If armies just ran at eachother with swords, that's what you'd expect. Therefore, we think that they did not do this. So yes, it would indeed be fair to say that battles mostly consisted of waiting.

Or rather, that is one of the major current theories. Goldsworthy, Sabin et. al. argue that battles consisted of what they term "pulses" of frantic activity, aggressive close combat, charges, etc, interspersed with much longer periods where the sides are in a stand-off, throwing projectiles and javelins at one another and presumably yelling insults, or where one side advances but the other falls back.

As this goes on, one side or the other will start to falter, lines start to get disorganised, soldiers will begin to waver, which will then encourage the other side to be more aggressive, until one of the aforementioned charges causes the other side to break and run from the field.

Casualties during the battle were relatively light. Figures of between 2 and 5% have been computed, though this is based on written sources that are never the most accurate when it comes to casualty figures. The actual large-scale killing must have mostly occurred after one side broke and ran. This is one of the major reasons why the above authors argue that the fighting must have been intermittent.

The reason for all this is the terrifying nature of close-combat. Running up to face a line of enemies with shields and swords and spears is one of the most difficult things for people to do. And whilst 2% casualties over several hours may not sound like much, it does constitute a considerable number of people in the front ranks who actually find the courage to run up and fight. (Most people in the back ranks are much safer... unless their army loses and they run away, pursued for miles by enemy cavalry.)

Goldsworthy argues that one of the most important factors in ancient armies' success was their cultivation of the bravest and most naturally aggressive fighters, whose task it was to lead the actual attacks against the enemy line, encouraging and motivating their fellows and taking the largest risk upon themselves. In the Roman army, this task fell to the Centurions, who therefore also took disproportionally heavy casualties in battle. (For the battle of Pharsalus, Caesar lists 30 centurions lost to 200 rankers at the battle of Pharsalus, which is 7 times as many proportionally. Ouch.)

The problem with this theory, and its competitors, is that it's highly speculative because none of our sources describes what exactly went on during a battle. Very frustrating. Still, I think this model does offer the best explanation we can get.

It does match what we know from later eras, though, when reading descriptions of bayonet charges in the Napoleonic wars. It also matches what we can observe nowadays when looking at the behaviour of rioters confronting police shield-walls and the like, where you'll see these lengthy stand-offs followed by charges this way and that led by the most aggressive ringleaders. None of these comparisons are perfectly accurate, of course. Rioters aren't soldiers, bayonets aren't hoplite spears. But it's the best we have.

Edit — on a tangent:

(such as the Romans) would have formations as thin as three men deep

Goldsworthy is actually the only historian I recall arguing for formations this thin, and I don't know anyone who actually agrees with him. Most formations were much deeper, and the same probably goes for the Romans. The typical An often mentioned depth for the Greek hoplite phalanx was 8 men.* Later pike phalanxes could be as deep as 50 men, though that's obviously exceptional.

Further Reading

This thread on AH: Did the people in the front lines of ancient armies basically know they are going to die? Among other things it has an excellent post by u/Iphikrates and another one by me saying most of the same stuff I do here.

If you can find it, read Philip Sabin's“The Face of Roman Battle” in The Journal of Roman Studies 90. It's the best article I know that actually tries to describe what an ancient battle was like.

Finally, if you want to know about some competing theories of ancient battle and why I don't believe they work as well, read Goldsworthy's “The Othismos, Myths and Heresies: The Nature of Hoplite Battle.” in War in History 4.

^*See ^follow-up ^post ^below