Why do some piezo elements have three wires?
Most piezos have two connections, but this type has three. They are called self drive types, and they’re meant to be used as part of the oscillator.
The piezo effect works both ways: if you apply a voltage the piezo stretches, but also if it stretches it creates a voltage. This principle is used to create a feedback signal which drives the oscillator.
The advantage of the self drive is that it will automagically work at its resonance frequency, where it produces the loudest sound.
You need:
- Piezo
- 2N2222A transistor (or some other NPN transistor)
- 150K ohm resistor
- 3.9K ohm resistor
- 620 ohm resistor
Demo:
Tauno Erik 
Schematic has three resistors while bread board has four
Yes, because I had no 620-ohm resistor. I had one 390 and 220 = 610 ohm.