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

From WikiWaves
Jump to navigationJump to search
m
Line 6: Line 6:
 
</center>
 
</center>
 
The water motion is represented by a velocity potential which is
 
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
+
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
 
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

Revision as of 22:50, 22 March 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(i\omega t)\, }[/math]

The water motion is represented by a velocity potential which is denoted by [math]\displaystyle{ \phi\, }[/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 denoted 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].