 Solution

The method of characteristics shows that a point is affected only after a westward characteristic from the forcing region reaches it. At a western observation point $X_1$, during the intervalthe fast barotropic signal has arrived but the first and higher baroclinic signals have not. Consequentlywhich is nearly independent of depth. At the eastern point $X_2$, no westward Rossby-wave characteristic arrives from the forcing region, so the disturbance remains zero under the stated initial and radiation conditions.

The propagation speed derived in part v is $C_n=\beta c_n^2/f_0^2$. Thus the question's notation $c_{0}t>L_X>c_{1}t$ must be read as a comparison with these modal Rossby signal speeds; using the gravity-wave speeds $c_n$ literally would not describe the long-wave equation derived above.