Standard Notation

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

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

Greek letters

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

Other notation, style etc.

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