Difference between revisions of "Template:Standard linear problem notation"

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We assume small amplitude so that we can linearise all the
 
We assume small amplitude so that we can linearise all the
equations (see [[Linear and Second-Order Wave Theory]]). We also assume that [[Frequency Domain Problem]] with frequency <math>\omega</math>
+
equations (see [[Linear and Second-Order Wave Theory]]).  
and we assume that all variables are proportional to
+
{{frequency definition}}
<center>
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{{velocity potential in frequency domain}}
<math>\exp(i\omega t)\,</math>
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The coordinate system is the standard Cartesian coordinate system
</center>
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with the <math>z-</math>axis pointing vertically up. The water surface is at
The water motion is represented by a velocity potential which is
 
denoted by <math>\phi</math>.  The coordinate system is the standard Cartesian coordinate system
 
with the <math>z</math> axis pointing vertically up. The water surface is at
 
 
<math>z=0</math> and the region of interest is
 
<math>z=0</math> and the region of interest is
 
<math>-h<z<0</math>. There is a body which occupies the region <math>\Omega</math>
 
<math>-h<z<0</math>. There is a body which occupies the region <math>\Omega</math>
and we denoted the wetted surface of the body by <math>\partial\Omega</math>
+
and we denote the wetted surface of the body by <math>\partial\Omega</math>
 
We denote <math>\mathbf{r}=(x,y)</math> as the horizontal coordinate in two or three dimensions
 
We denote <math>\mathbf{r}=(x,y)</math> as the horizontal coordinate in two or three dimensions
respectively and the cartesian system we denote by <math>\mathbf{x}</math>.  
+
respectively and the Cartesian system we denote by <math>\mathbf{x}</math>.  
 
We assume that the bottom surface is of constant depth at <math>z=-h</math>.
 
We assume that the bottom surface is of constant depth at <math>z=-h</math>.

Latest revision as of 04:14, 20 August 2009

We assume small amplitude so that we can linearise all the equations (see Linear and Second-Order Wave Theory). We also assume that Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] and we assume that all variables are proportional to [math]\displaystyle{ \exp(-\mathrm{i}\omega t)\, }[/math] The water motion is represented by a velocity potential which is denoted by [math]\displaystyle{ \phi\, }[/math] so that

[math]\displaystyle{ \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. }[/math]

The coordinate system is the standard Cartesian coordinate system with the [math]\displaystyle{ z- }[/math]axis pointing vertically up. The water surface is at [math]\displaystyle{ z=0 }[/math] and the region of interest is [math]\displaystyle{ -h\lt z\lt 0 }[/math]. There is a body which occupies the region [math]\displaystyle{ \Omega }[/math] and we denote the wetted surface of the body by [math]\displaystyle{ \partial\Omega }[/math] We denote [math]\displaystyle{ \mathbf{r}=(x,y) }[/math] as the horizontal coordinate in two or three dimensions respectively and the Cartesian system we denote by [math]\displaystyle{ \mathbf{x} }[/math]. We assume that the bottom surface is of constant depth at [math]\displaystyle{ z=-h }[/math].