Problem 2: Inertial oscillations on the $f$-plane

We consider two-dimensional flow in a rotating frame on the $f$-plane and linearize about a state of rest. Ignoring the pressure term (as instructed) we obtain the linearized horizontal momentum equations

$$\frac{\partial u}{\partial t} - f v = 0,$$ (11)

$$\frac{\partial v}{\partial t} + f u = 0,$$ (12)

where $f$ is constant (the Coriolis parameter) and $u,v$ are the horizontal velocity components in $x,y$.