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

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The first term in the expansion of the diffracted potential for the domain <math>x < 0</math> is given by
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The total velocity (scattered) potential now becomes <math>\phi = \phi + \phi</math>. The first term in the expansion of the diffracted potential for the domain <math>x < 0</math> is given by
 
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Revision as of 05:46, 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 for the domain of [math]\displaystyle{ x \lt 0 }[/math] above. 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 total velocity (scattered) potential now becomes [math]\displaystyle{ \phi = \phi + \phi }[/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.