Template:Incident potential for two dimensions: Difference between revisions
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\phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left( | \phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left( | ||
z\right) | z\right) | ||
</math> | |||
</center> | |||
== Expansion of the Potential == | |||
Therefore the potential can | |||
be expanded as | |||
<center> | |||
<math> | |||
\phi(x,z)=e^{-{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{{k}_{m}x}\phi_{m}(z), \;\;x<0 | |||
</math> | |||
</center> | |||
and | |||
<center> | |||
<math> | |||
\phi(x,z)=\sum_{m=0}^{\infty}b_{m} | |||
e^{-{k}_{m}x}\phi_{m}(z), \;\;x>0 | |||
</math> | </math> | ||
</center> | </center> | ||
Revision as of 01:16, 7 April 2010
Incident potential
The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. The incident potential can therefore be written as
[math]\displaystyle{ \phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left( z\right) }[/math]
Expansion of the Potential
Therefore the potential can be expanded as
[math]\displaystyle{ \phi(x,z)=e^{-{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{{k}_{m}x}\phi_{m}(z), \;\;x\lt 0 }[/math]
and
[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-{k}_{m}x}\phi_{m}(z), \;\;x\gt 0 }[/math]
