Hello John, Leonard, and Oxfordby,
The pump you almost certainly have is a Shureflo. These are 'displacement pumps' and they vibrate when they operate so can be quite noisy. Re fixing the pump to a rubber pad may reduce the noise.
The pump incorporates a non- return valve on the coldwater inlet side and a pressure switch. The pressure switch senses the water pressure in the caravan pipework. When you open a tap the pressure drops which turns the pump on. Also you will notice that when you turn the tap off, the pump runs for a few moments afterwards, this is while it re-pressurises the pipework until the pressure switch trips.
The momentary operation of the pump every so often is the pressure switch sensing a pressure drop in the pipework, and the pump running to make it up.
If the taps are turned off and this cycling this is happening quite often, it indicates there is a leak somewhere. It is worth checking that none of taps are dripping, and the pipework for weeping joints.
Sometimes the pumps non-return valve can become blocked or damaged and does not close properly. This allows the pipework pressure to vent water back to the tank. You cannot see this happening, but a service on the pump may help to relive the problem.
Usually these systems should go up to pressure and hold for a long time, so continued cycling does need to be investigated.