This paper deals with the perturbed solution of the problem of diffraction of surface waves by a thin, two-dimensional barrier. A train of simple harmonic progression waves is normally incident on the barrier of arbitrary shape

$x=\u03f5c\left(y\right)$, where

$\u03f5$ is small. The perturbed solution is obtained by using an integral formulation and Green’s function method. The first order corrections to the reflection and transmission coefficients are obtained in terms of two integrals of shape. Both high (ka

$\to \infty )$ and low (ka

$\to 0)$ frequency limiting solutions are found and compared with known analytical results. An equation for the second order correction to the velocity potential is obtained. It is shown that the method breaks down at the second order for general c(y). If, however, c(y) is chosen so that

$c\left(y\right)\sim {(a+y)}^{1/2}$ as

$y\to -a,$ then there is no difficulty. This means physically that the shape of the barrier must be more nearly vertical at the tip than for the first order solution. The perturbation analysis is also used for symmetric slender barriers which are non-oscillatory or oscillatory. This is an interesting paper in the theory of diffraction of water waves by barriers.