity of the piston

J_{1}( ) = Bessel function of the first
order for cylindrical coordinates)

The portion in brackets yields the directivity pattern {it is called
the Directivity Function).

When the circumference of the piston is less than ony-half the wavelength
(ka < 0.5), the piston behaves like a point source.

When ka becomes greater than 3, the piston is getting pretty directional.

When ka becomes around 10, things get highly directional.

**Basically, as the ratio of the circumference to wavelength increases,
the angular divergence of the beam decreases.**

When you plot the Directivity pattern on cylindrical coordinates, it looks like this:

As the ratio of the circumference to wavelength increases, the angular
divergence of the beam decreases.

But, on further increasing the circumference, small side lobes appear.

**Piezoelectric**

**Electrostatic**

In this case:

circumference of transducer = 0.1 m

wavelength = 0.0065 m

ratio = 16, therefore, highly directional, with large side-lobes