kolbrek2885 Horn Theory Part 2.pdf

Tube, Solid State,

Loudspeaker Technology

Article prepared for www.audioXpress.com

An Introduction, Part 2

By Bjørn Kolbrek

The author continues his look at the various horn types and how they work.

Spherical Wave horn

The spherical wave (or Kugelwellen)

horn was invented by Klangfilm, the mo-

tion picture division of Siemens, in the

late 1940s 26, 27 . It is often mistaken for

being the same as the tractrix horn. It’s

not. But it is built on a similar assump-

tion: that the wave-fronts are spherical

with a constant radius. The wave-front

area expansion is exponential.

To calculate the spherical wave horn

contour, first decide a cutoff frequency f c

and a throat radius y o ( Fig. 20 ). The con-

stant radius r 0 is given as

and the area of the wave-f ront with

height h is 2πr 0 h. Thus for the area to

increase exponentially, h must increase

exponentially:

h = h 0 e mx (20)

where x is the distance of the top of the

wave-front from the top of the throat

wave-front and

not very different from the throat im-

pedance of a tractrix horn.

m =

4 f

Now that you know the area of the wave-

front, you can find the radius and the

distance of this radius from the origin.

S = 2 π r 0 h

(18)

= π

FIGURE 21: Assumed wave-fronts in

spherical wave horns.

(21)

= −

The height of the wave-f ront at the

throat is

2 2

0 0 0 0

x h = x – h + h 0

(22)

h r

= − −

r y

(19)

The area of the curved wave-front at the

throat is

S 0 = 2 π r 0 h 0

The assumed wave-fronts in a spherical

wavehorn are shown in Fig. 21 . Notice

that the wave-fronts are not assumed to

be 90° on the horn walls. Another prop-

erty of the spherical wave horn is that it

can fold back on itself ( Fig. 22 ), unlike

the tractrix horn, which is limited to a

90° tangent angle.

The throat impedance of a 100Hz

spherical wave horn—assuming wave-

fronts in the form of flattened spheri-

cal caps and using the radiation imped-

ance of a sphere with radius equal to the

mouth radius as mouth termination—is

s h own in Fig. 23 . You can see that it is

FIGURE 20: Dimensions of a spherical

wave horn.

FIGURE 22: Spherical wave horn folding

back.

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Horn Theory:

π .

it reaches F2, the

wave-front area

has expanded,

and to account

for this, a small

triangular ele-

ment (or, really, a

sector of a circle)

b1 is added.

The wave-

front expansion

from b1 (line

3-4) continues in

element a3, and

an element b2 is

added to account

for further wave expansion at F3. The

process is repeated, and the wave-front

becomes a curved surface, perpendicular

to the axis and the walls, but without

making any assumptions regarding the

shape prior to the calculations. The wave-

fronts are equidistant from each other,

and appear to take the shape of flattened

spherical caps. The resulting contour of

the horn is shown in Fig. 25 .

The wave-front expands according to

the Salmon family of hyperbolic horns.

There is no simple expansion equa-

tion for the contour of the Le Cléac’h

horn, but you can calculate it with the

help of spreadsheets available at http://

ndaviden.club.fr/pavillon/lecleah.html

waveguide acts like a 1P horn for a re-

stricted frequency range. Above a certain

frequency dictated by throat radius and

horn angle, there will be higher order

modes that invalidate the 1P assump-

tions.

The contour of the oblate spheroidal

waveguide is shown in Fig. 26 . It follows

the coordinate surfaces in the coordinate

system used, but in ordinary Cartesian

coordinates, the radius of the horn as a

function of x is given as

= + θ

( )

(23)

FIGURE 23: Throat impedance of a spherical wave horn.

where

r t is the throat radius, and

θ 0 is half the coverage angle.

The throat acoustical impedance is not

given as an analytical function; you must

find it by numerical integration. The

throat impedance for a waveguide with

a throat diameter of 35.7mm and θ 0 = 30

is shown in Fig. 27 .

le cléac’h horn

Jean-Michel Le Cléac’h presented a horn

that does not rely on an assumed wave-

front shape. Rather, it follows a “natural

expansion.” The principle is shown in

Fig. 24 . Lines 0-1 show the wave-front

surface at the throat (F1). At the point

FIGURE 24: The principle of the Le

Cléac’h expansion.

oblate Spheroidal

Waveguide

This horn was first investigated by Free-

hafer 28 , and later independently by Ged-

des 6 , who wanted to develop a horn suit-

able for directivity control in which the

sound field both in-

side and outside the

horn could be accu-

rately predicted. To do

this, the horn needed

to be a true 1P-horn.

