Past exam of the fluids and waves course of the University of Oxford / 2023 / 3

For a plane wave with amplitude $\alpha$, wavenumber $k>0$ and angular frequency $\omega >0$, the dispersion relation $\omega =\omega (k)$ specifies which frequencies the governing equations permit. A point of fixed wave phase obeys $kx-\omega t=\text{constant}$, so the phase speed isIt is the speed of individual crests. For a narrow wave packet, two nearby plane waves with differences $\Delta k$ and $\Delta \omega$ have a slowly varying wave envelope depending on $\Delta kx-\Delta \omega t$. Taking the limit gives the group velocityThis is the envelope's propagation speed and, in the present linear lossless wave model, the energy transport speed for a narrow packet.

A dispersive wave has phase speed dependent on wavenumber, so different components can change their relative phases and alter the packet's shape. A nondispersive wave on the chosen propagation branch has constant phase speed over the relevant band:Indeed $d{c}_p/dk=(c_g-c_p)/k$, so equality throughout a band is equivalent to constant phase speed there. Equality at an isolated wavenumber does not make the entire wave family nondispersive.

The irrotational flow has a velocity potential $\varphi$ with $(u,w)=({\varphi }_x,{\varphi }_z)$. The incompressible flow satisfies $u_x+w_z=0$, hence the interior Laplace equation isThe ice supplies a boundary load through its bending stiffness; it does not alter this interior consequence of mass conservation.

At the rigid bed, the no-penetration boundary condition givesThe displaced surface is material, so taking the material derivative of $z-\eta (x,t)$ gives the exact kinematic boundary condition for a free-surface graph:With upward vertical coordinate $z$, the gravitational potential per unit mass is $gz$. The Unsteady Bernoulli equation, evaluated on the surface with the prescribed pressure $p=B{\eta }_{xxxx}$, givesHere the spatially uniform time-dependent Bernoulli term has been absorbed into the velocity potential. The pressure is measured relative to the constant exterior pressure. This dynamic boundary condition uses the specified massless bending-cover model, with no ice inertia or tension.

Linearize about the resting state $\eta =0$, $\varphi =0$. Introduce a dimensionless small parameter $ϵ$ and writeholding the depth, bending stiffness and variation scales fixed. Thus both boundary displacement and induced velocity are small. For any first-order derivative $D\varphi$, a Taylor expansion at the moving surface givesThe displacement correction to the leading field is ${ϵ}^2{\eta }_1{\partial }_z(D{\varphi }_1)(x,0,t)$, so evaluating it on $z=0$ changes it only at second order. Also ${\varphi }_x{\eta }_x=O({ϵ}^2)$ and $|\nabla \varphi {|}^2=O({ϵ}^2)$. The first-order kinematic boundary condition and dynamic boundary condition are thereforeDropping the subscript, the linearized boundary condition pair isThe fixed-bed condition remains ${\varphi }_z(x,-h,t)=0$. The source's shorthand $|\eta |\ll 1$ requires a length scale: for the subsequent plane wave, small surface slope $k|\alpha |\ll 1$ and displacement small compared with depth are sufficient conditions. Small height alone, with arbitrarily rapid spatial variation or a finite background disturbance, does not justify discarding the products.

Let $\chi =kx-\omega t$, with $k>0$. The surface elevation is $\eta =\alpha \cos \chi$, so ${\eta }_t=\alpha \omega \sin \chi$. Use separation of variables in the interior Laplace equation:The $\sinh (k(z+h))$ term is excluded by the bed no-penetration boundary condition. The linear surface kinematic boundary condition determinesAn irrelevant additive constant is omitted. At $z=0$,The linear dynamic boundary condition then reduces toConsequently the dispersion relation of a floating elastic plate with zero plate inertia isThe negative-frequency branch gives propagation in the opposite direction. For signed wavenumber, replace $k$ by $|k|$ in the squared-frequency formula. Setting $B=0$ recovers the finite-depth surface gravity wave relation, a useful sign check.

In the deep-water limit $kh\gg 1$, $\tanh (kh)\sim 1$. Put $\beta =B/\rho >0$. The squared phase speed of the flexural-gravity wave is thenIt diverges as $k\to 0^{+}$ and as $k\to \infty$ in this limiting model. Its unique stationary point satisfiesThe second derivative $2g/k^3+6\beta k$ is positive. Using $g/k_{\ast }=3\beta k_{\ast }^3$ gives the minimum phase speed of a flexural-gravity wave:This is the deep-water minimum: applying it to a finite depth requires $k_{\ast }h\gg 1$. Otherwise the $\tanh (kh)$ factor must be retained in the minimization. Within the deep-water approximation, phase speed and group velocity coincide at $k_{\ast }$, although the wave family remains dispersive.

Define the dimensionless parameterThe stipulated hierarchy $kh\ll \delta \ll 1$ implies both $kh\ll 1$ andThus the finite-depth correction uses $\tanh (kh)\sim kh$, while the restoring effect of bending stiffness is small compared with gravity. The dispersion relation becomesHence, to leading order, these surface gravity waves are nondispersive waves withFor clarity, the first corrections to the phase speed areThey exhibit the weak residual wave dispersion at finite wavenumber. The claimed nondispersive behavior is the asymptotic leading-order result, not exact equality for the complete relation.