Difference between revisions of "Template:Incident potential for two dimensions"

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===Incident potential===
 
===Incident potential===
  
To create meaningful solutions of the velocity potential <math>\phi</math> in the specified domains we add an incident wave term. The incident potential is a wave of amplitude <math>A</math>
+
To create meaningful solutions of the velocity potential <math>\phi</math> in the specified domains we add an incident wave term to the expansion in the domain <math>x < 0</math>. The incident potential is a wave of amplitude <math>A</math>
 
in displacement travelling in the positive <math>x</math>-direction. We would only see this in the time domain <math>\Phi(x,z,t)</math> however, in the frequency domain the incident potential can therefore be written as
 
in displacement travelling in the positive <math>x</math>-direction. We would only see this in the time domain <math>\Phi(x,z,t)</math> however, in the frequency domain the incident potential can therefore be written as
 
<center>
 
<center>
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</math>
 
</math>
 
</center>
 
</center>
The first term in the expansion of the diffracted potential for the region <math>x < 0</math> is given by
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The first term in the expansion of the diffracted potential for the domain <math>x < 0</math> is given by
 
<center>
 
<center>
 
<math>
 
<math>

Revision as of 05:22, 6 March 2012

Incident potential

To create meaningful solutions of the velocity potential [math]\displaystyle{ \phi }[/math] in the specified domains we add an incident wave term to the expansion in the domain [math]\displaystyle{ x \lt 0 }[/math]. The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. We would only see this in the time domain [math]\displaystyle{ \Phi(x,z,t) }[/math] however, in the frequency domain the incident potential can therefore be written as

[math]\displaystyle{ \phi^{\mathrm{I}}(x,z) =e^{-k_{0}x}\chi_{0}\left( z\right). }[/math]

The first term in the expansion of the diffracted potential for the domain [math]\displaystyle{ x \lt 0 }[/math] is given by

[math]\displaystyle{ a_{0}e^{k_{0}x}\chi_{0}\left( z\right) }[/math]

which represents the reflected wave.