= Ultraviolet catastrophe
The ultraviolet catastrophe is the divergent total thermal radiation energy predicted by classical <equipartition theorem> applied to the <electromagnetic mode density in a box>. Each mode has mean energy $k_BT$, while the number of modes per volume per frequency is $8\pi\nu^2/c^3$. The <Rayleigh-Jeans law> therefore gives $u_\nu=8\pi k_BT\nu^2/c^3$, whose integral over all positive frequencies diverges at every $T>0$.
<Planck's law> replaces the classical mode energy by $h\nu/(e^{h\nu/(k_BT)}-1)$. It agrees with equipartition when $h\nu\ll k_BT$ but suppresses high-frequency excitations exponentially. The resulting spectral energy density is integrable and yields the finite <Stefan–Boltzmann law>. The failure concerns the classical equilibrium distribution of energy among modes, not the existence of electromagnetic wave propagation itself.
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