Difference between revisions of "Template:Incident potential for two dimensions"
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To create meaningful solutions of the velocity potential <math>\phi</math> in the specified domains we add an incident wave term to the expansion for the domain of <math>x < 0</math> above. 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 for the domain of <math>x < 0</math> above. 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 | + | 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 be written as |
<center> | <center> | ||
<math> | <math> | ||
| − | \ | + | \phi_{\mathrm{I}}(x,z) =e^{-k_{0}x}\chi_{0}\left( |
z\right). | z\right). | ||
</math> | </math> | ||
</center> | </center> | ||
| + | The total velocity (scattered) potential now becomes <math>\phi = \phi_{\mathrm{I}} + \phi_{\mathrm{D}}</math> for the domain of <math>x < 0</math>. | ||
| + | |||
The first term in the expansion of the diffracted potential for the domain <math>x < 0</math> is given by | The first term in the expansion of the diffracted potential for the domain <math>x < 0</math> is given by | ||
<center> | <center> | ||
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</math> | </math> | ||
</center> | </center> | ||
| − | which represents the reflected wave. | + | which represents the reflected wave. |
| + | |||
| + | In any scattering problem <math>|R|^2 + |T|^2 = 1</math> where <math>R</math> and <math>T</math> are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock <math>|a_{0}| = |R| = 1</math> and <math>|T| = 0</math> as there are no transmitted waves in the region under the dock. | ||
Latest revision as of 21:24, 21 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 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_{\mathrm{I}} + \phi_{\mathrm{D}} }[/math] for the domain of [math]\displaystyle{ x \lt 0 }[/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.
In any scattering problem [math]\displaystyle{ |R|^2 + |T|^2 = 1 }[/math] where [math]\displaystyle{ R }[/math] and [math]\displaystyle{ T }[/math] are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock [math]\displaystyle{ |a_{0}| = |R| = 1 }[/math] and [math]\displaystyle{ |T| = 0 }[/math] as there are no transmitted waves in the region under the dock.
