Insight into cavity resonators

Each antenna is able to excite almost any mode, and will. The only modes that won't be excited by an antenna are those for which the antenna is at a node of the mode, a point where the standing wave of the mode has no amplitude.

Each antenna is at a different point in the cavity, so will excite different modes in different ratios.

If there only existed one mode how will the em field look inside the container for each of the 4 antennas?

The only way for there to be only one mode is if the cavity is much smaller, comparable to the half wavelength.

I'm am basically working backward in that I already have a cavity resonator and the frequency.

No, you don't have a cavity resonator, at least not what engineers would call a cavity resonator. You have a honking great box. A cavity resonator supports only one mode, it's built small enough or operated at the right frequency, to do that. You have a large box supporting many, many modes, most of which will be excited at essentially unpredictable levels by any radiation coming in from the antennae.

There is no need for you to visualise the modes in your box, all you need to know is that there are a lot of them. There are so many that the pattern you actually get will be unpredictable, given the accuracy with which you can characterise the box and the antennae.

A standard domestic microwave oven cavity is multimode. It will contain a number of high field and low field regions, which will result in over-cooked and cold food respectively. The three solutions to this are a) a turntable, to try to move the food through a variety of regions, b) a mode stirrer (very few have these) which is a metal vane rotating in the cavity to change the electrical geometry (roughly equivalent to your multiple antennae) and c) a single mode cavity with predictable power levels, these are only used in industrial applications where the cost of low frequency RF power generation is acceptable.

if I am correct these modes will be the same for each antenna

Not exactly. Modes themselves only depend on frequency and geometry of the cavity indeed.

However, which modes will be excited within your bandwidth of interest is a function of where you place your source of excitation. You can place it in a location where a maximum (or a minimum) of modes are excited. This is called mode coupling:

• If you place the antenna in a node of a mode (the point where the mode amplitude is zero) the coupling of your source to that mode is zero at that frequency (ideally).
• If you place the antenna in an anti-node of a mode (the point where the mode amplitude is maximum) the coupling of your source to that mode is maximum at that frequency (ideally).
• If you place the antenna somewhere else you will have something in between (coupling levels below maximum).

But does this mean that these will all have the same EM field pattern within the container or will each antenna have it's on EM field pattern?

Each antenna will couple differently to the modes according to its location. As a result, each antenna will probably result in very different overall frequency responses because of different coupling levels to whichever modes dominate at that location within your interest bandwidth. Unless, of course, you carefully choose the locations of each antenna so that the mode coupling is the same for all of them.

BONUS...

Wait, there's more to mode coupling than meets the eye:

• If you have two sources radiating the same signal, you can place them at phase-opposed anti-nodes of the same mode and expect cancelation of that mode. There are some applications (mainly audio) where this technique is used to flatten out the room response at bass frequencies. Several subwoofers are cleverly placed so they cancel out some room modes that cause 'boomy' reinforcement of some bass frequencies.

ADDITIONAL NOTE: Because of reciprocity, the same thing applies to a receiving antenna. You can place it to maximise (or minimise) the number of modes present at that location.