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?