Accurate simulation of non-linear free-surface wave interaction with structures as if in the open sea requires truncation of domains for computational feasibility. In absence of special boundary measures, truncation introduces artificial boundaries with spurious wave reflection back into the computational domain. The common way to prevent reflection is by means of dissipation zones, which are often larger than the domain of interest in order to achieve sufficient accuracy. Local absorbing boundary conditions pose a potentially more efficient alternative, but are limited in accuracy due to corners on Cartesian domains (advantageous for future implementations of Volume-of-Fluid methods). This work proposes a methodology that preserves a Cartesian structure of the interior grid, while mapping to a circular domain boundary through the use of curvilinear coordinates. On the circular boundary, higher-order absorbing conditions can be applied without corner limitations. The linearized shallow water equations are discretised using a staggered finite volume discretisation. Two second-order Engquist-Majda boundary conditions, extended with generating wave capabilities, are expressed in the coordinate system, implemented and benchmarked using a first-order Sommerfeld boundary condition, using a centred and off-centre source configuration. Results demonstrate that the methodology allows consistent implementation of higher-order absorbing boundary conditions with low spurious reflection.