Not quite spot on Speedbird and Finals…
The flap howl originates from the flap wing gap (slot) along most of the wingspan. The source is aerodynamic. Most high lift wing / flap combinations have such a noise, often heard from the ground. You can just hear a similar noise in the cabin on Airbus A320 when sitting near the wing leading edge (Kruger flap?). The difference between the 146 and Airbus is that the A320 has a low wing at cabin floor level that masks the flap noise, but the 146/RJ has a high wing position above the cabin. The 146/RJ also has more lift – still the highest CLMAX for any civil aircraft.
The noise is very speed dependant, hardly noticeable if the flaps are retracted at Vfto or lowered at min maneuvering speed, but you will get max howl when selecting flaps at flap limit speed.
BAe did test fly a partial fix, a wedge shaped spoiler on the underside of the wing; not too successful I guess, as it was not put into service.
Another other noise with flaps down was a mild drumming in the rear cabin; this was fixed by use of flap track shutters over the inboard flap track (fuselage gap; RJs only?). If you get a drumming noise at approx 180 kts, 18 flap then get the shutters regreased.