There are several problems that can cause water to surge or to go off and then come back on:
Bad Pressure Tank or Water Logged Tank
There are 2 kinds of pressure tanks
Standard tank- Water comes in contact with the air in the tank over time and the air is absorbed by the water. The tank needs air to compress for pressure. Without enough air, the tank becomes water logged causing water pressure to surge
Bladder pressure tank- The air and water are separated which eliminates water absorbing the air. If the bladder becomes ruptured or torn due to fatigue surging will occur.
Precharge on tank is incorrect
If the precharge is out of adjustment with the pressure switch surging could occur.
Pressure Switch/ Nipple
The Nipple or Pressure Switch could be plugged with corrosion, not allowing the switch to monitor pressure properly.
Production of pump is more than the well produces
When the pump is pumping faster than water is coming into the well, the pump will quit delivering water until the water rises in the well. Then the pump will start pumping water again causing it to surge.
If you notice your water from you water well surging, it is important to repair the situation quickly as it will cause premature pump failure.