= Linearized particle orbits of a bottom-forced gravity wave
{title2=$\xi=A\cos\Theta,\quad\upsilon=B\sin\Theta$}
For a <bottom-forced gravity wave> with signed surface coefficient $b$, evaluate the first-order <velocity field> at a parcel's mean position $(\bar x,\bar y)$ and integrate, with $\Theta=k(\bar x-ct)$. The horizontal and vertical displacement coefficients are
$$
A=b\left[\frac g{kc^2}\cosh(k\bar y)+\sinh(k\bar y)\right],\qquad
B=b\left[\frac g{kc^2}\sinh(k\bar y)+\cosh(k\bar y)\right].
$$
Thus the <pathline> is an <ellipse> with horizontal and vertical semiaxes $|A|,|B|$ whenever both are nonzero, and a line segment if one vanishes. The identity $A^2-B^2=b^2[(g/(kc^2))^2-1]$ determines which axis is longer. Also $B(0)=b$ and $B(-d)=a$, verifying the surface and bottom displacements. Closed first-order orbits do not rule out a higher-order mean drift.
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