= Causal pressure adjustment by internal waves
{title2=$p_\infty=[f(x)-\varepsilon/2]e^{imz}$}
A weak horizontal force $f'(x)e^{imz}$, with $m>0$ and $\int f'(x)\,dx=\varepsilon$, admits many steady pressures differing by an arbitrary function of height. Switching the force on in initially motionless, unbounded, uniformly stratified fluid selects $p_\infty=[f(x)-\varepsilon/2]e^{imz}$, 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 $-\varepsilon/2$ and $+\varepsilon/2$. The limit is at fixed $x$ after the fronts have passed; dispersive tails need not vanish at any finite time.
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