It is well known that standard finite element method (FEM) is unreliable to simulate the acoustic propagation problems addressed by the Helmholtz equation for high wavenumbers due to the ‘numerical dispersion error’. This dispersion error is essentially caused by the ‘overly-stiff’ FEM model. In order to depress the dispersion error, an edge-based smoothed finite method (ES-FEM) is proposed to solve the acoustic problems using the four-node quadrilateral elements. In the ES-FEM model, the gradient field of the problem is smoothed using gradient smoothing operations over the smoothing domain. Owing to this edge-based gradient smoothing operation, the ES-FEM model behaves much softer the standard FEM model and hence can reduce the numerical dispersion error significantly in solving the acoustic problems. Numerical examples have been studied and the results verify the excellent properties of the present ES-FEM.