Normal surface water waves, as generated by wind, do not have sine form but wave peak is higher and shorter than wave trough with different wave steepness. What parameters characterize such a surface water wave and how can one predict amplitude of water for given waves as function of time?
[Physics] How to model the form of a surface water wave
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Here are Kip Thorne's comments in Ch. 17 of The Science of Interstellar (note that when he refers to Miller's planet being "locked" to Gargantua, this refers to tidal locking in which the planet rotates at rate that always keeps the same face to the black hole, which minimizes the tidal stresses on the planet):
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 will naturally provide a sort of restoring force back to its preferred orientation, explaining why the planet would rock this way] ... 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
The first explanation might correspond to the explanation Neil DeGrasse Tyson gives in the quote from Kieran Hunt's answer, 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).
The use of complex numbers is just a mathematical convenience. It makes calculation of derivatives especially easy, it has nice properties when you do Fourier transforms, etc. You're correct that you can do it all using real numbers, so that's not wrong. It's just - in most people's view - more cumbersome.
EDIT In light of the back and forth in the comments, let me provide more detail.
First, starting with classical mechanics: Let $f$ be a (potentially) complex solution to the wave equation. The physically relevant (i.e. measurable) quantity here is the amplitude as a function of space and time. Any complex function can be rewritten in terms of two real-valued functions $g$ and $h$ such that $$ f = g + ih $$ The amplitude of $f$ is $\| f \| = (g^2 + h^2)^{(1/2)}$. We basically have two free functions here where we only need one to meet this constraint, so we're free to choose $h=0$, which means that $f$ is actually real-valued. You could choose some other values for $g$ and $h$ that have the same amplitude, but you don't need the complex part. (Note that I'm not dealing with plane wave solutions here, although you could build up your solution from them. I'm dealing with general solutions to the wave equation.)
For quantum mechanics, we have the Schroedinger equation: $$ i\hbar \partial_t \Psi = -\frac{\hbar^2}{2m} \nabla^2 \Psi$$ (where I set $V=0$ because it's not going to figure in the rest of the point). This is typically written with complex numbers, as shown above, but this is again a short-hand only. We could instead write the solution in terms of two real-valued functions: $$ \Psi = f + ig $$ and then, doing a little simplification, get two, coupled, real-valued PDEs: $$ \hbar \partial_t f = -\frac{\hbar^2}{2m} \nabla^2 g $$ $$ \hbar \partial_t g = +\frac{\hbar^2}{2m} \nabla^2 f $$ So, again, we can avoid complex numbers in the formulation. The price here is that we now have coupled PDEs for real functions instead of a single PDE over complex values. It turns out for practical reasons, that working with the single, complex-valued formulation is easier.
Best Answer
Deep water waves are often described as "cnoidal", with a mathematical description involving the Jacobian elliptic function cn(). This is an exact solution to the nonlinear Korteweg–de Vries differential equation. A more accurate equation is the Boussinesq. These are the basis for describing all water waves, whether stirred up by wind or otherwise. The basic parameters for a particular solution are wave height, period (or wavelength), depth of the water, and acceleration of gravity.
I hate to cite Wikipedia due to its propensity to change, but the best explanations I could find, including math, are there.
As for the details of wind pushing on the wave peaks, and the peaks disturbing the air flow, and big waves breaking over in ways that excite surfers, there are no nice mathematical forms I know of, but then I'm not expert on this. Numerical modeling is king in the area. Some original research was done for the movie "The Perfect Storm" on how to do better simulations and crunch the numbers faster.