A semi-analytical solution method, the so-called Scaled Boundary Finite Element Method (SBFEM), is developed for the two-dimensional Helmholtz equation. The new method is applicable to two-dimensional computational domains of any shape including unbounded domains. The accuracy and efficiency of this method are illustrated by numerical examples of wave diffraction around vertical cylinders and harbour oscillation problems. The computational results are compared with those obtained using analytical methods, numerical methods and physical experiments. It is found that the present method is completely free from the irregular frequency difficulty that the conventional Green's Function Method (GFM) often encounters. It is also found that the present method does not suffer from computational stability problems at sharp corners, is able to resolve velocity singularities analytically at such corners by choosing the structure surfaces as side-faces, and produces more accurate solutions than conventional numerical methods with far less number of degrees of freedom. With these attractive attributes, the scaled boundary finite element method is an excellent alternative to conventional numerical methods for solving the two-dimensional Helmholtz equation. (c) 2007 Published by Elsevier Ltd.