The definition of the skin depth is:
"Skin depth defines the distance a wave must travel before its amplitude has decayed by a factor of $1/e$."
My question why is the decay of 37% significant here. The EM wave will still have some penetration abilities after it has lost 37% of its initial amplitude, won't it? That is, it will still be able to penetrate the conductor after the skin depth is reached.
Because the decay of an electromagnetic wave is exponential, i.e. it decays as $A_0e^{-z/\delta}$, where $A_0$ is the initial amplitude, $z$ is the distance in the conductor, and $\delta$ is the skin depth. It feels straightforward to then write the skin depth in terms of the natural exponential function.
Of course the skin depth could also be given as a distance for which the field has decayed, say, 50%, like is common for the half-life of radioactive atoms. This is simply convention.
The mathematics (exponential decay) would suggest that infinite distance is needed for the amplitude to decay to zero. This would not be helpful, so an arbitrary agreed value is used. The choice of 1/e times the original amplitude gives a simpler form to the decay equation than another value would.
Not that the amplitude has not decayed by 37% but to 37% of the original value, i.e. it has lost 63% of its amplitude.
