Difference between revisions of "Template:Free surface submerged plate relations"

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<center>
 
<center>
 
<math>
 
<math>
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\chi_{m}^d(z) d z=B_{mn}
+
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\chi_{m}^d(z) \mathrm{d} z=B_{mn}
 
</math>
 
</math>
 
</center>
 
</center>
Line 10: Line 10:
 
<center>
 
<center>
 
<math>
 
<math>
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn}
+
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) \mathrm{d} z
 
</math>
 
</math>
 
</center>
 
</center>
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<center>
 
<center>
 
<math>
 
<math>
\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=B_{mn}
+
\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) \mathrm{d} z
 
</math>
 
</math>
 
</center>
 
</center>
 +
depending on whether the root <math>\mu_n</math> is above or below.

Latest revision as of 23:36, 8 August 2009

Inner product between free surface and submerged plate modes

We define

[math]\displaystyle{ \int\nolimits_{-d}^{0}\phi_{n}^h(z)\chi_{m}^d(z) \mathrm{d} z=B_{mn} }[/math]

where [math]\displaystyle{ B_{mn} }[/math] is either

[math]\displaystyle{ \int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) \mathrm{d} z }[/math]

or

[math]\displaystyle{ \int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) \mathrm{d} z }[/math]

depending on whether the root [math]\displaystyle{ \mu_n }[/math] is above or below.