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

Energy asymptotics of a Dirichlet to Neumann problem related to water waves