Standard Notation

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] is the wave phase velocity
• [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{ k }[/math] is the wave number
• [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{ \mathbf{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{ \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 }[/math] and [math]\displaystyle{ y }[/math] are in the horizontal plane with [math]\displaystyle{ z }[/math] pointing vertically upward and the free surface at [math]\displaystyle{ z=0 }[/math]

Greek letters

• [math]\displaystyle{ \rho }[/math] is the fluid density
• [math]\displaystyle{ \lambda \,(= 2\pi/k) }[/math] is the wave length
• [math]\displaystyle{ \phi\, }[/math] is the velocity potential in the frequency domain
• [math]\displaystyle{ \Phi\, }[/math] is the velocity potential in the time domain
• [math]\displaystyle{ \omega }[/math] is the wave/angular frequency
• [math]\displaystyle{ e^{i\omega t} }[/math] time dependence in frequency domain
• [math]\displaystyle{ \Omega\, }[/math] is the fluid region
• [math]\displaystyle{ \partial \Omega }[/math] is the boundary of fluid region
• [math]\displaystyle{ \zeta }[/math] is the displacement of the surface
• [math]\displaystyle{ \varepsilon }[/math] is the energy

Other notation, style etc.

• We always use [math]\displaystyle{ \partial_x\phi }[/math] etc. for all derivatives, never [math]\displaystyle{ \phi_, }[/math] or [math]\displaystyle{ \frac{\partial\phi}{\partial x} }[/math]

or [math]\displaystyle{ \phi^{\prime} }[/math]

• We use two equals signs for the first heading (rather than a single) following wikipedia style, then three etc.