Proof of rayleigh-jeans formula

I'm trying to understand proof of rayleigh-jeans formula from the book "Quantum Physics" by Resnick and Eisberg page 10 and i'm confused. I graduated in math but i'm a big fan of physics. I tried really hard and there are some questions. for example i don't understand three relations $E(x,t)$, $E(y,t)$ and $E(Z,t)$ (under page 10) are exactly what physical quantities and defined on what domain. i will be glad if i can a discussion about this proof maybe out of this site.

The equations in question (which you can see on page 28 of this link) describe the amplitude of electromagnetic radiation in a cavity. $E$ is the field, $x$, $y$ and $z$ are the orthogonal directions of the edges of this cubic cavity, and $t$ is the time dimension (these are standing waves in X, Y and Z).

Because the cavity is metallic, you know the boundary condition for the electric field (must be zero at the boundary). Thus the $\sin$ relationship requires an integer number of (half) wavelengths to fit in the box.

Since the box is cubic, $x, y, z$ etc are defined on $[0,~a]$, and $t>=0$

The standing wave in the cavity is set up in one dimension $x$ and is of the form

$E(x,t)=E_0 \sin\left(\dfrac {2\pi x}{\lambda} \right)\sin 2\pi \nu t$

This is equation 1-6 on page 8 of the textbook and the justification for this assuming metallic walls is explained starting on page 6.

Representing this wave as a "3-D" drawing is very difficult but the illustrator for the textbook has done a good job to show that the nodes which are points in one dimension become planes in three dimensions.

Going back to a simpler representation one could draw a standing wave as follows where there is no component of the electric field in the y-direction.

Using the ideas of superposition of waves this can be redrawn as follows:

where the standing wave is the sum of two component standing waves with $E^2=E^2_{\rm x} + E^2_{\rm z}$ and $\lambda^2 = \lambda^2_{\rm x} + \lambda^2_{\rm z}$.

The complexity of the situation and hence the difficulty in visualising the waves can be gauged by going back to what was called a one dimensional standing wave.

$E(x,t)=E_0 \sin\left(\dfrac {2\pi x}{\lambda} \right)\sin 2\pi \nu t$

This equation could equally well represent a two dimensional wave in the xy or xz plane where the nodal points in one dimension now become nodal lines perpendicular to the x-axis

and a three dimensional wave where the nodal points in one dimension become nodal planes orthogonal to the x-axis which is the textbook diagram but with the nodal planes being parallel to yz plane.

In the textbook derivation standing waves along directions which are orthogonal to the cube walls are forced to have a zero electric field at the boundaries which leads to a set of standing wave equations.

$E(x,t)=E_{0\rm x} \sin\left(\dfrac {2\pi x}{\lambda_{\rm x}} \right)\sin 2\pi \nu t$
$E(y,t)=E_{0\rm y} \sin\left(\dfrac {2\pi y}{\lambda_{\rm y}} \right)\sin 2\pi \nu t$
$E(z,t)=E_{0\rm z} \sin\left(\dfrac {2\pi z}{\lambda_{\rm z}} \right)\sin 2\pi \nu t$

These components are added together (the reverse of what I did above) to form a 3-D standing wave subject to $E^2=E^2_{\rm x} + E^2_{\rm y}+E^2_{\rm z}$ and $\lambda^2 = \lambda^2_{\rm x} + \lambda^2_{\rm y} + \lambda^2_{\rm z}$.