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For an inviscid flow of constant mass density $\rho$ and depth $h$, impose excess pressure $p_1\cos (k(x-ct))$ with $k>0$ on a free surface. Neglect surface tension. A travelling solution of the linearized free-surface boundary conditions isThe velocity potential satisfies the Laplace equation and the bed no-penetration boundary condition. Its surface time derivative is $-A{c}^{2}k\coth (kh)\cos (k(x-ct))$, so the dynamic condition gives the displayed signed response. The nonnegative amplitude is $|A|$. It is finite except atthe squared phase speed of a free surface gravity wave at that wavenumber. Equality produces resonance. This describes a travelling forced component; free surface gravity waves must be added to meet general initial conditions. At exact resonance, a solution of the initial value problem generally grows with time rather than having a bounded periodic amplitude.

A vortex line is an integral curve of the vorticity vector field: its tangent is parallel to $\mathbf{\omega }=\nabla \times \mathbf{u}$. Where vorticity vanishes, that direction does not define a unique line.

In two-dimensional, constant-mass density, inviscid flow with conservative body forces, the vorticity equation reduces to $D\omega /Dt=0$. There is no vortex stretching, because the vorticity is perpendicular to the plane. With nonzero kinematic viscosity $\nu$, the equation instead contains diffusion, $D\omega /Dt=\nu {\nabla }^2\omega$; incompressibility alone does not give material conservation.

A bundle of vortex lines forms a vortex tube. Under the conditions of Kelvin's circulation theorem, its material cross-section carries conserved vorticity flux, equal to circulation. In three dimensions that conserved flux is compatible with growing local vorticity when stretching reduces the tube's cross-sectional area.