 Causal pressure adjustment by internal waves

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.