Clarification of why Rayleigh-Jeans law failed to explain blackbody radiation

Question

I'm reading about why the Rayleigh-Jeans formula failed to explain blackbody radiation curves. The Rayleigh-Jeans formula: $$ E = kT\left(\frac{8\pi\nu^2}{c^3}\right) $$

I see explanations saying that the part of the equation in parentheses related to the "number of standing waves that are able to fit into a blackbody cavity". But is it really the number of waves or is it the number of "modes" at a given frequency? Or are they the same thing?

Secondly, I know that $kT$ is the amount of energy given to each "mode". Again, does that mean that each wave at $X$ frequency is given $kT$ energy? Or each "mode" is given $kT$ energy?

Any clarification would be great. I know there are other similar questions, but I feel like my question is a little different.

Answer 1

There is a bit of confusion here. Nominally, "the number of waves" could mean "the number of modes", though it is more likely that "the number of waves" means the available number of standing waves with frequency between $\nu$ and $\nu+d\nu$...and then, each of those waves has two polarization states, or modes. Whether a mode refers to a unique state or a unique frequency plus two polarizations is not always clear.

However, if each "mode" has an average energy of $kT$, then it is indeed a frequency plus 2 polarizations, as each degree of freedom has average energy:

$$ \bar E_{\nu, \epsilon} = \frac 1 2 kT $$

so that each mode has:

$$ \bar E_{\nu} = \sum_{\epsilon}\bar E_{\nu, \epsilon} = kT $$

Answer 2

I'm not particularly familiar with the concept of breaking this law into different standing waves, but the basic idea of why the Rayleigh-Jeans law failed is the so-called "Blackbody catastrophe." The catastrophe comes in when you consider the total amount of energy that must be emitted from a blackbody obeying the Rayleigh-Jeans law as you have written it. Namely, if we begin integrating over all the wavelengths (or equivalently, frequencies), we will immediately see that this integral has to diverge to infinity! It certainly doesn't make a lot of sense that any given blackbody should be giving off an infinite amount of energy, which is why this was considered a spectacular failure of classical physics. Planck essentially got lucky by "quantizing" radiation into discrete chunks and showing how this led to a modification of the blackbody radiation law which both converged to a finite energy emitted and was entirely consistent with the Rayleigh-Jeans law at low energies. This was essentially the birth of quantum mechanics, even if it wasn't appreciated as anything more than a mathematical trick at the time. Hope this was helpful!