When watching the part where they go down to Miller, I was bothered by the fact that the waves didn't break despite the fact that the water level is ostensibly only knee-deep (roughly, a wave starts to break when the depth of the water is less than the height of the wave). But then, a friend came up with the idea that the waves are not really waves. Given that the gravitational influence of Gargantua on Miller is, well, gargantuan (I'm not apologizing for that joke) the "waves" are actually mountains of water created by Gargantua's gravity. As such, the "waves" don't really move towards the crew; rather, they are stationary (relative to Gargantua), and it is the rotation of Miller that brings the crew towards the "waves". The reason the ocean is so shallow is because a substantial proportion of Miller's water is used up to make up the "waves".
Is this really a plausible idea? What bothers me about this is that, if it is really the planet that is rotating under the "waves", the crew should have experienced a night-day (or day-night) transition between one "wave" and the next.
Answer
In chapter 17 of Kip Thorne's explanation in The Science of Interstellar, he makes clear that Miller's planet is supposed to be tidally locked to Gargantua (the black hole), meaning its rotation period is the same as its orbital period so that one side of it is always facing Gargantua, while the other side is always facing away (specifically, Thorne writes in that chapter that 'In my science interpretation, the planet must always keep the same face pointing toward Gargantua'). Tidal locking is a well-understood idea in astrophysics, explained in terms of the gravitational tidal forces from the main body continually exerting a torque on tidal bulges in the orbiting body which decrease its rotation rate until it becomes locked; this is used to explain why the moon always presents the same face to the Earth, for example.
To explain the waves, Thorne says that although Miller's planet is almost perfectly tidally locked, it does rock back and forth slightly like a pendulum, with the tidal forces from Gargantua always acting as a restoring force to pull it back towards the orientation where the planet's tidal bulge is facing directly towards Gargantua. And given this rocking, he gives two possible explanations for the giant waves:
What could possibly produce the two gigantic water waves, 1.2 kilometers high, that bear down on the Ranger as it rests on Miller's planet (Figure 17.5)? I searched for a while, did various calculations with the laws of physics, and found two possible answers for my science interpretation of the movie. Both answers require that the planet be not quite locked to Gargantua. Instead it must rock back and forth relative to Gargantua by a small amount [snip Thorne's explanation of how Gargantua's tidal gravity would provide the restoring force to explain this rocking] ... The result is a simple rocking of the planet, back and forth, if the tilts are small enough that the planet's mantle isn't pulverized. When I computed the period of this rocking, how long it takes to swing from left to right and back again, I got a joyous answer. About an hour. The same as the observed time between giant waves, a time chosen by Chris without knowing my science interpretation.
The first explanation for the giant waves, in my science interpretation, is a sloshing of the planet's oceans as the planet rocks under the influence of Gargantua's tidal gravity.
A similar sloshing, called "tidal bores," happens on Earth, on nearly flat rivers that empty into the sea. When the ocean tide rises, a wall of water can go rushing up the river; usually a tiny wall, but occasionally respectably big. ... But the moon's tidal gravity that drives this tidal bore is tiny—really tiny—compared to Gargantua's huge tidal gravity!
My second explanation is tsunamis. As Miller's planet rocks, Gargantua's tidal forces may not pulverize its crust, but they do deform the crust first this way and then that, once an hour, and those deformations could easily produce gigantic earthquakes (or "millerquakes," I suppose we should call them). And those millerquakes could generate tsunamis on the planet's oceans, far larger than any tsunami ever seen on Earth
And in this interview he mentions that the wave is meant to be a soliton (short explanation of what that means here), a type of isolated wave that maintains a stable shape as it travels, often without turbulence or "breaking":
I don’t use this word in the book, but the waves appear to be solitons, solitary waves. They don’t break, and they are probably coming in from a region where the water is somewhat deeper. One possible explanation for them is that they are similar to tidal bores that can run up the long, gentle channels of rivers with the rising of a tide.
Here are some videos showing real-life solitons:
Astrophysicist Neil DeGrasse Tyson also offers an explanation for the giant wave that occurred to him in this interview:
Initially, I thought, “OK, they have to throw in a wave… that looks gratuitous.” My second thought was, “Well, if it’s a tsunami, the wave actually needs water to be the wave, and they would see the water rush from around their ankles to feed this wave as it came by.” That’s how you know to run. In this, I would later figure out that both of those concerns were unfounded. The planet is deep in the gravitational well of a black hole, and the black hole would surely have very high tidal forces. Also, a “tidal wave” is misnamed—it’s actually a “bulge” of water fixed in space. The bulge is always oriented in the same configuration in space, so you on the solid planet rotate in and out of that bulge. You interpret it as a wave coming towards you and away from you, but what actually happens is you’re rotating from a high tide part of the water to a low tide part of the water. The fact that the waves came every hour or so meant that the planet rotates once ever two of those—because you have two high tides for every rotation. If I were to say that there was something unrealistic about that, it was how spiky the wave was. A tidal bulge would be smoother than that, and they would just rise up, float over the top, and rise back down the way a duck floats up and down as a wave goes under it. This is where they’re taking dramatic liberties to turn the wave into something more menacing, and I don’t have a problem with that.
Tyson's answer might be the same as Kip Thorne's first possible explanation in the earlier quote, but I'm not sure--presumably tidal bores on Earth don't remain at a fixed orientation relative to the Sun while the Earth rotates under them, since that would require them to travel at over 1000 kilometers/hour at most latitudes, but then the Earth isn't nearly tidally locked to the Sun so it's possible that what Tyson describes would be a type of tidal bore as well.
Tidal locking also explains why there's no day/night cycle on the planet. The illumination is supposed to be coming from the accretion disk surrounding Gargantua (the glowing ring seen around it which is distorted in a strange way due to gravitational lensing, see my answer here for details on its appearance), and if Miller's planet is tidally locked to Gargantua then one side would always be facing the accretion disk in permanent daylight, and one side would always be facing away in permanent night. (It should technically be the side facing away from Gargantua that was facing towards the accretion disk--Thorne writes that 'since Miller's planet is the closest anything can live stably, without falling into Gargantua, the entire accretion disk should be outside the orbit of Miller's planet'--but he also notes elsewhere that they made artistic compromises with some of the visuals in the movie, one of which was depicting Miller's planet as much further from Gargantua than it really should be in order to avoid letting the audience see Gargantua in extreme closeup until the climax when Cooper falls into it.) For references on the accretion disk being the source of illumination, in chapter 9 Thorne says that Gargantua is supposed to have a relatively "anemic" accretion disk compared to known real-life quasars that have been observed (which are thought to be supermassive black holes like Gargantua), due to its not having swallowed any new large bodies in millions of years, so that it would emit light in the visible spectrum (temperature being related to peak light frequency by Wien's displacement law):
Instead of being a hundred million degrees like a typical quasar's disk, Gargantua's disk is only a few thousand degrees, like the Sun's surface, so it emits lots of light but little to no X-rays or gamma rays.
Then in chapter 19 on Mann's planet, he says:
Mann's planet can't be accompanied by a sun on its inward and outward journeys because, when near Gargantua, huge tidal forces would pry the planet and its sun apart, sending them onward in markedly different orbits. Therefore, like Miller's planet, it must be heated and lit by Gargantua's anemic accretion disk.
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