Wave propagation is a one of the most studied phenomenons in history due to the variety of applications such as quantum mechanics, electrodynamics and acoustics. In this thesis, the possibilities of improving numerical methods for solving the wave equation will be studied. More specifically, the dispersion relation will be used as a focal point. Generally there is a difference between the dispersion relation in the numerical solution and the analytic solution and the aim will be to decrease this difference and study the consequences. The numerical method that will be used and improved is the finite difference method (FDM). A dispersion relation for the numerical scheme will be derived including parameters from the spatial discretisation. These parameters will be optimised with the gradient descent method while retaining the second order accuracy of the derivative approximation. Performance is tested with numerical examples and the method of optimising for improved dispersion relation is proved to be successful. The optimised second order accurate schemes outperforms the standard second order accurate method in all simulated examples. When comparing the optimised stencil with the equally computationally expensive fourth order accurate method the optimised stencil performs better for sparse grids, especially when the spatial variation in the solution is high. For finer grids the fourth order accurate method quickly achieves smaller errors and is therefore preferable.