Geddes investigated

several coordinate

systems, and found

the oblate spheroi-

dal (OS) coordinate

system to admit 1P

waves. Putland 7 later

showed that this was

not strictly the case.

More work by Ged-

des 29 showed that

the oblate spheroidal

FIGURE 26: Contour of the oblate sphe-

roidal waveguide.

FIGURE 25: Contour of a Le Cléac’h horn.

FIGURE 27: Normalized throat impedance of a 60 ° included

angle infinite oblate spheroidal waveguide.

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r r tan x

The OS waveguide does not have a

sharp cutoff like the exponential or hy-

perbolic horns, but it is useful to be able

to predict at what frequency the throat

impedance of the waveguide becomes too

low to be useful. If you set this frequency

at the point where the throat resistance is

0.2 times its asymptotic value 30 , so that

the meaning of the cutoff frequency be-

comes similar to the meaning of the term

as used with exponential horns, you get

angle, and the contour lies inside that of

the plane-wave exponential horn, being

a little longer and with a slightly small-

er mouth flare tangent angle ( Fig. 28 ).

Unfortunately, the Wilson method only

corrects the wave-front areas, not the

distance between the successive wave-

fronts.

There is not much information avail-

able about the Iwata horn 32, 33 , just a

drawing and dimensions, but no descrip-

tion of the concept. It looks like a ra-

dial horn, and seems to have cylindri-

cal wave-fronts expanding in area like

a hypex-horn with T = √2. The ratio of

height to width increases linearly from

throat to mouth.

ity performance of a horn, you need the

polar plot for a series of frequencies. But

sometimes you also want an idea of how

the coverage angle of the horn varies

with frequency, or how much amplifica-

tion a horn gives. This is the purpose of

the directivity factor (Q) and the direc-

tivity index (DI) 34 :

Directivity Factor : The directivity factor

is the ratio of the intensity on a given axis

(usually the axis of maximum radiation)

of the horn (or other radiator) to the

intensity that would be produced at the

same position by a point source radiating

the same power as the horn.

Directivity Index : The directivity index

is defined as: DI(f ) = 10 log 10 Q(f ). It

indicates the number of dB increase in

SPL at the observation point when the

horn is used compared to a point source.

Because intensity is watts per square

meter, it is inversely proportional to area,

and you can use a simple ratio of areas 35 .

Consider a sound source radiating in all

directions and observed at a distance r. At

this distance, the sound will fill a sphere

of radius r. Its area is 4πr 2 . The ratio of

the area to the area covered by a perfect

point source is 1, and thus Q = 1. If the

sound source is radiating into a hemi-

sphere, the coverage area is cut in half,

but the same sound power is radiated,

so the sound power per square meter is

doubled. Thus Q = 2. If the hemisphere

is cut in half, the area is 1/4 the area cov-

ered by a point source, and Q = 4.

= π

sin

(24)

You see that the cutoff of the waveguide

depends on both the angle and the throat

radius. For a low cutoff, a larger throat

and/or a smaller angle is required. For

example, for a 1″ driver and 60° includ-

ed angle (θ 0 = 30), the cutoff is about

862Hz.

The advantages of the OS waveguide

are that it offers improved loading over a

conical horn of the same coverage angle,

and has about the same directional prop-

erties. It also offers a very smooth transi-

tion from plane to spherical wave-fronts,

which is a good thing, because most driv-

ers produce plane wave-fronts.

The greatest disadvantage of the OS

waveguide is that it is not suitable for

low-frequency use. Bass and lower mid-

range horns based on this horn type will

run into the same problems as conical

horns: the horns become very long and

narrow for good loading.

To sum up, the OS waveguide pro-

vides excellent directivity control and

fairly good loading at frequencies above

about 1kHz.

FIGURE 28: Comparison of the expo-

nential horn with the tractrix and the Wil-

son modified exponential horn 22 .

other hornS

Three other horn types assuming curved

wave-fronts that are worth mentioning

are: the Western Electric horns, the Wil-

son modified exponential, and the Iwata

horn. What these horns have in common

is that they do not assume curved wave-

fronts of constant radius.

