Two related wave problems are considered for a rotating sea of nearly uniform depth bounded by a coastline which is nearly straight. The depth changes are assumed to be independent of the distance from the coastline. The first problem, which is concerned with the origin of Kelvin waves in a coastal wave record, deals with a system of plane waves incident on the coastline and giving rise, in addition to reflected waves, to a Kelvin wave moving along the coast. Linearized theory is used to obtain details of the Kelvin wave for arbitrary perturbations in coastline and depth. Results suggest that the depth changes have their greatest effect in producing Kelvin waves if the incident wave crests are nearly parallel, but not exactly so, to the line of the depth changes. On the other hand when the wave crests are parallel to the coast, Kelvin waves are produced only by changes in the coastal boundary. In the second problem a Kelvin waye is assumed to be the incident wave. To find the energy propagated away from the coastline it is necessary to extend the theory to second order in the perturbations. It is shown that for a fixed wave period less than a pendulum day this energy has a maximum for a perturbation whose length is of comparable magnitude to the incident wavelength. Finally, the theory is applied to Kelvin waves propagating along the Californian coastline. Results obtained tend to confirm the suspicion that coastal irregularities are responsible for certain anomalies detected in tidal wave constituents by Munk, Snodgrass & Wimbush (1970).