= Solution
Consider the nonresonant traveling forced solution, with $k>0$ and $c\ne0$. Put $x=\bar x+\xi$, $y=\bar y+\upsilon$, where $(\bar x,\bar y)$ is the mean position of a <fluid element>. At <linear order>, evaluate the <velocity field> at this mean position: substituting $\xi,\upsilon$ into its spatial arguments would change the already first-order <velocity> only at second order. With $\Theta=k(\bar x-ct)$, differentiation of the <velocity potential> gives
$$
\begin{aligned}
\dot\xi&=b\left[\frac gc\cosh(k\bar y)+kc\sinh(k\bar y)\right]\sin\Theta,\\
\dot\upsilon&=-b\left[\frac gc\sinh(k\bar y)+kc\cosh(k\bar y)\right]\cos\Theta.
\end{aligned}
$$
Since $\dot\Theta=-kc$, integration, with zero mean displacements, yields
$$
\begin{aligned}
\xi&=A(\bar y)\cos\Theta,&A(\bar y)&=b\left[\frac g{kc^2}\cosh(k\bar y)+\sinh(k\bar y)\right],\\
\upsilon&=B(\bar y)\sin\Theta,&B(\bar y)&=b\left[\frac g{kc^2}\sinh(k\bar y)+\cosh(k\bar y)\right].
\end{aligned}
$$
If both coefficients are nonzero, eliminating the <wave phase> proves that the <pathline> is an <ellipse>:
$$
\boxed{\frac{(x-\bar x)^2}{A(\bar y)^2}+\frac{(y-\bar y)^2}{B(\bar y)^2}=1.}
$$
Its <principal axes of an ellipse> are horizontal and vertical, with respective <semi-axes>
$$
\boxed{|A(\bar y)|,\qquad|B(\bar y)|.}
$$
Thus the full axis lengths are $2|A|$ and $2|B|$. The signed coefficients also determine the direction of traversal. The useful checks $B(0)=b$ and $B(-d)=a$ reproduce the vertical displacements of the <free surface> and bottom.
To identify the major axis without comparing the two expressions separately, use $\cosh^2s-\sinh^2s=1$:
$$
A^2-B^2=b^2\left[\left(\frac g{kc^2}\right)^2-1\right].
$$
Consequently the major axis is horizontal if $c^2<g/k$ and vertical if $c^2>g/k$; equality gives <circles>. If $A$ or $B$ vanishes at a particular depth, the first-order orbit degenerates to a line segment. These are <linearized particle orbits of a bottom-forced gravity wave>, not exact closed trajectories at all orders: a higher-order calculation may introduce a mean drift.
\Image[/past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2017-forced-wave-orbits.png]
{title=First-order particle orbits above a traveling corrugated bed}
{description=Examples with $k=d=g=1$ and $a=0.025$ compare $c^2=0.3$ below resonance and $c^2=2$ above resonance. The surface reverses phase in the first case. Orbit sizes are shown without magnification, with equal horizontal and vertical scales. Arrows indicate traversal as time increases; dashed lines indicate the mean surface and bottom.}
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