Interstellar: non-breaking waves on Miller

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.