physics - Fundamental property of an EM wave

Imagine there's an electromagnetic wave propagating in vacuum with a frequency \$f\$ and wavelength \$\lambda\$. Its speed is \$c\approx3\cdot10^8{{m}\over{s}}=f\cdot\lambda\$.

Now, the wave has to travel through a material with relative permittivity and permability \$\epsilon_r\$ and \$\mu_r\$. The speed of the wave will change in the new material, the new speed is$$v={{c}\over{\sqrt{\epsilon_r\cdot\mu_r}}}=f^{'}\cdot\lambda^{'}$$

There are three possibilities to fulfill the equation:

1. The frequency changes.


2. The wavelength changes.


3. Both the frequency and the wavelength change.

My guess is that one of these two properties is more "fundamental" than the other and remains unchanged. Is that assumption correct, and if it is, which is the fundamental property?

Answer

The frequency will not change. That's because you state upfront implicitly that the material has relative permittivity and permeability constants \$\epsilon_r, \mu_r\$, i.e. it is linear. There is no way that a linear medium will generate frequencies that are not in the excitation signal. Therefore it's the wavelength that will change.

BTW, in what you say there are a lot of implicit assumptions: not only the material is linear (\$\epsilon_r, \mu_r\$ don't vary with the signal intensity), the material is homogeneous (\$\epsilon_r, \mu_r\$ don't vary with the specific point in space), the material is not dispersive (\$\epsilon_r, \mu_r\$ don't depend on the frequency of the signal), the material is isotropic (\$\epsilon_r, \mu_r\$ are scalars and not tensors - i.e. matrices to put it simple), the material is time-invariant (\$\epsilon_r, \mu_r\$ don't vary with time).

Just to make an example of a non-linear material: the active medium used to generate LASER light. For example, in old LASERS ruby crystals were excited with flashes of light and as a byproduct you got the stimulated emission at a different frequency from the frequency components of the excitation light. Non-linear media can generate/change the frequency of the excitation signal.

Another, more common, example of a nonlinear material is the phosphor coating used to make white LEDs out of blue ones. When hit with the blue light generated by the underlying junction of the LED it will downconvert (convert to a lower frequency) a part of its energy and emit yellowish light, which mixes with the remaining blue light producing white light as a result.

EDIT (to integrate and expand a comment of mine which seems to have helped the OP)

To summarize: there is no "more fundamental" quantity in general. What happens to an EM wave in general can be inferred solving Maxwell's equations taking into account the constitutive relations of the materials involved.

In the simple case of linear, time-invariant, etc., material the frequency happens to be "more fundamental" in the sense that it cannot change, but this fact depends heavily on the properties of the material.

Another example in which you can see how the properties of the material influence the frequency content of the incident wave: a piece of colored glass, say a green window pane. This is a medium which is linear, but frequency-dispersive, i.e. its \$\epsilon_r, \mu_r\$, depend on the frequency. After the incident wave (sunlight) has passed through the window, components with frequency far from the green light frequency will have been greatly attenuated, therefore the frequency content of the emerging light will have changed.

Of course this is not directly related to your example, since you explicitly stated that the incident wave was monochromatic, but I reported it to show you how different the behavior of a material can be depending on the assumptions you make.

As yet another example: have you ever seen one of those American movies where a truck is seen running along the road coming toward the camera from a distance? You'll see the shape of its front distorted and blurred in a random, time-varying way. I guess you know that's the effect of the flow of hot air rising from the road surface. What you may not know is that this is an example of a time-varying material: the air changes its refraction index (i.e. its \$\epsilon_r, \mu_r\$ "constants") with time. Therefore the light passing through it is sort of "modulated" by the material, giving rise to that blurring.