Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / K (Fluid Dynamics and Waves)

The momentum of a material volume $V(t)$ is ${\int }_{V(t)}\rho \mathbf{u}dV$. Newton's second law equates its time derivative to the pressure force and gravity:Apply the given Reynolds transport theorem componentwise to the left side, and the divergence theorem to the pressure term. ThenSince this holds for arbitrary material volumes and the fields are continuous, the integrand vanishes pointwise. Dividing by positive density gives the Euler equations for an inviscid fluidAt a fixed physical time $t_0$, a streamline is a curve tangent to $\mathbf{u}(\cdot ,t_0)$, which may be parametrized by $d\mathbf{x}/ds=\mathbf{u}(\mathbf{x}(s),t_0)$. The chain rule gives $df(\mathbf{x}(s),t_0)/ds=\mathbf{u}\cdot \nabla f=0$, proving constancy along it. The time is held fixed; a time-dependent streamline statement does not by itself assert material conservation along particle trajectories.

Seek nonzero time-harmonic waves of the form $\varphi =\mathrm{Re}[F(y)e^{i(kx-\omega t)}]$ and $\eta =\mathrm{Re}[A{e}^{i(kx-\omega t)}]$, taking $k>0$. Laplace's equation reduces to $F^{\prime \prime }-k^{2}F=0$. The impermeable-bed condition $F^{\prime }(-h)=0$ gives $F(y)=C\cosh (k(y+h))$.

The linearized kinematic and dynamic linearized free-surface boundary conditions at zero becomeEliminating $A$ and the nonzero amplitude $C$ givesConversely, if this dispersion relation holds, either amplitude can be chosen nonzero and the other set by the second boundary equation, producing a wave satisfying every condition. The expression also holds for negative $k$ because $k\tanh (kh)$ is even.