Theory of continuously phased planar sources for sound reinforcement

Open Access

Algorithm 1

Find parameters a and b.

procedure ParamAB(x_r0, z₀, H, V, N, η, Y)
constant:
maxit = 1000
κ₂ = 0.2
κ₂ = 0.2
Δv = −V/N
Δh = Δv
ν = f(x_r,0, z₀)	according to Eq. (19)
ϑ₀ = ν − η
$M = - \frac{H}{2 Δ v} + 1$ $M=-\frac{H}{2\enspace \Delta v}+1$
$r = \frac{z_{0}}{\sin (η + ϑ_{0})}$ $r=\frac{{z}_0}{\mathrm{sin}(\eta +{\vartheta }_0)}$	according to Fig. 2
a_min = 0.3
b_min = 0.01
initialization:
a = 1.2 a_min
b = 1.2 b_min
W = zeros(M, 1)
Wh = zeros(M, 1)
Wv = zeros(M, 1)
Wv[1] = −sinϑ₀
Whh = a
repeat
for m = 1…M do
W[m + 1] ← W[m] + Wh[m] Δh	num. int.
v = n Δv
r ← f(v, η, Wh[m], Wv[m], z₀)	cf. Eq. (12)
$Whh = \frac{Wh [m]^{2}}{r} + a$ ${Whh}=\frac{{Wh}[m{]}^2}{r}+a$	cf. Eq. (16)
$Whv = \frac{Wh [m] Wv [m]}{r} + b$ ${Whv}=\frac{{Wh}[m]\enspace {Wv}[m]}{r}+b$	cf. Eq. (17)
Wv[m + 1] = Wv[m] + Whv Δh	num. int.
Wh[m+1] = Wh[m] + Whh Δh	num. int.
end for
$\tilde{Y} \leftarrow - \frac{H}{2} - r Wh [m]$ $\mathop{Y}\limits^\tilde\leftarrow -\frac{H}{2}-r\enspace {Wh}[m]$
$uz \leftarrow - Wv [m, n] \cos η + \sqrt{1 - Wv [m]^{2} - Wh [m]^{2}} \sin η$ ${uz}\leftarrow -{Wv}[m,n]\enspace \mathrm{cos}\eta +\sqrt{1-{Wv}[m{]}^2-{Wh}[m{]}^2}\enspace \mathrm{sin}\eta$
$\tilde{X} \leftarrow r \sqrt{1 - Wh [m]^{2} - u z^{2}}$ $\mathop{X}\limits^\tilde\leftarrow r\enspace \sqrt{1-{Wh}[m{]}^2-u{z}^2}\enspace$
$α_{2} \leftarrow {(\frac{Y}{\tilde{Y}})}^{κ_{2}}$ ${\alpha }_2\leftarrow {\left(\frac{Y}{\mathop{Y}\limits^\tilde}\right)}^{{\kappa }_2}$
$α_{3} \leftarrow {(\frac{X}{\tilde{X}})}^{κ_{3}}$ ${\alpha }_3\leftarrow {\left(\frac{X}{\mathop{X}\limits^\tilde}\right)}^{{\kappa }_3}\enspace$
$a \leftarrow a_{\min} {(\frac{a}{a_{\min}})}^{α_{2}}$ $a\leftarrow {a}_{\mathrm{min}}\enspace {\left(\frac{a}{{a}_{\mathrm{min}}}\right)}^{{\alpha }_2}\enspace$
$b \leftarrow b_{\min} {(\frac{b}{b_{\min}})}^{α_{3}}$ $b\leftarrow {b}_{\mathrm{min}}\enspace {\left(\frac{b}{{b}_{\mathrm{min}}}\right)}^{{\alpha }_3}$
until $0.99 \leq \frac{\hat{X}}{X} \leq 1.01$ $0.99\le \frac{\widehat{X}}{X}\le 1.01$ and $0.99 \leq \frac{\hat{Y}}{Y} \leq 1.01$ $0.99\le \frac{\widehat{Y}}{Y}\le 1.01$ or it > maxit
return a, b
end procedure

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