Difference between revisions of "Template:Incident plane wave"

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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}}({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h}
+
\phi^{\mathrm{I}}({r},z)=Ae^{-k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h}
 
</math></center>
 
</math></center>
 
where <math>A</math> is the wave amplitude and <math>k_0</math> is  
 
where <math>A</math> is the wave amplitude and <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>).

Revision as of 02:27, 1 September 2009

The equation is subject to some radiation conditions at infinity. We usually assume that there is an incident wave [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction

[math]\displaystyle{ \phi^{\mathrm{I}}({r},z)=Ae^{-k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h} }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude and [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]).