Surface gravity wave forced by a traveling bottom ($\frac{b}{a}=\frac{(kc{)}^2\mathrm{sech}(kd)}{(kc{)}^2-gk\tanh (kd)}$)

Consider fluid at rest with mean depth $d$ and a bottom displacement $a\sin (k(x-ct))$. The linearized free-surface boundary conditions at $y=0$ and moving-bottom kinematic boundary condition at $y=-d$ give a periodic particular response $\eta =b\sin (k(x-ct))$ and velocity potentialThe bottom derivative fixes the signed response coefficient to the title formula. Here $k>0$, $c\ne 0$, and the forcing must avoid $k^2{c}^2=gk\tanh (kd)$. This equality is resonance with the natural finite-depth surface gravity wave; nonzero undamped resonant forcing has no bounded periodic solution. The response reverses phase below resonance. General initial data can add free surface gravity waves, which do not decay in this inviscid model. Near resonance the resulting wave steepness and particle excursions, rather than the bottom displacement alone, determine whether the linearization remains valid.

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 areThus 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 $A^2-B^2=b^2[(g/(k{c}^2){)}^2-1]$ 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.