The dynamics of small perturbations on a buoyant coastal current is investigated. The system is described using a one and a half-layer model where the active upper layer vanishes at a certain distance from the coast, forming a front. Perturbations are imposed on a steady basic state with no along-coast variation. Analytical solutions are discussed for two special configurations of the basic state: (i) constant along-shore velocity, i.e. a coastal current with triangular cross-section, and (ii) a constant potential vorticity current. Two wave modes are found in both cases: a slowly moving frontally trapped wave, and a coastally trapped wave that moves with approximately the internal Kelvin wave speed plus the speed of the current at the coast. However, these two wave modes are not sufficient to construct a generally shaped initial perturbation. The part of the initial perturbation not covered by the two wave modes will in case (i) split into an infinite number of higher wave modes all travelling faster than the frontal wave and in case (ii) be advected and slowly smeared out by the current. Under the assumption that the current is unidirectional we find that the perturbations always move in the direction of a Kelvin wave, i.e. in the same direction as the coastal current, for all physically relevant cases.