Template:Free surface submerged plate relations: Difference between revisions
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<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> | ||
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<center> | <center> | ||
<math> | <math> | ||
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z | \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 | \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.
