This thesis considers the use of the wave approach for damage detection in beams. It is proposed that the existence of damage can be predicted and its location and depth estimated by measurement of the wave scattering coefficients in the frequency domain. Importantly, the wavelengths considered here are much longer than the dimensions of the cross-section of the beam. Here, a damaged beam with a transverse notch or slot is modelled using finite element (FE) analysis. This model offers a more detailed description of the shape of the damage and its dynamic characteristics than conventional analytical models. The FE model is assembled to semi-infinite spectral elements, which impose infinite boundary conditions at each end of the FE model. A wave superposition method is then used to estimate the scattering coefficients. The scattering coefficients are estimated experimentally for a number of beams containing slots. These were made by cutting through the width of the cross-section. The measured reflection coefficients are compared with numerical results to estimate the slop depth and good agreement is found between the actual and estimated slot depths. Experimental errors and noise can make it difficult to estimate the scattering coefficients, particularly at low frequencies or when the reflection coefficient is small (e.g. a small slot). It is shown how the location of the slot can be estimated, either from the reflection coefficient or the phase of the point frequency response function. The results in this thesis show that the reflection coefficient offers a useful feature for detecting damage in beams. The main limitation lies in the fact that experimental error and noise make it difficult to detect small slots. From the results given here, the method works best when the slot depth, and hence the reflection coefficient, is large.