Difference between revisions of "Standard Notation"

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* <math>\Phi\,</math> is the  velocity potential in the time domain
 
* <math>\Phi\,</math> is the  velocity potential in the time domain
 
* <math>\omega</math> is the wave/angular frequency  
 
* <math>\omega</math> is the wave/angular frequency  
* <math>e^{i\omega t}</math> is the time dependence in frequency domain
 
 
* <math>\Omega\,</math> is the fluid region  
 
* <math>\Omega\,</math> is the fluid region  
 
* <math>\partial \Omega</math> is the boundary of fluid region, <math>\partial\Omega_F</math> is the free surface, <math>\partial\Omega_B</math> is the body surface.  
 
* <math>\partial \Omega</math> is the boundary of fluid region, <math>\partial\Omega_F</math> is the free surface, <math>\partial\Omega_B</math> is the body surface.  

Revision as of 07:38, 25 March 2009

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{ d }[/math] is a water depth parameter
  • [math]\displaystyle{ D }[/math] is the modulus of rigidity
  • [math]\displaystyle{ \mathcal{E} }[/math] is the energy
  • [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{ \mathcal{L} }[/math] is the linear operator at the body surface
  • [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{ \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_n }[/math] is the normal derivative of the moving surface of a volume
  • [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 at [math]\displaystyle{ z=0 }[/math]

Greek letters

  • [math]\displaystyle{ \alpha }[/math] is free surface constant [math]\displaystyle{ \alpha = \omega^2/g }[/math]
  • [math]\displaystyle{ \rho }[/math] is the fluid density (sometimes also string density).
  • [math]\displaystyle{ \rho_i }[/math] is the plate 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{ \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.
  • [math]\displaystyle{ \zeta }[/math] is the displacement of the 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.