We consider a class of inviscid free surface flows where the free surface is of finite length and in which the pressure on the free boundary
p
b
is different from the free stream pressure
p
∞
. The aim of the paper is to determine the shape of the free surface as a function of the velocity ratio parameter
λ
. The free boundary problem is tackled by seeking a mapping
z
═
f
(ζ) such that the flow past a circle in the ζ-plane maps to a flow with constant pressure
p
b
on the free surface in the
z
-plane. The formulation leads to an infinite system of coupled nonlinear equations for the coefficients in the mapping function. Remarkably, the system can be solved exactly to yield two families of free surface flows of the form
z
═ ζ +
λ
2
/ζ +
a
(
λ
) ln (ζ +
b
(
λ
)/ζ ─
b
(
λ
)). The nature of the solutions, their limitations and possible extensions to them are discussed.