There are four outlets that are 120 volts each. Two that will run two water pumps on the right side of the aquarium, and two for the left.
OK, are the pumps actually the things that make the waves? If not, please let us know what sort of mechanism creates the waves.
I custom build aquariums and have never touched code writing so please be patient. When I say one for one second, I'm referring to the time the the 120 volt outlet is on.
OK.. that's a fairly easy one. You will not want to put anything inside the outlet itself, so the best thing to use to switch 120VAC is a Solid State Relay. You can get versions that activate with fairly low current and voltage, yet will switch fairly high current devices on/off. For example, I just did a search on eBay using "solid state relay" without the quotes, and saw quite a few. The one at the top of the list was eBay item number 310702225734, "Output 24V-380V 25A SSR-25 DA Solid State Relay For PID Temperature Controller", (being sold by a vendor I have purchased many items from), and the Buy It Now price is $3.88 USD. You'll wait about three weeks for delivery.
You would want to test the input current with a multimeter to ensure that it was within the Arduino limits, but it most likely i. Worst case, you buy one more transistor to drive it.
You would also want to mount the SSRs in a secure enclosure, perhaps ventilated if they produce much heat, and bring the outputs out to proper AC sockets.
As far as what type of wave, it does not matter as it is a water wave. I also have no idea what parts to use so any guidance there would also be greatly appreciated.
Well, it matters in that there are several ways to make waves, and in order to advise you on parts requirements, we really do need to know how those waves are made.