Lets dispel a few myths:

Fire has to have three things to burn, Fuel, Oxygen, and Heat. This is called the fire tetrahedron, and is the primary theory behind fire fighting. If you eliminate one, then the fire is unable to maintain a chemical reaction, and goes out.

With forest fires you're constantly running out of water, and the only way to ship it the location it is needed is often multi-hour drives (which consume a lot of fuel).

Most forest fires are not fought with water, they use fuel reduction to control the fire and eventually starve it of fuel. This is why forest fires are multi-week, or even seasonal events, because you essentially have to dig a fuel free line around the entirety of a fire to starve it of fuel, so that it eventually burns it's self out. On the largest of fires, it usually is a game of trying to drive the fire in the direction you want it to burn, and waiting for a weather event, such as a rain storm, to either put the fire out, or bed the fire down enough that you can put a line around it and contain it.

Now, yes, fires can reignite. However that is also true with water, water evaporates extremely quickly from an area that was on fire both because that area is extremely dry, because of the ambient heat, and also because it likely started out hot to begin with.

With water in direct firefighting you are removing one side of the fire tetrahedron, heat. In a small fire, say a house, you are able to use enough water, to overcome the number of BTUs that the fire is creating, via steam conversion. Water expands to steam at a ratio of 1:1700. In structure firefighting, you can put out a small room fire with a quick burst of water, say 20 - 30 gallons, and through steam conversion(because steam is still much cooler then fire) put the fire out. You will then have to follow up and cool of the remainder of the room.

In large urban fires, the use of water is less about putting the fire out, as much as controlling the expansion of the fire, and limiting it to the original building/location, and waiting for the fire to burn it's self out. Even with 20 or 30 fire trucks pumping 1000s of gallons of water, large fires produce more heat then we are capable of dissipating, so we again wait for fuel to be starved, and the chemical reaction to reduce.

Fire retardants help keeping fires out, but they're bad for the environment and are also hard to keep supplied in remote areas, and when you're refilling directly from a river.

Most fire retardants that are used in active firefighting are 100% biodegradable. Almost all toxic(and even the ones that are still legal, have a diminishing lifespan as they have been banned from being further produced, see Halon fire extinguishers for an example.) have been removed, both for environmental, and firefighter safety.

The problem that I see with device, is that is does not have the capability to scale. To be able to remove 100% of the oxygen from an environment, even the small size that they are testing on, is not realistic. When you begin to get to large fires, you are talking about having to move millions of cubic feet of oxygen, and what are you replacing that air with? They are using this on a very small fire with very volatile fuel(rubbing alcohol). My guess would be that they are not only using sound waves, but that they are getting a certain amount of air movement that is also helping with extinguish, as they are also changing the fuel to air ratio that the fire needs to burn, thus putting the fire out.

There is billions of dollars spent each year to perfect fire extinguish, and I would believe that if it was as easy as pointing a sub woofer at a fire and putting it out, we would all have sub woofers in our ceilings, not water pipes and sprinklers.

Experience: I spent 10 years in the structural and wildland fire service as a volunteer where we had an average of 100 house fires a year, and 30 wildland fires a summer.