The buzzing is the brake pump - the P38 uses a positive pressure booster system rather than a vacuum assisted system on other vehicles. Partly because of the traction control I guess, and in part I guess it made manufacturing more straightforward between diesel and V8 varients, as the diesel would have needed a vacuum pump running on the engine just for the brakes.
When the car is left, the pressure bleeds out over time, so when you next start up, the pump has to run to build it back up to the point the pressure switch turns the pump off. Then periodically (after a good number of brake applications) the pump will run again briefly to top off the pressure. The pump is quite loud, but without comparing with another or recording it etc, its a bit hard to say if yours sounds normal.
How many brake applications you should get before the pump runs is hit and miss - 4-5 decent applications is probably realistic for a good/new accumulator. If the pump runs every time you press the pedal, your accumulator is no longer holding pressure and needs replacing. Usually accompanied by the three lights lighting up briefly.
Three lights is ABS, TC and the hand brake light
Southampton with a Nanocom for Thor/GEMS P38s
1996 4.6 HSE, then became a 4.0, now cubed.
1997-8 4.6 HSE Motronic/Wabco prototype vehicle. Now M57 powered. Still auto.