Analytic solutions are obtained for forced, barotropic circulation at subinertial frequencies over a bilinear continental margin (shelf and slope) in situations where bottom friction is important. Three different alongshore forces are considered: wind-stress, offshore oceanic pressure gradients and offshore currents. Forcing functions are assumed to vary sinusoidally in time and in space alongshore. Steady models are found to perform adequately provided that the forcing functions do not move in the same direction as the free modes (continental shelf waves) propagate. Near resonance, when the alongshore velocity of the forcing approximates that of a free mode, the response is dominated by the mode. In the case of wind forcing, signals are trapped nearshore. If the shelf break occurs within this trapping length (as occurs near resonance) the shelf width becomes the elective trapping length. In this instance there can be significant horizontal shear in the alongshore velocity on the shelf near the shelf break.
When the velocity of an oceanic, alongshore pressure gradient signal approximates that of a free mode, the signal can be amplified towards the coast. For example, near a mode 2 resonance the signal is a maximum near the coast with a secondary maximum on the continental slope, near the shelf break. This amplification is in stark contrast to the solution forced by a signal which is either stationary or moving in a direction opposite to that in which the free modes propagate, which simply fall away from their maximum values offshore, resulting in weak coastal circulations.
Bottom friction affects the free continental shelf waves in three ways: their phase speeds are reduced, they decay with time and their altered structures exhibit phase differences across the continental margin whereby the flow nearshore leads that offshore in time. As a result, increased bottom friction reduces the response at resonance, broadens the range of frequencies over which responses are increased and detunes, or shifts, the frequency at which resonance occurs to a lower value. At practical parameter values, the reduction is minimal for the first mode but can he substantial for the second.