The JICF is studied using direct numerical simulation of the linearized Navier-Stokes equations, as well as their adjoint, at a Reynolds number of 2000, and two jet-to-cross-flow velocity ratios: R = 2 with an absolutely unstable upstream shear-layer, and R = 4 with a convectively unstable upstream shear-layer. Linear stability analysis of the JICF reveals that the dominant eigenmodes are shear-layer modes whose frequencies match frequencies of the upstream shear-layer observed in simulation and experiment. Asymmetric modes are shown to be more important to the overall dynamics at higher jet-to-cross-flow ratios. Adjoint modes show that the upstream shear-layer is most sensitive to perturbations along the upstream side of the jet nozzle exit. Wavemaker results are shown to be consistent with the transition of the upstream shear-layer from absolute to convective instability.