Template:Free surface submerged plate relations

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.