Sorry for the late response, but in my experience and observations, you'll need to find a way to keep the roots damp until they reach into the reservoir. this is normally done one of three ways:
using a raft, so the cups holding the plants are always in contact with the water.
setting a wick into the cup so the plants get water that way until they grow down into the reservoir
setting up a drip system or other means of directly watering the root base until they extend into the media.
I have no experience with rafts though I understand they're not good for large heavy plants.
My problem with wick is; if the wick isn't sterile, you could introduce bacterial growth into your medium and/or reservoir. I lost several plants this years to that end.
That's why I use a modified aeroponic/deep water culture technique. the aero side keeps them wet until they reach the water. it keeps the upper roots damp as well. and it provides decent aeration of the nutrients.
As far as deep versus shallow reservoirs go, I like deeper better. for the same volume, a deeper reservoir will have less surface area which means less area to absorb heat from the sun. Now if you're using a raft system you have to weigh the depth of the reservoir against the height of the walls. when the reservoir level is on the low side, the walls of the tanks could block some light from your plants. This of course assumes you're growing outside. under lights. . .moot point.
And always remember, a larger reservoir is better. you don't have to change the nutrient as often and the larger volume will help buffer against giant pH and EC swings.
Here is a link to my smaller, one or two plant DWC/Aero mutant.
here is a link to My Hydroponic Tomatoes thread which shows the larger version of this system. BTW, the plants in that thread are now almost 6 feet tall.

grizzman,
How about some pics of those MONSTERS!!!!

I have been doing what you are looking to do for quite some time now. It does not matter how deep the tank, bucket or container is. The secret is to get the air to rise directly beneath each plant. If you have a wide rubber maid one, and you have more than one plant, you will want a air stone for each plant where each plant sits directly above where the air rises.

Also, the more air the better. A good rule of thumb is buy a air pump that is rated 10X more than the amount of water you are trying to oxygenate. So for a 5 gallon bucket, buy an air pump that is rated for 50 gallons.

I have two 5 gallon buckets, one with a large Cucumber plant, and the other with a 5' tall tomato plant, and one 100 gallon Whisper Air pump (Dual Outlet) proving the air. Nothing else runs off that air pump. The more air the better, and I also use the 4" air stones. I get the air stones at Petco, they are cheap and work great.

Here is a link that might be useful: My Hydro Pics

I've got a few things to add.

First off, I've never liked wicks much. If they're made of synthetic fibers they don't seem to wick very well, and if they're made of natural fibers they tend to rot.

Second, it's not much of a problem to keep things damp in this kind of system while the roots are developing. You just fill the reservoir to just below the cups the plants are in. The bubbles bursting on the surface throw a surprising amount of water upward. That will keep things plenty wet.

Third, keep in mind the strength of the container. Many of them seem really strong until you fill them up with water. Be careful, they like to bow outward or simply collapse when completely full, so test it out either in the yard or in the bathtub first.

Once your plants have decent-sized roots you want to make sure you've got some air for the roots to hang in, and plenty of water.