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I realize this is a very hard question. In the very least I'd like to know if there is a way to do this or not.

Say you have a summation of plane waves in a 3d volume, with longitudinal and transverse components.

water

Actually I'm trying to model water. What I'm trying to do is figure out the "cubic period" - the size of the 3d volume that repeats.

This document says the period of the sum of sinuosoids is the LCM of the denominator of their periods, e.g.

The period of

$$ y(t) = \sin\left( \frac{2\pi t}{6} \right) + \cos\left( \frac{2\pi t}{6} \right) + \sin\left( \frac{2\pi 7 t}{2} \right) + \sin\left( \frac{2\pi t}{4} \right) $$

is

$$ LCM( 6,6,2,4 ) = 12 $$

Indeed a quick sketch shows it appears to work,

I believe it

Now my question is, how would I go about extending this to three dimensions?

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Your waves look like something like $\exp k \cdot z$? Should be straightforward. –  Alice Jul 24 '12 at 20:16
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up vote 2 down vote accepted

For water, you should probably be thinking in two dimensions, because the surface of the water is two dimensional. The waves are then amplitudes in the $z$ direction. If you have two orthogonal waves, it will be like $z=\cos \left(\frac{2\pi x}5\right)+ \cos \left(\frac{2\pi y}3\right)$. The repeat is a rectangle, $5$ units in $x$ by $3$ units in $y$. If you have two non-orthogonal waves, the repeat is a parallelogram with sides in the two directions. If you have waves in more than two directions, there may not be a repeat.

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Actually 2 dimensional waves don't give you the "shifting" (longitudinal motion) along the surface of the sea. What I'm worried about is yes, there might not be a repeat when the waves are allowed arbitrary 3-space direction. –  bobobobo Jul 25 '12 at 1:56
    
@bobobobo: You can get the shifting by making the amplitude be a vector that can point in various directions. The waves still propagate only in two dimensions. There will be a repeat as long as there are no more independent waves than dimensions. Even in one dimension, there may not be a repeat if the period of two waves are not rationally related. I don't know what condition applies if you have more waves than dimensions to make a repeat. –  Ross Millikan Jul 25 '12 at 2:01
    
Interesting approach. So perhaps I allow waves to have amplitude on both longitudinal and traverse axes. –  bobobobo Jul 25 '12 at 16:44
    
In a usual wind wave, the motion of the particles of water is known to be (approximately) a circle centered on the rest location. So for a wave in the $x$ direction, you have the position $p=\hat z \cos (\omega t)+\hat x \sin (\omega t)$ –  Ross Millikan Jul 25 '12 at 18:17
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