# Standard Notation

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

• $\displaystyle{ A }$ is the wave amplitude
• $\displaystyle{ c \,(=\omega / k) }$ or sometime $\displaystyle{ c_p }$ is the wave phase velocity
• $\displaystyle{ c_g = \frac{\mathrm{d} \omega}{\mathrm{d} k} }$ is the wave group velocity
• $\displaystyle{ d }$ is a water depth parameter
• $\displaystyle{ D }$ is the modulus of rigidity for a plate
• $\displaystyle{ e^{i\omega t} }$ is the time dependence in frequency domain
• $\displaystyle{ E }$ is the Young's modulus
• $\displaystyle{ \mathcal{E}(t) }$ is the energy density
• $\displaystyle{ g }$ is the acceleration due to gravity
• $\displaystyle{ h }$ is the water depth (with the bottom at $\displaystyle{ z=-h }$)
• $\displaystyle{ \mathbf{i} }$ is the unit vector in the $\displaystyle{ x }$ direction
• $\displaystyle{ \mathrm{Im} }$ is the imaginary part of a complex argument
• $\displaystyle{ \mathbf{j} }$ is the unit vector in the $\displaystyle{ y }$ direction
• $\displaystyle{ \mathbf{k} }$ is the unit vector in the $\displaystyle{ z }$ direction
• $\displaystyle{ k }$ is the wave number
• $\displaystyle{ k_n }$ are the roots of the dispersion eqution
• $\displaystyle{ \mathcal{L} }$ is the linear operator at the body surface
• $\displaystyle{ \mathcal{M} }$ is the momentum
• $\displaystyle{ \mathbf{n} }$ is the outward normal
• $\displaystyle{ \frac{\partial\phi}{\partial n} }$ is $\displaystyle{ \nabla\phi\cdot\mathbf{n} }$
• $\displaystyle{ P }$ is the pressure ($\displaystyle{ P_1 }$, $\displaystyle{ P_2 }$ etc are the first, second order pressures)
• $\displaystyle{ \mathcal{P}(t) }$ the energy flux is the rate of change of energy density $\displaystyle{ \mathcal{E}(t) }$
• $\displaystyle{ \mathbf{r} }$ vector in the horizontal directions only $\displaystyle{ (x,y) }$
• $\displaystyle{ R }$ is the radius of a cylinder
• $\displaystyle{ \mathrm{Re} }$ is the real part of a complex argument
• $\displaystyle{ S_F }$ is the free surface
• $\displaystyle{ t }$ is the time
• $\displaystyle{ T \,(= 2\pi / \omega) }$ is the wave period
• $\displaystyle{ U }$ is the forward speed
• $\displaystyle{ U_n }$ is the normal derivative of the moving surface of a volume
• $\displaystyle{ V_n = \mathbf{n} \cdot \nabla \Phi }$ is the flow in the normal direction for potential $\displaystyle{ \Phi }$
• $\displaystyle{ \mathbf{v} }$ is the flow velocity vector at $\displaystyle{ \mathbf{x} }$
• $\displaystyle{ \mathbf{x} }$ is the fixed Eulerian vector $\displaystyle{ (x,y,z) }$
• $\displaystyle{ x }$ and $\displaystyle{ y }$ are in the horizontal plane with $\displaystyle{ z }$ pointing vertically upward and the free surface is at $\displaystyle{ z=0 }$
• $\displaystyle{ \bar{x} }$ is the $\displaystyle{ x }$ coordinate in a moving frame.
• $\displaystyle{ X_n(x) }$ is an eigenfunction arising from separation of variables in the $\displaystyle{ x }$ direction.
• $\displaystyle{ Z(z) }$ is an eigenfunction arising from separation of variables in the $\displaystyle{ z }$ direction.

## Greek letters

• $\displaystyle{ \alpha }$ is free surface constant $\displaystyle{ \alpha = \omega^2/g }$
• $\displaystyle{ \mathcal{E} }$ is the energy
• $\displaystyle{ \zeta }$ is the displacement of the surface
• $\displaystyle{ \xi }$ any other displacement, most usually a body in the fluid
• $\displaystyle{ \eta }$ any other displacement, most usually a body in the fluid
• $\displaystyle{ \lambda \,(= 2\pi/k) }$ is the wave length
• $\displaystyle{ \rho }$ is the fluid density (sometimes also string density).
• $\displaystyle{ \rho_i }$ is the plate density
• $\displaystyle{ \phi\, }$ is the velocity potential in the frequency domain
• $\displaystyle{ \phi^{\mathrm{I}}\, }$ is the incident potential
• $\displaystyle{ \phi^{\mathrm{D}}\, }$ is the diffracted potential
• $\displaystyle{ \phi^{\mathrm{S}}\, }$ is the scattered potential ($\displaystyle{ \phi^{\mathrm{S}} = \phi^{\mathrm{I}}+\phi^{\mathrm{D}}\, }$)
• $\displaystyle{ \phi_{m}^{\mathrm{R}}\, }$ is the radiated potential (for the $\displaystyle{ m }$ mode
• $\displaystyle{ \Phi\, }$ is the velocity potential in the time domain
• $\displaystyle{ \bar{\Phi}\, }$ is the velocity potential in the time domain for a moving coordinate system
• $\displaystyle{ \omega }$ is the wave/angular frequency
• $\displaystyle{ \Omega\, }$ is the fluid region
• $\displaystyle{ \partial \Omega }$ is the boundary of fluid region, $\displaystyle{ \partial\Omega_F }$ is the free surface, $\displaystyle{ \partial\Omega_B }$ is the body surface.

## Other notation, style etc.

• We prefer $\displaystyle{ \partial_x\phi }$ etc. for all derivatives or $\displaystyle{ \frac{\partial\phi}{\partial x} }$. Try to avoid $\displaystyle{ \phi_x\, }$ or $\displaystyle{ \phi^{\prime} }$
• We prefer $\displaystyle{ \mathrm{d}x\,\! }$ etc. for differentials. Avoid $\displaystyle{ dx\,\! }$
• $\displaystyle{ \mathrm{Re}\,\! }$ and $\displaystyle{ \mathrm{Im}\,\! }$ 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.