Directivity of speakers

big speakers tense to beam the sound

Main idea using Huygens principle to describe the angle

Intensity equation

$$I(\theta )=I_{0}sin{c}^2(\frac{d\pi }{\lambda }\sin \theta )$$

where $d$ is the slit width $\lambda$ is a wave length $\theta$ is a angle

$$sinc(x\pi )=\frac{\sin (x\pi )}{x\pi }$$

values of $sinc(x\pi )$

x	$sinc(x\pi )$	$sin{c}^2(x\pi )$
1	0	0
0.99	0.01	0.0001
0.603	0.5	0.25
0.443	0.707	0.5
0.25	0.90	0.81
0.1	0.984	0.967
0.075	0.99	0.98
$\lim (x\to 0)$	1	1

let say, we’re only interested in 90° from slit, so $\sin \theta =1$

If we want to calculate the wave length that reduce an intensity by half (-6 dB) we can use

$$\begin{array}{rl}I(\theta )& =I_{0}sin{c}^2(\frac{d\pi }{\lambda }\sin \theta )\\ \frac{1}{2}{I}_0& =I_{0}sin{c}^2(\frac{d\pi }{\lambda }\cdot 1)\\ \frac{1}{2}& =sin{c}^2(\frac{d\pi }{\lambda })\\ 0.707& =sinc(\frac{d\pi }{\lambda })\\ \frac{d}{\lambda }& =0.443\end{array}$$

reduce an intensity by forth (-12 dB)

$\frac{d}{\lambda }=0.603$