[Physics] Why the energy of electromagnetic waves is directly proportional to frequency whilst for mechanical waves this is not true

Question

The energy of a photon is directly proportional to its (angular) frequency:
$$
E=\hbar \omega.
$$
The energy of a classical mechanical wave is, however, proportional to the square of $\omega$:
$$
E=\frac{1}{2}\mu A^2\omega^2 \lambda
$$
per wavelength.

I struggle to see why there is $\omega$ in one equation but $\omega^2$ in another.

It is quite mysterious, though, because the $E$ in two equations mean different things. How many photons per wavelength do we have?

Answer 1

The difference isn't electromagnetic versus mechanical. All classical waves behave the same way, as do all quanta. The difference arises because you're treating the light as quantum and the mechanical wave as classical.

• The energy of a single quantum is always proportional to $\omega$. This is true for photons, but it's also true for the quantized excitations that make up, say, classical waves on a string.
• For any wave described by a generalized coordinate $\phi(t)$ obeying the ideal wave equation and with standard normalization, the energy density of a classical plane wave at fixed amplitude is always proportional to $\omega^2$. This is true for both classical waves on a string (where the coordinate is the height $y(x, t)$) and for electromagnetic waves (where the coordinate is the vector potential $\mathbf{A}(\mathbf{x}, t)$).

Combining these two results shows that the density of quanta making up a classical ideal plane wave of fixed amplitude and frequency $\omega$ is proportional to $\omega$.