= 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 interval
$$
C_0t>L_X>C_1t,
$$
the fast barotropic signal has arrived but the first and higher baroclinic signals have not. Consequently
$$
\psi(X_1,y,z,t)\simeq\phi_0(X_1,y,t)P_0(z),
$$
which 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 $c_0t>L_X>c_1t$ 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.
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