I'd say that while disadvantage is a good place to start, there are a few considerations that would change the potential results in ways that could be more fun. I don't think over-the-top physics-yness is necessary, more just some common sense and some extrapolation of existing mechanics.
First, can the fighter see the target? Water's pretty good at hiding things that are underneath it, especially if the water is murky or the target is far away. The farther away, the more water is in between the archer and target, and the more reflection off of the surface. So depending on those factors (as you deem appropriate) the target would be one of: visible, lightly obscured, heavily obscured or functionally invisible. If invisible, require a difficult Intelligence (Search) check to locate signs of passing.
Second, how far under water is the target? If deep enough underwater, even if the target is visible, there would be no reasonable path for the projectile to take and still do damage. At lesser depths, it may slow the projectile. I'd personally treat a deep target as having total cover, giving the projectile no chance of inflicting damage. For a shallow target, I'd confer resistance to piercing and immunity to slashing and bludgeoning. For a target close enough to the surface (say within 5 feet) I'd confer no resistance to piercing and resistance to slashing & bludgeoning.
(For fun, you could add in immunity to nonmagical fire, resistance to magical fire and vulnerability to lightning damage.)
Third, how skilled of a swimmer is the target? If he's floundering in the water, give advantage on the attack rolls. If the target has a natural swim speed, you can assume this is not the case. Note that a target that is drowning is restrained which also confers advantage.
Fourth, does the vantage confer a benefit to the archer? This is a judgment call: shooting straight down at someone underwater might be considered a very easy shot. I'd personally apply advantage if the shot is point blank, sort of like with a prone target.
Fifth, are there extraordinary abilities or magical powers at play that might influence the results? DM's call of course, but maybe the Archery Master feat could negate a disadvantage. It is all about making the shots that are otherwise impossible, after all.
So say some fleeing ruffian jumps in the water to hide, and your fighter whips out a bow to take pot shots at him as he swims away. Your fighter is on a dock, aiming straight downwards at the ruffian who's just below the surface, and has a clear shot. Maybe the ruffian is just an adequate swimmer, and is reasonably impeded or disoriented by being in the water. You might even say that in this circumstance the ruffian is a sitting duck, and confer advantage on the shot.
Next round, the ruffian puts some distance, but is still right near the surface. Your fighter takes a few more pot shots. Treat the target as being lightly obscured. If there are other factors at play, like mist rolling over the water, maybe heavily obscured (with disadvantage). So maybe for this one, with the ruffian 30 feet away from having hustled, getting 15 feet on the move and 15 more on the hustle, with half speed for swimming, the vantage is no longer advantageous. If you deem the obscuring to be just light, then that's also not a disadvantage.
Next round, the ruffian, with two arrows sticking out of various uncomfortable places, takes a gasp of air and dives deeper to try to avoid further injury. He's deeper now, and harder to see. The fighter loses sight of the target, drat! Not knowing where he is, he fires blindly and wastes an arrow.
Next round, the rogue points out the spot (assisting a Search check with his action) enabling the fighter to locate his victim. He makes a long-odds shot with disadvantage and resistance. It's a hit! Bubbles float to the surface as the ruffian falls unconscious, drowning. If the party needed information from him, quick action will be necessary, or some means of talking to a soggy corpse!
Note: Some types of arrows float.