Difference between revisions of "Template:Incident plane wave 2d definition"

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is a plane wave travelling in the <math>x</math> direction,
 
is a plane wave travelling in the <math>x</math> direction,
 
<center><math>
 
<center><math>
−
\phi^{\mathrm{I}}(x,z)=A \left\{ \frac{\cos k_0(z+h)}{\cos k_0 h} \right\} e^{-k_0 x}
+
\phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{-k_0 x}
 
</math></center>
 
</math></center>
−
where <math>A </math> is the wave amplitude (in potential) and <math>k_0 </math> is  
+
where <math>A </math> is the wave amplitude (in potential) <math>k_0 </math> is  
 
the positive imaginary solution of the [[Dispersion Relation for a Free Surface]]
 
the positive imaginary solution of the [[Dispersion Relation for a Free Surface]]
−
(note we are assuming that the time dependence is of the form <math>\exp(\mathrm{i}\omega t) </math>).
+
(note we are assuming that the time dependence is of the form <math>\exp(\mathrm{i}\omega t) </math>)
 +
and
 +
<center><math>
 +
\phi_0(z) =\frac{\cos k_0(z+h)}{\cos k_0 h}
 +
</math></center>

Revision as of 03:31, 26 November 2009

[math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,

[math]\displaystyle{ \phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{-k_0 x} }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ k_0 }[/math] is the positive imaginary solution of the Dispersion Relation for a Free Surface (note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(\mathrm{i}\omega t) }[/math]) and

[math]\displaystyle{ \phi_0(z) =\frac{\cos k_0(z+h)}{\cos k_0 h} }[/math]
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