# Ripple equation

I want to formulate an approx 2D ripple equation μ(x,y). It should satisfy following:

1) it should have highest amplitude on y axis at x=0. It should be symmetrical about y axis. It should fade away at some x on both positive and negetive x axis symmetrically. Fading in a way of decreasing amplitude. It should not give negetive value of y.

2) most important. The square of norm of μ should be normalized in infinity.

I am not good at math but this is integral part of my project. It can be visualized as : consider x axis as still surface of water and you drop a pebble on it. But it should be shifted upwards such that it should not fall in negetive y axis. Also it should be preferrably in cartesian co ordinates

Do you mean a "ripple" like function on a plane surface that models a water wave?
1.symmetric across the y axis (simply solve in polar coordinates, with the condition that the function obeys ψ(r,θ)=ψ(r,θ+pi)
2.Falls off at infinity, that's fine we'll just pick a solution that does this

And then simply solve the wave equation in polar coordinates with these conditions.
Ta-da.

However, could you specify how you want it?
There are a few things you have to consider.
1.Source function?
2.Initial displacement?

Also, you cannot limit the function to be positive, that would be non-physical. Real water ripples go both ways.

Edit: Now I'm a bit confused, do you want a plane solution or a one dimensional solution? I thought you meant the former, seeing as you referred to the function as a "ripple equation" and specified two arguments for it (mu(x,y)), but then I read your conditions again and you also imply you want a one dimensional one. (i.e. y(x))

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Do you mean a "ripple" like function on a plane surface that models a water wave?
1.symmetric across the y axis (simply solve in polar coordinates, with the condition that the function obeys ψ(r,θ)=ψ(r,θ+pi)
2.Falls off at infinity, that's fine we'll just pick a solution that does this

And then simply solve the wave equation in polar coordinates with these conditions.
Ta-da.

However, could you specify how you want it?
There are a few things you have to consider.
1.Source function?
2.Initial displacement?

Also, you cannot limit the function to be positive, that would be non-physical. Real water ripples go both ways.

Edit: Now I'm a bit confused, do you want a plane solution or a one dimensional solution? I thought you meant the former, seeing as you referred to the function as a "ripple equation" and specified two arguments for it (mu(x,y)), but then I read your conditions again and you also imply you want a one dimensional one. (i.e. y(x))

I did a silly mistake there, I want 1D solution of it. I want it in form y(x)

$y(x) = Ae^{-bx}\cos{x}$ fits the bill. What exactly are you trying to model? You're better off deriving a differential equation and trying to solve it than just picking out a function from random that meets certain requirements.