How do they study the mathematics of nature ? Apparently there is mathematic of oceans and waves? How and what do they study mathematically?

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Anything can be described by mathematical models. Some things we have no other way of describing than maths, because we can’t touch them or see them. Like forces, or anything quantum.

There are equations modelling heat transfer, motion, mechanical stresses, etc.

That super realistic water in the new Avatar movie? It was made on a computer, with maths, because that’s the language computers speak. People found a way to model the behaviour of water with equations of motion and forces, then had a computer crunch them and show the results as visuals.

Mathematics is all about studying systems that have well-defined rules and working out what the consequences of those rules are. It appears that nature follows some well-defined rules. For a simple example, we know from experiments that at ground level, dense, compact objects (i.e. those that don’t experience much drag from the air) experience constant acceleration of about 9.8 m/s^2 . We can develop a mathematical model of a hypothetical object that experiences constant acceleration of 9.8 m/s^2 , and use it to answer questions like “How many seconds does it take the hypothetical object to fall 1 m?” Since we know that the rules of our mathematical model are very close to the rules followed by real objects, we can be confident that the conclusions we derived in our model will also apply to the real world. If we really want to be sure, we can also test whether the conclusions match the real-world results.

Oceans and waves are complicated, and many different mathematical models are used to describe different aspects of them. But the general principle is the same. You do experiments to find some rules that always seem to be followed, then you use maths to ask what would happen in a system that did follow those rules.

Any time you have a pattern, or any time you can identify some force that acts on a system, you have math. The math part is just relating that math to the math you already know. All math is is the idea that the same patterns show up in lots of different systems, and that it’s useful to study those patterns in their own right. Then, when you can find patterns in one system, you can use everything you’ve learned about the patterns to immediately tell you a lot about that system.

To address your specific example: a mathematical pattern called the [wave equation](https://en.wikipedia.org/wiki/Wave_equation) turns out to describe lots of different things: waves in water, vibrations in a drum, the behavior of light, and the behavior of an electron in an atom. We know a lot about the wave equation, so any time we can show that a system seems to be described by the wave equation, we can instantly apply everything we know about the wave equation to that system.