Directivity & coverage

Mouth directivity

Polar pattern, beamwidth and directivity index of a mouth treated as a piston. Find where a horn starts to beam.

Sheet: 18 / 25
Units: Metric
Air: 20 °C

Fig. 1Polar response, one octave either side

2.00 kHz1.00 kHz4.00 kHz6 dB per ring

Beamwidth: 35.2° at −6 dB
Size in wavelengths $ka$: 7.32
Directivity index $DI$: 17.4dB
Directivity factor $Q$: 55.1
First null: 31.6°
Wavelength $\lambda$: 17.2cm

Fig. 2Beamwidth of this mouth against frequency

−6 dB beamwidthdrag on plot to read

A flat line at 180° means the mouth is too small to narrow the beam at all.

How it works

However the flare is shaped, the mouth of a horn sets a floor under its directivity: an opening cannot hold a beam narrower or wider than its size in wavelengths allows. The simplest useful model treats the mouth as a flat, rigid piston of radius $a$ in a large baffle. Its far-field pressure at an angle $\theta$ off axis, relative to the axis, is

D(\theta) = \left|\frac{2\,J_1(ka\sin\theta)}{ka\sin\theta}\right|, \qquad ka = \frac{2\pi f a}{c} = \frac{\pi d}{\lambda}

where $J_1$ is the Bessel function of the first kind and order one.

Numbers that fall out

The response is 6 dB down where $ka\sin\theta = 2.215$ . Below $ka = 2.2$ that never happens and the mouth radiates into the whole half space.
The first null sits at $ka\sin\theta = 3.83$ , the first zero of $J_1$ ; side lobes follow.
The directivity factor compares on-axis intensity with that of a point source of equal power:

Q = \frac{(ka)^{2}}{1 - J_1(2ka)/ka}, \qquad DI = 10\log_{10} Q

At low frequencies $Q$ tends to 2 (3 dB, the baffle alone); at high frequencies it grows as $(ka)^2$ , which is 6 dB per octave of on-axis gain bought with a beam that halves in width.

Using it for horns

A real horn mouth is not a flat piston. The wave-front arriving at it is curved and weaker at the edges, which widens the main lobe and suppresses the side lobes. So read this sheet as a limit: below the frequency where the piston beamwidth equals your intended coverage angle, the horn can no longer hold its pattern, whatever its walls do. Above it, the walls take over. Exponential and tractrix horns keep narrowing with frequency much as the piston does, which is why they are said to beam.

Loading sheet…

How it works

D(\theta) = \left|\frac{2\,J_1(ka\sin\theta)}{ka\sin\theta}\right|, \qquad ka = \frac{2\pi f a}{c} = \frac{\pi d}{\lambda}

where $J_1$ is the Bessel function of the first kind and order one.

Numbers that fall out

The response is 6 dB down where $ka\sin\theta = 2.215$ . Below $ka = 2.2$ that never happens and the mouth radiates into the whole half space.
The first null sits at $ka\sin\theta = 3.83$ , the first zero of $J_1$ ; side lobes follow.
The directivity factor compares on-axis intensity with that of a point source of equal power:

Q = \frac{(ka)^{2}}{1 - J_1(2ka)/ka}, \qquad DI = 10\log_{10} Q

At low frequencies $Q$ tends to 2 (3 dB, the baffle alone); at high frequencies it grows as $(ka)^2$ , which is 6 dB per octave of on-axis gain bought with a beam that halves in width.

Mouth directivity

Inputs

How it works

Numbers that fall out

Using it for horns

Inputs

How it works

Numbers that fall out

Using it for horns

§ How it works

Numbers that fall out

Using it for horns

§ How it works

Numbers that fall out

Using it for horns

How it works

How it works