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