# 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\omega 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
- $ \mathrm{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
- $ k_n $ are the roots of the dispersion eqution
- $ \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
- $ \mathrm{Re} $ is the real part of a complex argument
- $ S_F $ is the free surface
- $ 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 $ \Phi $
- $ \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 is at $ z=0 $
- $ \bar{x} $ is the $ x $ coordinate in a moving frame.
- $ 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 $ z $ direction.

## Greek letters

- $ \alpha $ is free surface constant $ \alpha = \omega^2/g $
- $ \mathcal{E} $ is the energy
- $ \zeta $ is the displacement of the surface
- $ \xi $ any other displacement, most usually a body in the fluid
- $ \eta $ any other displacement, most usually a body in the fluid
- $ \lambda \,(= 2\pi/k) $ is the wave length
- $ \rho $ is the fluid density (sometimes also string density).
- $ \rho_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
- $ \bar{\Phi}\, $ is the velocity potential in the time domain for a moving coordinate system
- $ \omega $ 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.