We present theoretical calculations and experimental measurements which reveal finite-size effects in the tunneling between two parallel quantum wires, fabricated at the cleaved edge of a GaAs/AlGaAs bilayer heterostructure. Observed oscillations in the differential conductance, as a function of bias voltage and applied magnetic field, provide direct information on the shape of the confining potential. Superimposed modulations indicate the existence of two distinct excitation velocities, as expected from spin-charge separation.