Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 1

The velocity potential gives horizontal and vertical velocity components $({\varphi }_x,{\varphi }_z)$. The kinematic boundary condition at each fixed impermeable wall is zero normal fluid velocity, henceLet $\eta (x,t)$ be the surface elevation. To first order in its amplitude, the kinematic boundary condition for a free-surface graph gives ${\eta }_t={\varphi }_z$ at $z=0$. The dynamic boundary condition for an inviscid interface fixes the surface pressure to atmospheric pressure. The unsteady Bernoulli equation, after choosing the time-dependent gauge of $\varphi$, then gives ${\varphi }_t+g\eta =0$ at $z=0$: the quadratic velocity term is second order in amplitude. Differentiate this last equation in time and eliminate ${\eta }_t$ to obtain the linearized free-surface boundary conditions in combined form,

Apply separation of variables to Laplace's equation. The side-wall Neumann boundary conditions give horizontal factors $\cos (k_{n}x)$ with $k_n=n\pi /L$. The bottom condition fixes the vertical factor to $\cosh (k_n(z+h))$. Substitution into the surface condition in i gives the normal modes of surface gravity waves in a rectangular tank:The corresponding surface elevation follows from ${\eta }_n=-{\partial }_t{\varphi }_n(x,0,t)/g$. The spatially uniform $n=0$ mode has zero frequency; its constant elevation changes total fluid volume, so it is absent when that volume is fixed. An additive constant in $\varphi$ changes no fluid velocity.

The wall position is $x_w(t)=\mathrm{Re}(c{e}^{-i\omega t})$, so its velocity is $\mathrm{Re}(-i\omega c{e}^{-i\omega t})$. The kinematic boundary condition equates the normal fluid velocity to the normal wall velocity. Linearizing the evaluation point to the mean wall position givesThe other wall and bottom retain $f_x(L,z)=0$ and $f_z(x,-h)=0$, while the free surface gives $f_z(x,0)=({\omega }^2/g)f(x,0)$.

Put $\lambda ={\omega }^2/g$ andThis polynomial is a harmonic function: $Q_{xx}+Q_{zz}=0$. Also $Q_x(0,z)=i\omega c$, $Q_x(L,z)=0$ and $Q_z(x,-h)=0$. Thus $G$ satisfies Laplace's equation with homogeneous Neumann boundary conditions on both side walls and the bottom. Its Fourier cosine series consequently has the formThe surface condition becomes $G_z-\lambda G=H(x)$, whereFor a nonzero forcing frequency away from the natural frequencies, the coefficients of the oscillating-wall forcing of a rectangular tank areHere $k_0=0$ in the definition of $H_0$. In fact $H_n=2i\omega c\lambda /(L{k}_n^2)$ for $n\ge 1$, since ${\int }_0^L(L-x{)}^2\cos (k_{n}x)dx=2L/k_n^2$. The zero-mode coefficient ensures that the mean surface elevation is $hc/L$ in complex amplitude, exactly the first-order volume conservation adjustment to the moving wall.

For the $n$th normal mode, the denominator just obtained isIts forcing coefficient is nonzero for $c\ne 0$ and $\omega >0$. Therefore the periodic response grows like $({\omega }_n^2-{\omega }^2{)}^{-1}$ as $\omega$ approaches ${\omega }_n$: this is resonance. At exact resonance there is no bounded periodic particular solution in the linear inviscid model; the resonant amplitude instead contains a secular term proportional to $t\sin ({\omega }_{n}t)$ or $t\cos ({\omega }_{n}t)$. Physical damping and nonlinear effects eventually matter. Away from resonance, undamped free oscillations can persist alongside the periodic particular solution.