Why are 'low frequency' EM waves attenuated by a single layer of kitchen foil? Can someone explain why my am radio doesn't work when covered by a layer of foil that is less than one 'skin depth' at the appropriate frequency? 
According to wikipedia and other websites on the subject of 'skin depth' kitchen foil which is 0.01mm thick is less than half of the skin depth of a 200khz em wave, so according to the graph and article should not completely attenuate said wave.
why then, does it attenuate it in my radio(s)?
 A: The skin depth of a good conductor is given by the expression
$$ d = \sqrt{\frac{2}{\mu_r \mu_0 \sigma \omega}},$$
where $\omega$ is the angular frequency of the EM wave and $\sigma$ is the conductivity. For an angular frequency of $6\times 10^5$ rad/s and aluminium foil with $\mu_r = 1$ and $\sigma \sim 4\times 10^6$ S/m, then $d \sim 0.8$ mm and much bigger than the thickness of aluminium foil.
However, don't forget that there is an important reflection effect from the surfaces of conducting materials too. For a good conductor, the modulus of the transmission coefficient (from air/vacuum into the conductor) is approximately 
$$ |T| = 5.3\times 10^{-3} \sqrt{\frac{\mu_r \mu_0 \omega}{\sigma}}$$
Thus a more conductive material will transmit less field at the interface between the conductor and air/vacuum. This effect is more important for attenuating the E-field at low frequencies or cases where the conductor is not thick compared with the skin depth.
For the situation of Al foil and am radio, I get $|T| \sim 2\times 10^{-6}$. This transmission coefficient is for the field strength, so almost all the power is reflected.
A: The concept of "skin depth" only applies for conductors whose thickness is significantly greater than the skin depth at the AC frequency of operation. For the case where the conductor is thinner than what the skin depth would be for a thick conductor, this does not mean the current is somehow protruding out of the conductor. It instead means the current is bunched within the available thickness and does not follow the exponential density curve for the skin effect. 
This means that stopping the transmission of an EM wave through a conductive shield does not require the shield to be thicker than the skin depth. This is why a single layer of aluminum foil can stop radio waves. 
