SNR contains $\log N$ term where $N$ stands for number of samples
In the additive model $y=s+n$, when the signal is deterministic, it adds coherently over the "realizations". Hence, its variance $V(\sum s_n) = V(N s) = N^2  V( s)$. And when the noise $w$ is independent identically distributed (IID), then $V(\sum w_n) = NV( n) $. This is a classical result on the  Variance of Uncorrelated Variables.

You can find this classical result detailed in Deriving the SNR for averaged signals. Warning, this does reach a limit when data is quantized, in other words the noise floor does not follow $1\sqrt{N}$, but plateaues.