| Perturbation methods are used to calculate nonlinear interaction of waves, however most analyses skip the question as to whether the zeroth order solutions exist. The dispersion relation for internal gravity waves does not relate the magnitude of the wave vector and its frequency, rather it relates the frequency and direction of the wave vector. Thus, spatially columnated beams of internal waves are made of a continuum of plane waves with different wavelengths, but the same magnitude of frequency. For two parent beams to create a daughter, the plane waves within the parent and daughter beams must obey the triad condition (the spatial wave vector of the daughter equals the sum of the parents' vectors, and temporal frequency of the daughter equals the sum of the parents' frequencies) and the dispersion relation. Contrary to what is assumed implicitly, these conditions cannot always be satisfied. If they could, then the interaction of two beams of gravity waves would produce 8 daughter beams, consisting of two St. Andrew's crosses (each with 4 beams). The beams in one cross have a frequency equal to the sum of the frequencies of the parents and the beams in the other have a frequency equal to the difference. At least two daughter beams cannot exist for each cross according to the selection rules derived in this work.;Similar selection rules are obtained for the interaction among inertia-gravity conical waves. When two parent conical waves intersect, unlike the two-dimensional beams, the interaction area is not a single point. The intersection produces spatial continuous curves and there are no more than two intersection points in any horizontal plane with a normal vector parallel to gravity direction. Each intersection point is acted like the collision point of two-dimensional beam-beam interaction. As a result, at most two harmonic beams with a frequency either equal to the sum of the frequencies of the parents or to the difference, not a conical wave, are produced from the nonlinear interaction. |
