# Wave equation for water waves?

• TriKri
In summary, the person is looking for a model that can simulate waves with different wavelengths accurately, where the speed of a wave is dependent on it's wavelength according to some specific equation, for example this formula presented Wikipedia: Wind wave: Science of waves.
TriKri
Hi,

Has there to your knowledge been developed any wave equation for for water waves?

what kind of water waves? There are quite complex models for waves in fluids and fluid like substances.

Like ocean waves for an arbitrary water depth

A bit more specifically, I'm looking for a computationally cheap model, that can simulate waves with different wavelengths accurately, where the speed of a wave is dependent on it's wavelength according to some specific equation, for example this formula presented Wikipedia: Wind wave: Science of waves. I essentially want the time complexity for evolving the system in time to be of as low order as possible, for example O(n) or O(n ln n) or something like that, where n is the number of surface elements. Thank you!

So it sounds like you want to do a Fast Fourier Transform f(x) -> F(k), propagate each wavelength according to its phase velocity, then FFT back again. FFT algorithms are O(N ln N).

Bill_K said:
So it sounds like you want to do a Fast Fourier Transform f(x) -> F(k), propagate each wavelength according to its phase velocity, then FFT back again. FFT algorithms are O(N ln N).

Yes, but I want it to be able to handle varying water depths, just like the Schrödinger equation can handle varying potentials. This is because one wavelength may travel in one speed at deep water, reaching a maximum speed when the depth approaches infinity, but slow down as the water gets shallower. So the speed doesn't depend on the wavelength only, but on the wavelength, and the depth of the water which can unfortunately not be determined when working in the frequency domain.

A very nice treatment can be found in the excellent book

A. Sommerfeld, Lectures on Theoretical Physics, Vol. II

The U.S. Army Corps of Engineers publishes a volume in their Coastal Engineering Manual series which discusses water waves in deep and shallow water quite completely. Volume II can be downloaded from the link below in Acrobat format:

"[URL

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vanhees71 said:
A very nice treatment can be found in the excellent book

A. Sommerfeld, Lectures on Theoretical Physics, Vol. II

Do you know where can I find that book? I can't even find it when googling for it! All I can find is Volume 1, 5, 13, etc. Are these books earlier/later versions of the same one, or are they simply all different books?

SteamKing said:
The U.S. Army Corps of Engineers publishes a volume in their Coastal Engineering Manual series which discusses water waves in deep and shallow water quite completely. Volume II can be downloaded from the link below in Acrobat format:

"[URL

I have taken a look in it, and it looks quite comprehensive, but I think that it is going to take a long time for me to find what I'm looking for by simply browsing through the documents or browsing through the tables of contents. Is there any specific equation or section in some of the documents you can hint about?

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Chapter 1 of Vol. II of the CEM is what you should study. I don't know how computationally cheap a simulation you are looking for, but there are a variety of wave theories to choose from.

The Sommerfeld Lectures on Theoretical Physics have 6 vols. I'm pretty sure you find them in any university library. I'm not sure whether you can find these books online...

I think this is the data you are looking for.

go well

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What I'm looking for is not an equation describing the speed of different wavelengths, but the PDE (partially differentiating and integrating equation?) for evolving the water surface in time in a simulation.

You mean the wave equation in water? Something like this:

$$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{g h} \frac{\partial^2 \psi}{\partial t^2}$$

where h is the depth of water, and g gravity... this is only 1D of course. I imagine it generalises to 2D by replacing the derivative wrt x, with the laplacian. I'm not an expert, I just dug this up in some old lecture notes.

I guess that is the wave equation for shallow water waves? I probably works good for that case, but it is still not exactly what I want... This equation makes sure every wave travels with the speed $$\sqrt{gh}$$, but that only holds when the wavelength is significantly greater than the depth of the water. When the water gets deeper, a wave with a certain wavelength in a real ocean will however reach a maximum speed which is equal to $$\sqrt{gL/2\pi}$$, where L is the wavelength of the wave (see the attached papers in Studiot's post).

I don't know what this is for and I certainly don't intend to do someone elses' work for them or post larger extracts.
If I had been asking the question and supplied with this information, not only would I have said a thank you, even if it was not quite what I was looking for, but I would have used it as a jumping off point.
The two pages I posted were the culmination and summary of a large amount of work about marine hydraulics. Derivations and explanations are further back in the text.

