Standard Notation

From WikiWaves

This is a list of standard notation with definitions. If you find notation which does not appear here or non-standard notation please feel free to highlight this, or better still try and fix it. The material on these webpages was taken from a variety of sources and we know the notation is currently not always consistent between pages.

Latin Letters

$A$ is the wave amplitude
$c \,(=\omega / k)$ or sometime $c p$ is the wave phase velocity
$c_g = \frac{\mathrm{d} \omega}{\mathrm{d} k}$ is the wave group velocity
$d$ is a water depth parameter
$D$ is the modulus of rigidity for a plate
$e i ω t$ is the time dependence in frequency domain
$E$ is the Young's modulus
$\mathcal{E}(t)$ is the energy density
$g$ is the acceleration due to gravity
$h$ is the water depth (with the bottom at $z = - h$ )
$\mathbf{i}$ is the unit vector in the $x$ direction
$Im$ is the imaginary part of a complex argument
$\mathbf{j}$ is the unit vector in the $y$ direction
$\mathbf{k}$ is the unit vector in the $z$ direction
$k$ is the wave number
$\mathcal{L}$ is the linear operator at the body surface
$\mathcal{M}$ is the momentum
$\mathbf{n}$ is the outward normal
$\frac{\partial\phi}{\partial n}$ is $\nabla\phi\cdot\mathbf{n}$
$P$ is the pressure ( $P 1$ , $P 2$ etc are the first, second order pressures)
$\mathcal{P}(t)$ the energy flux is the rate of change of energy density $\mathcal{E}(t)$
$\mathbf{r}$ vector in the horizontal directions only $(x, y)$
$R$ is the radius of a cylinder
$Re$ is the real part of a complex argument
$t$ is the time
$T \,(= 2\pi / \omega)$ is the wave period
$U$ is the forward speed
$U n$ is the normal derivative of the moving surface of a volume
$V_n = \mathbf{n} \cdot \nabla \Phi$ is the flow in the normal direction for potential $Φ$
$\mathbf{v}$ is the flow velocity vector at $\mathbf{x}$
$\mathbf{x}$ is the fixed Eulerian vector $(x, y, z)$
$x$ and $y$ are in the horizontal plane with $z$ pointing vertically upward and the free surface at $z = 0$
$X n (x)$ is an eigenfunction arising from separation of variables in the $x$ direction.
$Z (z)$ is an eigenfunction arising from separation of variables in the $x$ direction.

Greek letters

$α$ is free surface constant $α = ω 2 / g$
$\mathcal{E}$ is the energy
$ζ$ is the displacement of the surface
$ξ$ any other displacement, most ususually a body in the fluid
$η$ any other displacement, most ususually a body in the fluid
$\lambda \,(= 2\pi/k)$ is the wave length
$ρ$ is the fluid density (sometimes also string density).
$ρ i$ is the plate density
$\phi\,$ is the velocity potential in the frequency domain
$\phi^{\mathrm{I}}\,$ is the incident potential
$\phi^{\mathrm{D}}\,$ is the diffracted potential
$\phi^{\mathrm{S}}\,$ is the scattered potential ( $\phi^{\mathrm{S}} = \phi^{\mathrm{I}}+\phi^{\mathrm{D}}\,$ )
$\phi_{m}^{\mathrm{R}}\,$ is the radiated potential (for the $m$ mode
$\Phi\,$ is the velocity potential in the time domain
$ω$ is the wave/angular frequency
$\Omega\,$ is the fluid region
$\partial \Omega$ is the boundary of fluid region, $\partial\Omega_F$ is the free surface, $\partial\Omega_B$ is the body surface.

Other notation, style etc.

We prefer $\partial_x\phi$ etc. for all derivatives or $\frac{\partial\phi}{\partial x}$ . Try to avoid $\phi_x\,$ or $\phi^{\prime}$
We prefer $\mathrm{d}x\,\!$ etc. for differentials. Avoid $dx\,\!$
$\mathrm{Re}\,\!$ and $\mathrm{Im}\,\!$ for the real and imaginary parts.
We use two equals signs for the first heading (rather than a single) following wikipedia style, then three etc.

Standard Notation

From WikiWaves

Latin Letters

Greek letters

Other notation, style etc.

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