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Are there any good functions to model tidal waves other than the standard trigonometric functions? I think Fourier series might work, but I am not entirely sure yet so I want to know if there are some good sources out there where it demonstrates modelling waves with Fourier series, or any trigonometric modelling.

Like in particular I want to know like how the shape of the Fourier series changes as we keep adding the sine functions, or chaining the period of series of sine and cosine functions. Are there any good sources which demonstrate the graph behavior of sine series and cos series when changing periods for beginners? Like are there any sources which can help me get more familiar with the patterns of trigonometric series just like how we got familiar with the transformation and shapes of ordinary trigonometric functions in high school? Because most online Fourier pdfs give more of a formal treatment of the matter proving various properties of Fourier's series such as orthogonality, but doesn't treat so much on the shape of such series.

(I have been playing around with the shapes on demos like I tried inputting https://www.desmos.com/calculator/ukomfyigco however my biggest confusion is what is the pattern with the number of peaks formed and the shape depending on how many terms we add on to the series).

Aurora Borealis
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  • First of all, Fourier series are the same as that of the standard trigonometric functions. This is a powerful approach for long waves. A book can be recommended https://pdfs.semanticscholar.org/582a/dc3d80e4a9a5f2affe6a6994201793ea2633.pdf or a more advanced https://www.amazon.com/Sloshing-Odd-M-Faltinsen/dp/1107646731. – MathArt Sep 20 '19 at 10:51
  • Not that it matters for the question you're asking, but by "tidal waves" do you mean the rising and falling of the surface of the ocean at a scale of about 1/2 the circumference of the earth, or do you mean tsunamis, which are not tidal at all, but are often called "tidal waves". – John Hughes Sep 20 '19 at 10:53
  • I mean just ordinary tidal waves not tsunamis. – Aurora Borealis Sep 21 '19 at 00:33
  • @MathArt thank you for the sources. – Aurora Borealis Sep 21 '19 at 00:42

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I think most modeling of tidal fluctuations is done with a number of Fourier series based on the periods of the forcing functions. The sun contributes a drive at one cycle every $24$ hours, for example, so we put that in and its harmonics. The moon contributes a drive at a slightly different frequency because of its orbital motion, so we put that in and its harmonics. The moon provides a drive at once per lunar revolution because of its orbital inclination and so on. These frequencies are not multiples of any base frequency, but they allow us to represent the tides with many fewer terms than if we blindly picked a single base frequency for a Fourier series.

Ignoring the above, suppose we have a Fourier series with one base period like $24$ hours. The harmonics have periods of $12,8,6,\frac {24}5,4,\ldots$ hours. You can get one peak per period that you add in, but do not have to. I think a better way to approach it is that each period you add in allows you to fit finer structure in the function, like you are reducing the pixel size in a picture. The highest harmonic you add in represents the shortest wavelength available, so the finest detail you can represent. For a nicely smooth function you have large Fourier coefficients as long as there is structure at wavelengths of the harmonic. After that the coefficients are just working on the detailed shape of the function and they fall off rapidly. You cut off the series wherever it is convenient.

Ross Millikan
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  • Hello sir. Could you please help me with this question: https://math.stackexchange.com/q/3380406/394202 – Soumee Oct 04 '19 at 17:54