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