Template:Numerical calculation of Q

Numerical Calculation of [math]\displaystyle{ \mathbf{Q} }[/math]

We begin by truncating to a finite number ([math]\displaystyle{ N }[/math]) of evanescent modes,

[math]\displaystyle{ \mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0} f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s \frac{\phi _{m} \left( z\right)}{A_m} . }[/math]

We calculate the integral with the same panels as we used to approximate the integrals of the Green function and its normal derivative . Similarly, we assume that [math]\displaystyle{ f(s) \, }[/math] is constant over each panel and integrate [math]\displaystyle{ \phi _{m}\left( s\right) }[/math] exactly. This gives us the following matrix factorisation of [math]\displaystyle{ \mathbf{Q} }[/math]

[math]\displaystyle{ \mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f]. }[/math]

The components of the matrices [math]\displaystyle{ \mathbf{S} }[/math] and [math]\displaystyle{ \mathbf{R} }[/math] are

[math]\displaystyle{ s_{jm} = -k_m\phi _{m}\left( z_{j}\right) , }[/math] [math]\displaystyle{ r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi _{m}\left( s\right) \mathrm{d}s }[/math]