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Math Help - Waves on water

  1. #1
    Newbie Anthony Appleyard's Avatar
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    Nov 2009
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    Waves on water

    Please correct me if I am wrong anywhere.

    I get the impression from what I have read that, for a simple moving wave train on water,

    X = horizontal rest coordinate of a bit of the water surface,
    x = horizontal displacement of a bit of the water surface,
    y= vertical displacement of a bit of the water surface,
    t = time

    a = amplitude/2
    b = constant
    c = 2*pi/wavelength
    f = constant
    speed of waves varies as sqrt(wavelength)

    x = a*sin(b+c*X-v*t), y = a*cos(b+c*X-v*t), v=f*sqrt(c) :: eqns (1)

    This leads to a hypocycloid with the sharper curves on top.

    If a is too big compared to 1/c, the curve becomes an epicycloid with loops on top, but water cannot go through itself, so the waves develop foam crests (sometimes called "white horses").

    I have been unsuccessfully trying to make equations (1) into a differential equation. Please is there known a differential equation, or set of differential equations, for waves moving on the (2-dimensional) surface of 3-dimensional water, including explaining the usual mixed disorderly wave patterns seen on water in nature?
    Last edited by Anthony Appleyard; November 26th 2009 at 07:18 AM.
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