Theory of continuously phased planar sources for sound reinforcement

Open Access

Algorithm 2

Find parameter g.

procedure ParamG(x_r0, x_rV, z₀, H, V, N, η, β, a, b, X)
constant:
κ₁ = 3
maxit = 50
Δv = −V/N
ν = f(x_r,0, z₀)	according to Eq. (19)
ϑ₀ = ν − η	according to Fig. 2
$r = \frac{z_{0}}{\sin (η + ϑ_{0})}$ $r=\frac{{z}_0}{\mathrm{sin}(\eta +{\vartheta }_0)}$
X = x_r,0 − x_Rv
initialization:
W = zeros(N, 1)
Wh = zeros(N, 1)
Wv = zeros(N, 1)
Wv[1] = −sinϑ₀
Whh = a
g_max = f(z₀, η, Wv[1], Wh[1])
$g = \sqrt{0.99 g_{\max}^{2}}$ $g=\sqrt{0.99\enspace {g}_{\mathrm{max}}^2}$
repeat
for n = 1…N do
W[n + 1] ← W[n] + Wv[n]Δv	num. int.
v = n Δv
r ← f(v, η, Wh[n], Wv[n], z₀)	cf. Eq. (12)
Wvv ← f(r, g, β, a, b, Wv[n], Wh[n])	cf. Eq. (18)
Wv[n + 1] ← Wv[n] + Wvv Δv	num. int.
end for
$uz \leftarrow - Wv [n] \cos η + \sqrt{1 - Wv [n]^{2} - Wh [n])^{2}} \sin η$ ${uz}\leftarrow -{Wv}[n]\mathrm{cos}\eta +\sqrt{1-{Wv}[n{]}^2-{Wh}[n]{)}^2}\enspace \mathrm{sin}\eta$
$\tilde{X} \leftarrow x_{r 0} + V \sin η - r \sqrt{1 - Wh [n]^{2} - Wv [n]^{2}}$ $\mathop{X}\limits^\tilde\leftarrow {x}_{\mathrm{r}0}+V\enspace \mathrm{sin}\eta -r\enspace \sqrt{1-{Wh}[n{]}^2-{Wv}[n{]}^2}$
$α_{1} \leftarrow {(\frac{X}{\tilde{X}})}^{κ_{1}}$ ${\alpha }_1\leftarrow {\left(\frac{X}{\mathop{X}\limits^\tilde}\right)}^{{\kappa }_1}$
$g \leftarrow \sqrt{g_{\min} {(\frac{g}{g_{\min}})}^{α_{1}}}$ $g\leftarrow \sqrt{{g}_{\mathrm{min}}\enspace {\left(\frac{g}{{g}_{\mathrm{min}}}\right)}^{{\alpha }_1}}$
until $0.99 \leq \frac{\hat{X}}{X} \leq 1.01$ $0.99\le \frac{\widehat{X}}{X}\le 1.01$ or it > maxit
return g
end procedure

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