Causal pressure adjustment by internal waves ($p_{\infty }=[f(x)-\epsilon /2]e^{imz}$)

A weak horizontal force $f^{\prime }(x)e^{imz}$, with $m>0$ and $\int f^{\prime }(x)dx=\epsilon$, admits many steady pressures differing by an arbitrary function of height. Switching the force on in initially motionless, unbounded, uniformly stratified fluid selects $p_{\infty }=[f(x)-\epsilon /2]e^{imz}$, up to a spatially constant pressure gauge. For horizontal wavenumber $q$, the transient frequency is $\Omega (q)=Nq/\sqrt{q^2+m^2}$. Its long-wave group velocity is $N/m$, and oppositely propagating outgoing fronts leave pressure limits $-\epsilon /2$ and $+\epsilon /2$. The limit is at fixed $x$ after the fronts have passed; dispersive tails need not vanish at any finite time.

For a weak horizontal forcing proportional to $e^{imz}$ in an ideal uniformly stratified nonrotating fluid, the low-frequency hydrostatic approximation gives the displayed one-dimensional wave equation for the streamfunction amplitude. Two oppositely propagating characteristic integrals determine the response. A source confined to $|x|\le a$ and switched on at time zero produces no response beyond $|x|\le ct+a$ in this reduced equation. The bound on mixed forcing derivative $|f_{xt}|\ll |mN{f}_{\max }|$ controls the neglected vertical acceleration in characteristic magnitude.