Why are electromagnetic waves attenuated by thick barriers? Atoms have a specific absorption spectrum that determines what frequencies of EM radiation they are most likely to absorb. This is why visible light cannot pass through a wall, but radio waves can. However, if you make the wall thick enough, the radio waves will eventually be attenuated. Since the radio frequencies are not in the absorption spectrum of the atoms in the wall, why does increasing the thickness of the wall attenuate the radio signal?
 A: Each atom has a well-defined absorption spectrum $\alpha(\lambda)$. The most prominent features of these spectra are the absorption resonances at $\{\lambda_1,\lambda_2, \ldots\}$. However, this does not imply that absorption becomes zero between two resonances. Instead, the absorption for a wavelength between two resonances is merely "much smaller" than at the resonances. Hence, by increasing the width of the wall the exponential intensity law (Lambert's law)
$$
I(z) = I_0 e^{-\alpha(\lambda)\cdot z}
$$
kicks in and the absorption becomes non negligible.
Mathematically there exists a very simply argument which shows that the absorption is not allowed to be zero within any finite wavelength interval: If the absorption would be zero within the interval
$[\lambda_0, \lambda_0+\epsilon]$ for $\epsilon>0$ it must be zero everywhere.
Physically, the atomic transition can be modelled as an externally driven harmonic oscillator with eigenfrequency $\omega_0$, and damping constant $\gamma$. This leads to the complex part of the refractive index $n$ to be
$$
\textrm{Im}\{n\}  =:\kappa 
= \frac{\lambda \alpha}{4\pi} 
\propto %= \frac{N q^2}{2\epsilon_0 m}
\frac{\gamma/\omega_0}{(\omega^2 - \omega_0^2)^2 + (\gamma/2 )^2}
$$
From this we see that the absorption never becomes truly zero.
A: To answer Matthew's second question: If the substance of a wall is slightly electrically conductive, the impingement of radio waves upon it will induce currents to flow in it, which are dissipated by any electrical resistance that the wall material possesses. This mechanism does not invoke discrete energy level transitions occurring within the electronic orbital structure of the atoms comprising the wall.
