This subject has got me a bit engaged as you can tell by my several posts.
Though I believe from recorded interviews that Capa had stated he elected to go in with Easy Company, the 2nd Battalion, 16th Infantry Regiment, US Army 1st Division which he believed would be in the first wave, it also would not surprise me to find that there may have been some delays or changes to the order of battle that mean that this unit did not hit the beach till later - the photos do show for plenty of action ahead of him when he landed. It is of course also possible that Capa was not being strictly accurate in saying he elected to go with the first wave (he was known for exaggeration) and perhaps this was compounded by his loose interpretation of English something he was also known for - "one of the first waves" sounds awfully like "the first wave" if you are of that mindset.
Never the less I felt that one way of checking further on his time of landing was from tide times. Planners had elected for the landings to start shortly after low tide - one reason the 5th, 6th or 7th of June were selected as possibles was that low tide occurred at around dawn. The reason for selecting low tide (not high tide which army planners wanted in order to get the men off the sand as quickly as possible) was that Rommel had planted thousands of underwater staked obstacles. Landing at low tide meant they could be avoided by incoming landing craft and blown up by engineers. A further downside to going at low tide incidentally was that many landing craft grounded off shore in sand bars forcing the men to struggle ashore through the deeper water between the sand bars and get shot down (or drowned) in the process. If anyone has had this kind of experience (and I have though decidedly not under fire attacking defended beaches) they will understand its quite scary enough when the bullets are not flying.
But Capa's images show those obstacles clearly well out of the water. If they are not under water it was because the time was not too far off absolute low tide - perhaps an hour or a bit more, just as the planners planned. Whatever the exact time he made his photos it could not have been very late in the piece as the obstacles were designed to be "underwater" obstacles. You can see some of the them here:
Here is a report on the tidal conditions which when read in conjunction with Capa's photos suggest to me that he went onto the beaches, if not in the first wave, then reasonably soon thereafter.
"The tidal range from one low water to the next high water along the entire French coast of the English Channel was never less than six metres. At low tide, those large tidal ranges exposed long stretches of beach that Allied soldiers would have to cross under heavy German fire......
.....From low water, the water would rise at a rate of at least a metre per hour, sometimes even faster due to shallow-water effects. Five landing beaches had been identified (code-named Utah, Omaha, Gold, Juno, and Sword) and the timing of the tidal conditions varied between them. Between the farthest of them, separated by only about 100km, the difference was more than an hour – so the landing time on each beach, had to be staggered according to the tidal predictions.
Initial landings needed to be soon after low tide so that demolition teams could blow up enough obstacles to open corridors through which the following landing craft could navigate to the beach. To allow enough time for the demolition teams to blow up a sufficient number of beach obstacles, the times of low water and the speed of the tidal rise had to be known precisely. The tide had to be rising, because the landing craft had to unload troops and then depart without danger of being stranded by a receding tide."
So: First wave? One of the first waves? Does it really matter when what is clear he was there, there was an awful many men dying that day and he got the pictures and got them back for publication. He did his job.