Difference between revisions of "Standard Notation"

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A list of standard notation with definitions. If you find notation which does not appear here or non-standard notation please
+
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
 
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.
 
know the notation is currently not always consistent between pages.
Line 6: Line 6:
  
 
* <math>A</math> is the wave amplitude
 
* <math>A</math> is the wave amplitude
* <math> c \,(=\omega / k)  </math>  is the wave phase velocity
+
* <math> c \,(=\omega / k)  </math>  or sometime <math>c_p</math> is the wave phase velocity
 +
* <math> c_g = \frac{\mathrm{d} \omega}{\mathrm{d} k} </math> is the wave group velocity
 
* <math>d</math> is a water depth parameter
 
* <math>d</math> is a water depth parameter
* <math>D</math> is the modulus of rigidity
+
* <math>D</math> is the modulus of rigidity for a plate
* <math>\mathcal{E}</math> is the energy
+
* <math>e^{i\omega t}</math> is the time dependence in frequency domain
 +
* <math>E</math> is the Young's modulus
 +
* <math> \mathcal{E}(t) </math> is the energy density
 
* <math>g</math> is the acceleration due to gravity  
 
* <math>g</math> is the acceleration due to gravity  
 
* <math>h</math> is the water depth (with the bottom at <math>z=-h</math>)
 
* <math>h</math> is the water depth (with the bottom at <math>z=-h</math>)
 +
* <math>\mathbf{i}</math> is the unit vector in the <math>x</math> direction
 +
* <math>\mathrm{Im}</math> is the imaginary part of a complex argument
 +
* <math>\mathbf{j}</math> is the unit vector in the <math>y</math> direction
 +
* <math>\mathbf{k}</math> is the unit vector in the <math>z</math> direction
 
* <math> k </math> is the wave number  
 
* <math> k </math> is the wave number  
 +
* <math>k_n</math> are the roots of the dispersion eqution
 +
* <math>\mathcal{L}</math> is the linear operator at the body surface
 +
* <math>\mathcal{M}</math> is the momentum
 
* <math>\mathbf{n}</math> is the outward normal  
 
* <math>\mathbf{n}</math> is the outward normal  
 
* <math>\frac{\partial\phi}{\partial n}</math> is <math>\nabla\phi\cdot\mathbf{n}</math>  
 
* <math>\frac{\partial\phi}{\partial n}</math> is <math>\nabla\phi\cdot\mathbf{n}</math>  
 
* <math>P</math> is the pressure (<math>P_1</math>,  <math>P_2</math> etc are the first, second order pressures)
 
* <math>P</math> is the pressure (<math>P_1</math>,  <math>P_2</math> etc are the first, second order pressures)
 
* <math>\mathcal{P}(t)</math> the energy flux is the rate of change of energy density <math> \mathcal{E}(t) </math>
 
* <math>\mathcal{P}(t)</math> the energy flux is the rate of change of energy density <math> \mathcal{E}(t) </math>
* <math>\mathbf{Re}</math> is the real part of a complex argument
+
* <math>\mathbf{r}</math> vector in the horizontal directions only <math>(x,y)</math>
 +
* <math>R</math> is the radius of a cylinder
 +
* <math>\mathrm{Re}</math> is the real part of a complex argument
 +
* <math>S_F</math> is the free surface
 
* <math>t</math> is the time  
 
* <math>t</math> is the time  
 
* <math> T \,(= 2\pi / \omega)</math> is the wave period
 
* <math> T \,(= 2\pi / \omega)</math> is the wave period
 +
* <math>U</math> is the forward speed
 
* <math>U_n</math> is the normal derivative of the moving surface of a volume
 
* <math>U_n</math> is the normal derivative of the moving surface of a volume
 +
* <math> V_n = \mathbf{n} \cdot \nabla \Phi </math> is the flow in the normal direction for potential <math>\Phi</math>
 
* <math>\mathbf{v}</math> is the flow velocity vector at <math>\mathbf{x}</math>  
 
* <math>\mathbf{v}</math> is the flow velocity vector at <math>\mathbf{x}</math>  
* <math>\mathbf{x}</math> is the fixed Eulerian vector  
+
* <math>\mathbf{x}</math> is the fixed Eulerian vector <math>(x,y,z)</math>
* <math>x</math> and <math>y</math> are in the horizontal plane with <math>z</math> pointing vertically upward and the free surface at <math>z=0</math>
+
* <math>x</math> and <math>y</math> are in the horizontal plane with <math>z</math> pointing vertically upward and the free surface is at <math>z=0</math>
 +
* <math>\bar{x}</math> is the <math>x</math> coordinate in a moving frame.
 +
* <math>X_n(x)</math> is an eigenfunction arising from separation of variables in the <math>x</math> direction.
 +
* <math>Z(z)</math> is an eigenfunction arising from separation of variables in the <math>z</math> direction.
  
 
== Greek letters ==
 
== Greek letters ==
  
* <math>\rho</math> is the fluid density
+
* <math>\alpha</math> is free surface constant <math>\alpha = \omega^2/g</math>
 +
* <math>\mathcal{E}</math> is the energy
 +
* <math>\zeta</math> is the displacement of the surface
 +
* <math>\xi</math> any other displacement, most usually a body in the fluid
 +
* <math>\eta</math> any other displacement, most usually a body in the fluid
 +
* <math> \lambda \,(= 2\pi/k) </math> is the wave length
 +
* <math>\rho</math> is the fluid density (sometimes also string density).
 
* <math>\rho_i</math> is the plate density
 
* <math>\rho_i</math> is the plate density
* <math> \lambda \,(= 2\pi/k) </math> is the wave length
 
 
* <math>\phi\,</math> is the  velocity potential in the frequency domain
 
* <math>\phi\,</math> is the  velocity potential in the frequency domain
 +
* <math>\phi^{\mathrm{I}}\,</math> is the  incident potential
 +
* <math>\phi^{\mathrm{D}}\,</math> is the  diffracted potential
 +
* <math>\phi^{\mathrm{S}}\,</math> is the  scattered potential (<math>\phi^{\mathrm{S}}
 +
= \phi^{\mathrm{I}}+\phi^{\mathrm{D}}\,</math>)
 +
* <math>\phi_{m}^{\mathrm{R}}\,</math> is the  radiated potential (for the <math>m</math> mode
 
* <math>\Phi\,</math> is the  velocity potential in the time domain
 
* <math>\Phi\,</math> is the  velocity potential in the time domain
 +
* <math>\bar{\Phi}\,</math> is the  velocity potential in the time domain for a moving coordinate system
 
* <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</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>\zeta</math> is the displacement of the surface
 
  
 
== Other notation, style etc. ==
 
== Other notation, style etc. ==

Latest revision as of 19:59, 26 July 2012

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{ S_F }[/math] is the free surface
  • [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 usually a body in the fluid
  • [math]\displaystyle{ \eta }[/math] any other displacement, most usually 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{S}} = \phi^{\mathrm{I}}+\phi^{\mathrm{D}}\, }[/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.