Meylan: /* Numerical Calculation of \mathbf{Q} */

2010-02-10T21:32:29Z

Numerical Calculation of \mathbf{Q}

← Older revision		Revision as of 21:32, 10 February 2010
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	_{m}\left( s\right) \mathrm{d}s		_{m}\left( s\right) \mathrm{d}s
	</math></center>		</math></center>

			{{reflection and transmission coefficients BEM}}

Meylan: Created page with '===Numerical Calculation of $\mathbf{Q}$ === We begin by truncating to a finite number ( $N$ ) of evanescent modes, $\mathbf{Q}\left[ f \right]…'$

2010-02-10T21:30:56Z

Created page with '===Numerical Calculation of <math>\mathbf{Q}</math> === We begin by truncating to a finite number (<math>N</math>) of evanescent modes, <center><math> \mathbf{Q}\left[ f \right]…'

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===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\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></center>
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>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
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></center>

Template:Numerical calculation of Q - Revision history

Meylan: /* Numerical Calculation of \mathbf{Q} */

Meylan: Created page with '===Numerical Calculation of \mathbf{Q} === We begin by truncating to a finite number (N) of evanescent modes, \mathbf{Q}\left[ f \right]…'

Meylan: Created page with '===Numerical Calculation of $\mathbf{Q}$ === We begin by truncating to a finite number ( $N$ ) of evanescent modes, $\mathbf{Q}\left[ f \right]…'$