Many accounts of the history of quantum physics explain how Planck resorted to quantizing energy in an "act of desperation" while attempting to solve blackbody radiation, only to discover by surprise that a nonzero value of $h$ in $E=nh\nu$ reproduced experimental results.
- What was Planck's motivation behind the $\nu$ dependence in this expression?
- Did classical physics provide any hints for this frequency dependence?
Einstein used this same relation to help explain the photoelectric effect, but that came later.
Finally, to emphasize why I have this question, consider these seemingly contradictory facts:
- Planck was treating the quantized EM waves as harmonic oscillators. However, the relation between energy and frequency for a classical harmonic oscillator has a square dependence: $E=\frac{1}{2}m\omega^2A^2=2\pi^2m\nu^2A^2$, where $A$ is the amplitude.
- In classical electromagnetic theory, the average energy density of a plane wave in vacuum has no frequency dependence: $u=\frac{1}{2}\epsilon_oE_o^2$, where $E_o$ is the amplitude of the electric field part of the wave.
- It's easy to imagine postulating $E=nh$ as a first attempt to quantize energy. The $n$ part of this expression is the quantization piece, which was a new idea that I can understand as a hopeful guess or mathematical trick—but the $\nu$ part seems a priori unmotivated, and this isn't addressed in any of the sources I looked through.