Petrol gauge vibrates madly even with car not moving

The gauge in my Mk I showing the fuel level in the tank madly vibrates.
This happens even with the car stationary. It appears to be triggered by the small voltage fluctuations when the fuel pump goes through a pumping cycle. But once triggered, the needle in the gauge vibrates for a long time.

Is there an easy fix?

Have you lost your vibrator recently?

If not I suspect that the sender unit in your tank may be faulty.

A relatively easy thing to try is substitute a known resistance across the sender return

You would have to purchase resistors or a variable resistor,

I have an idea the earlier senders are in the order of 30-300 Ohms

If you place the resistance in circuit in place of the sender, and the needle settles, that will be the culprit

If not, maybe a voltage regulator if the MK1 has one

The senders can be taken out, apart, and cleaned, often to good effect

They have a point that wipes across a copper coil

Since this is related to the fuel pump, I expect a ground issue somewhere is the trunk.
Fuel pump and fuel gauge do not share wiring.
My MK1 Fuel gauge is steady, but all wiring is new (wire loom from SNGBarratt )