Abstract: We study a two-dimensional problem that models sound propagation in a narrow water wedge near a seashore. For the Helmholtz equation, an adiabatic normal mode propagating shoreward along the water wedge is discussed. We describe the phenomena arising when the mode reaches the critical depth and afterwards. Prior to this, the acoustic field is localized in the water wedge. When the critical depth is reached, the energy of the field radiates into the sea bottom. Thereafter, a surface wave propagates inside the bottom along the water-bottom interface, occasionally leaking back into the water wedge.