Difference between revisions of "Standard Notation"
From WikiWaves
Jump to navigationJump to searchm (→Greek letters) |
|||
| (46 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| − | + | 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>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> k </math> is the wave number | + | * <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_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>\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>\mathbf{Re}</math> is the real part of a complex argument | + | * <math>\mathcal{P}(t)</math> the energy flux is the rate of change of energy density <math> \mathcal{E}(t) </math> |
| + | * <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> 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>\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>\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. |
| − | |||
| − | |||
== Other notation, style etc. == | == Other notation, style etc. == | ||
| − | * We | + | * We prefer <math>\partial_x\phi</math> etc. for all derivatives or <math>\frac{\partial\phi}{\partial x}</math>. Try to avoid <math>\phi_x\,</math> or <math>\phi^{\prime}</math> |
| + | * We prefer <math>\mathrm{d}x\,\!</math> etc. for differentials. Avoid <math>dx\,\!</math> | ||
| + | * <math>\mathrm{Re}\,\!</math> and <math>\mathrm{Im}\,\!</math> for the real and imaginary parts. | ||
* We use two equals signs for the first heading (rather than a single) following [http://www.wikipedia.org wikipedia] style, then three etc. | * We use two equals signs for the first heading (rather than a single) following [http://www.wikipedia.org wikipedia] style, then three etc. | ||
[[Category:Administration]] | [[Category:Administration]] | ||
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.
