There’s a sound associated with falling bombs, a long steady whistle that gradually decreases in pitch. What’s the origin of this association?
Did most WWII bombs make this sound? Do modern bombs? If there was such a sound, could a person on the ground hear the bombs approaching?
IIRC bombs that did that were made to do so to scare people.
Most bombs I do not think make a sound (or rather not intentionally).
Weren’t Luftwaffe planes designed the same way?
Yes.
The Stuka dive bomber (do not know about any others) had a siren affixed to it to make a noise when it dove on a target to scare the people below.
The Ju-87 Stuka dive-bomber was equipped with a siren which was used during the dive to frighten the attackees. As if the bombs wouldn’t be enough.
Modern mortars do, indeed, whistle. And yes, it’s scary - for about a second.
Artillery rounds like those fired from a M110 are supersonic so they make a sonic boom like sound. That also means that you’ll hear the sound some seconds after the round has already dropped on you 
Are mortar rounds supersonic?
I’m not an artillery expert, but I don’t think a mortar could possibly be supersonic. They are made to drop onto the target using gravity alone, so they should hit with some subsonic terminal velocity.
Is it even possible for freefall terminal velocity to be supersonic?
One thing I always giggle at in movies is when a falling bomb makes the descending whistle noise. It’s completely backwards, if you think about it- the sound should doppler shift the opposite way.
A bomb dropped from a plane would only make that descending (high-to-low pitch) sound from the point of view of someone on the plane. Someone on the ground would hear the bomb whistle low-to-high.
I have heard of some very heavy, deep-penetrator bombs that supposedly have a supersonic terminal velocity, but I couldn’t find anything that I would consider conclusive. (some forums, a few pics with opinions/statements, nothing official or authoritative)
<<Edited to add: I am assuming we’re all talking about the earth here, and not some other planet with a thinner atmosphere or plasma jets in an astronomical event, etc.>>
I think a lot of what we think bombs sound like come from Bugs Bunny, so take that with a grain of salt. 
V-1 rockets did make a really loud noise, though. It often stopped before it hit, because (I might remember this incorrectly) early in their development the engine often ran out before it was supposed to, but later on they’d figured them out so they hit under power.
The V-1’s had a small propeller in its nose which was turned by the air that the V-1 was moving through. This propeller turned a counting device, which was how the V-1 measured how far it had gone. When it had reached its target distance according to this counter, the V-1’s control system would throw it into a steep dive. This steep dive tended to make the fuel stop flowing to the engine, so the engine would cut out at that point.
Later V-1’s fixed this problem and dove under full power.
Yes—it’s even been achieved by a skydiver.
I don’t think the high-to-low is because it’s approaching you. It’s because it’s going from “toward you” to “over your head” or “parallel to you”. If you intend to catch it on the fly (thus putting the artillery gunner “out” :p), then you’d hear a low-to high. But if you’re alive to tell the tale, it’s still high-to-low, just like a speeding 18-wheeler.
Good to know I was in the neighborhood. 
At any rate, I’ve read the sound was very distinctive - it seems YouTube comes through.
There was a bomb developed for the Gulf War (I or II?) which was designed to take the deeply buried bunkers used by the Iraqi high command. It had to go through 50 feet of dirt or gravel then a few feet of reinforced concrete. According to Time Magazine, some wag at the munitions plant even painted the name on it, “The Saddamizer”. It had a time delay fuse, a nose cone that was 25 feet of solid steel and was dropped from 50,000 feet. I wonder if that hit supersonic speeds?
The V1 was a pulse-jet, which apparently are VERY noisy; also called a buzz-bomb. They were subsonic, and the time to panic from what I read was when the noise stopped. What I read was that they ran out of fuel. George Orwell describes setting up lawn chairs on the hill at Hampstead Heath in north London to watch them coming in over London;I thought he said the sound stopped before they dived. You could see them flying in, the noise stopped, then they dived and where they hit a big explosion happened.
The V2’s OTOH, were parabolic flight rockets much faster than sound. When they first began to hit with no warning, the authorities blamed random explosions on gas mains. Some wags started calling the V2’s “flying gas mains”.
On the contrary, your link explicitly states that the peak speed attained in those jumps was “614 miles an hour, nine-tenths the speed of sound at my altitude.”
Fuzzy memory - but weren’t there some ‘dropped’ bombs that employed a propeller like that? Or maybe it was one of the atomic bombs, that they wanted to detonate before it hit the ground?
Some types of bomb fuses had/have (sorry, it’s been many years since I was current) a vane that can be confused as a propeller. It’s purpose is counting revolutions before it’s armed, to let the bomb fall a safe distance from the aircraft.
Many dropped bombs had similar sorts of things. They were usually used to arm the bomb, not detonate it. The mechanism would prevent the firing pin from striking the bomb’s detonator until the propeller had rotated a certain number of times. This prevented the bomb from accidentally blowing up too close to the plane that dropped it.
Here’s a good diagram of one:
http://www.fototime.com/BAF4773FDF98642/standard.jpg
Note the small propeller marked as “ARMING VANE”.
I don’t know how much these mechanisms contributed to the whistling sound when the bomb was dropped.
Fat man and little boy used radar altimeters to trigger their detonation.
ETA: Technically, as Harry1945 pointed out, it’s a vane, not a propeller.