This paper defines the linearized problem of free-surface flow past a shallow two-dimensional body in terms of complex variables and provides an asymptotic solution for the Froude number tending to zero. The simple case of flow past a rectangular box with a triangular extension in front is worked out as an example. The low-Froude-number solution for ‘double-body’ flow in a fully infinite fluid past the same box with its reflection in the upper half-plane is also calculated. The upstream free-surface height for the linear solution is compared to the equivalent free-surface height for the double-body flow. Experimental results for this body are also presented and compared.