Past exam of the fluids and waves course of the University of Oxford / 2021 / 3 / b

Seek the travelling forced component of the surface elevation in the form $\eta =A\cos \chi$, where $\chi =x-ct$ and $A$ is a signed coefficient. The forcing has wavenumber one. Separation of variables in the Laplace equation, followed by the bed no-penetration boundary condition, givesSince ${\eta }_t=Ac\sin \chi$, the surface kinematic boundary condition requires $B\sinh h=Ac$, soAt $y=0$, ${\varphi }_t=-A{c}^2\coth h\cos \chi$. The dynamic member of the linearized free-surface boundary conditions therefore becomesConsequently the signed response and the nonnegative amplitude areThe travelling-pressure response of finite-depth surface gravity waves is opposite in phase to the imposed pressure below the resonant speed and in phase above it. These expressions describe the travelling forced component; freely propagating surface gravity waves can be superposed to meet additional initial conditions.

The denominator of the travelling response vanishes atFor the positive speeds in the paper, a finite travelling response exists whenever $c\ne c^{\ast }$. At $c=c^{\ast }$, the dynamic condition would reduce to $p_1/\rho =0$, which is impossible. The finite-depth dispersion relation of a surface gravity wave with wavenumber $k$ is ${\omega }^2=gk\tanh (kh)$. At $k=1$, its phase speed $\omega /k$ equals $c^{\ast }$: the moving pressure pattern drives a natural surface gravity wave at its own speed, producing resonance.

The restriction concerns a bounded travelling response. To distinguish this from existence of a solution of an initial value problem, set $T=\tanh h$, $P=p_1/\rho$, and ${\omega }_{\ast }^2=gT$. Ifthe linearized free-surface boundary conditions reduce to the forced harmonic oscillatorFor example, if the forcing is switched on at $t=0$ from a flat surface with zero initial velocity, then, for $c\ne {\omega }_{\ast }$,This satisfies $H(0)=\dot{H}(0)=0$ and contains both free oscillations and the forced component. At $c={\omega }_{\ast }$ a solution of the initial value problem still exists, but its particular solution containswhich grows linearly with time. Thus the ideal inviscid flow model predicts resonant growth rather than a finite constant amplitude. Additional free oscillations can set the initial conditions but cannot remove that secular growth.

For $0<c<c^{\ast }$, the nonnegative amplitude isIts first two derivatives are positive in this interval:The graph is increasing and convex, starts with a horizontal tangent at the limiting value $a(0)=p_1/(\rho g)$, and has a vertical asymptote at $c=c^{\ast }$. Therefore the hydrostatic lower bound for a travelling-pressure response isBecause the paper assumes $c>0$, this is an infimum, approached as $c\downarrow 0$, rather than an attained minimum. If the stationary limit $c=0$ is admitted, equality holds: the hydrostatic equilibrium condition is $\rho g\eta =-p_1\cos x$. Near resonance,The divergence describes the linear model; sufficiently close to $c^{\ast }$ the small-disturbance assumptions fail. The lower bound and sketch refer to the travelling forced component, since adding free oscillations can change the instantaneous surface elevation.