Airy Wave Theory - Velocity Potential

[Double_Tyme]

Hi everyone, I am a sixth form student in year 13 currently studying my extended project. For my extended project i have decided to base it around fluid mechanics, more specifically I am looking into how you can use equations, laws and formulas to mathematically describe the characteristics of waves (water waves).

 

Right now i am looking into how it is possible to use the velocity potential of a fluid to work out it's flow velocity. I understand that the flow velocity is equal to the gradient of the velocity potential which is a scalar quantity and how the flow velocity of a fluid can be used to describe it's motion (i.e. Steady flow, Incompressible flow and irrational flow). I know how, vector calculus, Del operators and a tiny bit about how vector field work. HOWEVER I really need help with the equation for velocity potential in the first place as I know that the velocity potential and that however what i really need to know for research purposes, Who actually came up with the equation for velocity potential in the first place? and how was it derived? or Is their a research paper or text which can show me how this is derived? and if there is any other information on this that you think would be useful for me please don't hesitate to share it please?

 

P.s. I do understand that this is very advanced for someone in sixth form but i'm determined. Also i have asked but no maths or physics teachers in my school have any clue about this stuff so i'm on my own

 

Many thanks to anyone that can help me

Edited September 26, 2013 by Double_Tyme

[Enthalpy]

One has to distinguish between

- a pressure and velocity wave within water

- a wave at the surface of water

 

Waves at a free surface are quite a bit more complicated; for instance, they cannot depend only on t and x; the formula you give, with t, x, z, h, corresponds to them.

 

In the free-surface case even more than usually, finding a solution (here for the velocity potential) results from no general method. It takes one genius in history to find it from the differential equation, and then people learn it from course or encyclopaediae.

http://en.wikipedia.org/wiki/Airy_wave_theory

 

What students learn is how to reproduce the derivation of the solution, how to prove it fits the equation... but not how to find (invent, I'd say) the solution of an even so little different problem, because what professors and book teach are methods, but there is no method for inventing.

 

You know, Euler gave the general method to compute beam buckling, but 1.5 century later all books give still the same five examples as he did... Green and Fourier found general methods to solve the diffusion equation, but books give still the solutions to the same problems the ancestors found.

Edited September 26, 2013 by Enthalpy

[studiot]

Fluid mechanics is tough and then some.

 

You will need some additional knowledge to work through from first principles to your end result.

 

There are two types if differentials used in fluids.

 

Differentials track the variation of something (some quantity, say H).

 

We are interested in two possibilities.

 

1) What happens at a particular point in space, as the fluid streams past.

 

This type of calculus leads to the familiar ordinary and partial diffs you are used to plus the vector operator del.

 

2) What happens to a particular particle of fluid as it moves along ie how it changes velocity, direction etc.

 

This is known as "differentiation following the fluid" and is awarded the symbol D.

 

Various relationships between D and d can be developed.

 

Your route from velocity potential to your result requires several supporting equations or theorems.

 

Continuity or the conservation of mass (fluid)

 

An expression for external forces acting on the fluid (gravity and surface) This is done in the D form.

 

These lead to Bernoulli's Theorem in vector form.

 

This leads to a differential equation that can be solved and manipulated to yield the expressions you have posted.

 

A PM (private message) with an Email address capable of receiving attachments will get you scans from a text I think will be accessible (with some work) if you wish. It is too much to reproduce here.

 

Alternatively fluid maths is Bignose speciality, he may help, again try a PM.