[Pseudoscience-is-malarkey (OP)]

Another cringe-worthy question from yours truly: During World War II when bombs were being dropped on cities, what mechanics were at work that produced the whistling sounds they made?

[Malamute Lover]

Another cringe-worthy question from yours truly: During World War II when bombs were being dropped on cities, what mechanics were at work that produced the whistling sounds they made?

Mostly Hollywood.

The stereotype falling pitch is just plain wrong, It would be an increasing pitch due to Doppler shift as the bomb accelerated. The explanation that this was really the sound heard by bomber crews as the bombs fell does not hold water either. They would not hear anything like that over the engine noise.

But a bomb is not going to make a great deal of noise as it falls since it is intentionally streamlined to make the trajectory as invariant as possible and thereby maximize accuracy. The thousand pound bomb that was the favorite of the US 8th Army Air Force in Europe would reach a speed of close to Mach 1 before hitting the ground,which means whatever sound might be perceived on the ground would be of very short duration before the impact.

The Germans did put whistling devices on their bombs during the Blitz for psychological effect, but as I said before, it would be increasing pitch not falling pitch. The German Ju-87 'Stuka' dive bomber had a air speed driven siren that could be activated for use in a dive, again for psychological effect.

[syhprum]

I can vouch for the psychological effect the nearest one that fell close to me was about .5 km away and I was terrified

[Petrochemicals]

I imagine due to position of release relative to you whistleing bombs could  have made various sound patterns due to the fact bombs dont fall straight down as they are travelling at the same pace as a place. If its released above you it will fall to the side of you in a straight line of diagonal.

[Anad Rutner]

It was the stabilizing fins at the rear of the bombs.   The Germans tweaked theirs to really shriek adding to the terror but American bombs were not manufactured to do anything but fall straight down,

[alancalverd]

Beware of Hollywood. Even in the most factual of documentaries you will see and simultaneously hear the explosion of bombs filmed from the air. If you were flying at 30,000 ft and 300 mph, (a) you wouldn't hear much above the noise of the engines inside your pressurised cockpit or full-face oxygen mask and headset, and (b) by the time the sound travelled 30,000 ft upwards you would be 2.3 miles away. On the receiving end of an artillery shell, you might just hear the gun fire a few milliseconds before the shell hit you. The psychological effect of a V1 flying bomb  engine cutting out 10 - 20 seconds before impact is worth comparing with the supersonic (thus silent) ballistic descent of a V2 rocket. Fortunately I have no memory of either.

[Bill S]

Fortunately I have no memory of either.

Nor I; but one of my earliest memories is of a large chunk of chimney pot landing in the hearth, having been hit by a passing incendiary.  I guess, if it had been a few feet lower, I might not be posting this now. 

[science12]

I would say it's the bomb, or their fins, "cutting" the air and further on their sound waves.
The germans also put jericho devices on their bombs to make them whistle to demoralize the people being in the target.