Well, I'm sorry, thank you for your answer. Of course I appreciate it. I appreciate every answer I get that is trying to help me and I want to make that clear to anyone else who also might have perceived me as ungrateful. Except from that, I must say I'm sorry if you feel that you have written here in vain; people should write here because of interest, and not because they feel they have to.

I'm definitely not asking others to do my work, I'm just checking to see if there is anyone who knows about such an equation, so I'm not reinventing the wheel. I have already tried to derive a wave equation myself - from the equations you posted - before I started this thread but I found myself trapped, not knowing how to continue or if it was possible at all in the way I wanted to do it. However, I found my work to be too much to post here before I had checked if there already existed a solution or not.

All the answers that, as you said, were not quite what I was looking for only got me thinking that maybe I hadn't posed the question right. Perhaps you're right when you say it's time to end the thread; maybe people don't know. But at least I feel I have to give it a chance by asking a well posed question and I hope that is fine with everyone.

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I think what is puzzling us is the use of the term "wave equation" and also what you are trying to do with it.

There is only one basic 'wave equation' and JesseC has already given that, but you have rejected it.
The wave equation connects distance and time, however the general solution contains and arbitrary continuous function of both distance and time.

The procedure in applying the wave equation to a given situation is to infer (guess) a suitable arbitrary function that fits the boundary conditions.

You cannot directly apply the wave equation to water waves, you also need to include some of fluid mechanics.

Airy's solution (in one dimension) works like this:

If x is the horizontal distance and z is the vertical distance, assume the surface profile forms waves according to

z = h + acos(vx)

If we assume steady irrotational flow of -v, both the surface and the sea bed form streamlines.
So we can use the fact that the stream function is constant along these surfaces.

Taking Airy's inspired guess assume a stream function of the form

$$\psi = c\cos (vx)\sinh (vz)$$

This meets the boundary conditions that $$\psi$$ is constant on z=0 (the sea bed) and z=cos(vx) - the surface.

If we plug these into Bernouilli's equation we arrive, after some manipulation at the relationships I posted earlier.

Is this enough to get you going and is it going in the right direction?

I think I should look into Airy wave theory a bit, and it is possible that it will get me going in the right direction, so thanks.

I basically wanted something like the Schrödinger equation but for ocean waves, that allows waves with different wavelength to travel with different speeds. I wanted to have it to accurately be able to simulate surface waves quickly on a computer without also having to simulate everything that happens underneath the surface. I my mind, a PDE for waves became a wave equation, although this is maybe not the case, so I can understand your confusion here.

I think I will have to look into this a bit more on my own though. Thanks everyone for the help, I think I have a lot of material now. Maybe I can post something here again if I come to any conclusion with the subject.

Marine waves are traveling waves, they transport energy. Further they do this in a moving medium, possibly against the flow.

You cannot accurately simulate the surface profile without reference to the bottom profile, this basic to fluid mechanics.
Those of us who live near the Severn Bore, or those unfortunate enough to live in the recent Tsunami zones are only to well aware of this.

Schrodingers equation is not really about waves, and is usually seen in the time independent form which is more akin to a standing wave where energy is not transported beyond the next node.

I wish you well in your endeavours, so please post back if you have further thoughts.

go well

## 1. What is the wave equation for water waves?

The wave equation for water waves is a mathematical equation that describes the behavior of water waves. It is a partial differential equation that takes into account factors such as the water's depth, gravity, and surface tension.

## 2. How is the wave equation derived?

The wave equation for water waves is derived by applying the principles of fluid dynamics and using the Navier-Stokes equations. These equations describe the conservation of mass and momentum in a fluid, which can then be applied to water waves.

## 3. What is the significance of the wave equation for water waves?

The wave equation for water waves is significant because it allows scientists and engineers to accurately model and predict the behavior of water waves. This is important for understanding and predicting ocean currents, tides, and other water-related phenomena.

## 4. Are there limitations to the wave equation for water waves?

Yes, there are limitations to the wave equation for water waves. It assumes an idealized scenario and does not take into account factors such as wind, temperature, and water density variations. It also assumes a uniform and unobstructed body of water, which may not always be the case in the real world.

## 5. How is the wave equation used in practical applications?

The wave equation for water waves is used in various practical applications, such as predicting wave heights and patterns for marine structures like offshore oil platforms, designing ships and boats to withstand rough seas, and understanding the impact of tsunamis and other natural disasters on coastal areas.

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