# Standard Notation

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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.