We investigate a model which shows how the introduction of a perturbing dielectric close to an electromagnetic surface wave leads to radiation away from the surface through the dielectric. This resembles a surface waveguide passing through a wall or being deployed underground. Our theory, which is based on the mode-matching technique, allows quantitative determination of losses from a bound surface wave mode up to the point of its complete extinction. For a surface wave supported by a coated, conducting sheet the attenuation due to the perturbing dielectric is calculated for a number of frequencies, permittivities of the perturbation and separations between the sheet and the perturbing dielectric. The accuracy of our results is verified by simulation of the system with a full-wave numerical solution. Finally, we report experimental data of perturbed surface waves on a cable, which are in qualitative agreement with our model.