EM wave decomposition to axis components in the Rayleigh-Jeans cube

[aliinuur]

TL;DR

why axis components of the wave must fit as integer, but wave itseld does not?

$$\lambda_x = \frac{L}{n_x} , \lambda_y=\frac{L}{n_y} . \lambda$$
does not fit the cube as integer. In the figure
$$n_x=4, n_y=3$$

 

[SSJWarrior]

I don't know what you mean exactly, but wave equation must be satisfied for the EM wave (Derived from Maxwell's equations). Boundary conditions are that the fields die at the edges, which leaves you exactly with the two formulas that you provided. (You can do for 3d case directly)