What is an O gauge "Whistle Controller" box please?

You cannot whistle a Lionel on dc track.  You have to mount the whistle on the layout and feed it with power, a button is ideal vs toggle.

Lionel did not include whistle tenders with dc only trains.

Normally 1 ac cycle has two equal voltage waves, one +, one - (+~-←1 wave)and measured from the 0 point middle. (osilisope graph view).  Adding a single diode , one of those equal waves gets it's peak cut off. Which wave depends on which way the diode points.

So one ac wave half now has a lower volt reading than the other half. This is an offset.

Move our starting measuring point to either wave peak and measure the offset voltage to the next. Measured this way, you have a small + or - value that is dc+ or (-) piggybacked on an ac wave(always equal).  Our whistle relay ingores equal waves, it only acts on a difference. (special relay coil w/copper field shunt).

Edit: dc+ or (-) piggyback

was dc+ or + piggyback.

A whisrle control box has diodes inside and adds them or removes them as needed.

Older units use a special disc that predates the first diodes but does the same thing. A rectifier disc has a coating on it that if scratched by the contact nut, won't block the power, and conducts like plain metal.

The disc looks like a big washer with a copper paint misted on it, bolted to the frame

I did an edit above

The whistle controllers 'have connections for AC in, and correct amount of DC out, to send the correct amount of DC (current)  voltage to a free standing whistle building or billboard''.

The postwar whistles use AC voltage to operate the whistle motor, and about 5 volts DC to pull in the motor relay.

Larry

The post war transformers have a boost coil in them to raise the track voltage when the whistle is blowing so the train does not slow down.  The stand alone whistle controllers use a choke to reduce the voltage to the train. When the whistle is blown the choke is bypassed to raise the voltage to the track to prevent the train from slowing down.   The use of DC is to operate the whistle relay.  The whistle motor is a universal motor and will run fine on either AC or DC.

So, in summary,  if I buy a whistle controller, and run AC current lines to it directly from my AC transformer, and then run the DC current wires out of the controller to a billboard whistle, . . . will the dang whistle blow??  And if so, will my DC train slow down?

(Alternative solution:  For $58, I can buy a real airhorn, made for mounting on the cabs of tractor trailers, which sound exactly like a train whistle, mount it under the train table, hook it to a 12 volt rechargeable battery, and let'er rip!)

Mannyrock

You skipped rectification. All whistle controllers are not identical. They have thrown you into a loop because they don't see all the cards in this post. They lay elsewhere too. 

You have left a lot of wiggle room for any answer ro be wrong.

You plan on ac to dc for the track. Fine, covered.

Now consider this your output from now on. You may power a whistle off it too. It needs a spring return off/on, N. O. normally open in catalogs.(means pressed the ciruit is connected/closed.)

Or

Now, add another BR to the same ac and use that just to run a whistle motor(no relay)

So it will stop whistling, use a button with Normally open and common on a button.  No power flows.

or a constant ac to a br.

Or if the whistle motor is ac, you can run off the var. ac or constant ac.... still need a button to interrupt power until pressed.

The button here is simple off and on when pressed, sprung return. A #90 Lionel will work. 

Now... your loco is going to slow some when you whistle. You are suddenly sharing power, what would you expect?

We can hold back a few volts and save those for a boost that kicks in only during whistle time.  ...interested?

Don't pass more power thru the BR if you are using an ac motor. Why bother? work=heat for a BR.

You only need the offset nonsense for the Lionel Whistle RELAY, you're gonna replace the relay (a switch with a motor) with a finger powered switch/button for a whistle shack/billboard. (on ac track your shack may use a whistle relay)

New age air whistles use a dc can motor with much less draw than a heavy duty old school ac/dc version.  Both impellers are tiny and the can motor spins it easy. It's actually an improvement if oportuninity knocks... these don't need a boost, or just a small one vs 5v.  I wont try to sell you on can motor locos, but some other uses, think about it.

Sometimes a relay us used to seperate control and action power types. Sometimes it's to shorten high amp delivery(always good) using just a tiny bit of power and /or from a distance. You can use a reley to save wire costs and gain efficiency, control tricks, etc. (latching relays is how elevators "think".)

Actually, I see opportunity with dc here. One or two diodes on its own dc whistle leads, to separate it from the track power more, and you could add a capacitor/resistor to that little whistle sub-curcuit. That will give a nice amp and volt supply to get the whistle to get it spinning fast, and that's what slows the train.(its like a strong battery that drains fast.)  One is also on one of your hvac motors most likely   a resitor to control charge rate or discharge.. a maybe.