Beam Width

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:
uo = rms velocity of the piston
J1( ) = 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

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