  We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a
model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the
expression of the operator in terms of the Fourier transform, highlighting a
local behavior for small frequencies and a nonlocal behavior for large
frequencies. We further investigate the $ \Gamma $-convergence of the energy
associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a
double-well potential. When $a\in(-1,0]$ the energy $\Gamma$-converges to the
classical perimeter, while for $a\in(0,1)$ the $\Gamma$-limit is a new nonlocal
operator, that in dimension $n=1$ interpolates the classical and the nonlocal
perimeter.