The Western Electric type horn 17 uses

wave-fronts of constantly increasing ra-

dius, all being centered around a vertex a

certain distance from the throat ( Fig. 29 ).

In the Wilson modified exponential

horn 31 , the waves start out at the throat

and become more and more spherical.

The horn radius is corrected in an it-

erative process based on the wall tangent

FIGURE 29: Wave-fronts in the Western

Electric type exponential horn 17 .

directivitY control

Control of directivity is an important

aspect of horn design. An exponential

horn can provide the driver with uniform

loading, but at high frequencies, it starts

to beam. It will therefore have a cover-

age angle that decreases with frequency,

which is undesirable in many circum-

stances. Often you want the horn to radi-

ate into a defined area, spilling as little

sound energy as possible in other areas.

Many horn types have been designed to

achieve this.

For the real picture of the directiv-

FIGURE 30: Contour of the Iwata horn 32 .

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0 . 2c

For a horn with coverage angles α and

β as shown in Fig. 31 , you can compute

Q as

where

f I is the intercept frequency in Hz where

the horn loses directivity control,

x is the size of the horn mouth in mm in

the plane of coverage, and

θ is the desired coverage angle in degrees

in that plane.

You thus need a large horn to control di-

rectivity down to low frequencies.

Most methods of directivity control

rely on simulating a segment of a sphere.

The following different methods are list-

ed in historical order.

beam widths of a typical multicellular

horn are shown in Fig. 33 . The fingering

at high frequencies is shown in Fig. 34 .

The beam width of a multicellular

horn with different number of cells is

shown in Fig. 35 34 . The narrowing in

beam width where the dimensions of the

horn are comparable to the wavelength is

evident.

180

(25)

 

 

 

sin sin sin

2 2

−

Most constant directivity horns try to

act as a segment of a sphere. A sphere

will emit sound uniformly in all direc-

tions, and a segment of a sphere will emit

sound uniformly in the angle it defines,

provided its dimensions are large com-

pared to the wavelength 11 . But when the

wavelength is comparable to the dimen-

sions of the spherical segment, the beam

width narrows to 40-50% of its initial

value.

A spherical segment can control direc-

tivity down to a frequency given as

25 10



Multicellular hornS

Dividing the horn into many conduits

is an old idea. Both Hanna 36 and Slepi-

an 37 have patented multicellular designs,

with the conduits extending all the way

back to the source. The source consists

of either multiple drivers or one driver

with multiple outlets, where each horn

is driven from a separate point on the

diaphragm.

The patent for the traditional mul-

ticellular horn belongs to Edward C.

Wente 38 . It was born from the need to

accurately control directivity, and at the

same time provide the driver with proper

loading, and was produced for use in the

Bell Labs experiment of transmitting the

sound of a symphonic orchestra from

one concert hall to another 39 .

A cut view of the multicellular horn, as

patented by Wente, is shown in Fig. 32 .

In this first kind of multicellular horn,

the individual horns started almost paral-

lel at the throat, but later designs often

used straight horn cells to simplify man-

ufacture of these complex horns. As you

can see, the multicellular horn is a cluster

of smaller exponential horns, each with a

mouth small enough to avoid beaming in

a large frequency range, but together they

form a sector of a sphere large enough to

control directivity down to fairly low fre-

quencies. The cluster acts as one big horn

at low frequencies. At higher frequencies,

the individual horns start to beam, but

because they are distributed on an arc,

coverage will still be quite uniform.

The multicellular horn has two prob-

lems, however. First, it has the same lower

midrange narrowing as the ideal sphere

segment, and, second, the polar pattern

shows considerably “fingering” at high

frequencies. This may not be as serious

as has been thought, however. The -6dB

(26)



FIGURE 33: -6dB beam widths of Elec-

tro-Voice model M253 2 by 5 cell horn 42 .

FIGURE 31: Radiation into a solid cone

of space defined by angles α and β .

FIGURE 34: High frequency fingering of

EV M253 horn at 10kHz 42 .

FIGURE 32: Multicellular horn 38 .

radial hornS

The radial or sectoral horn is a much

simpler concept than the multicellular

horn. The horizontal and vertical views

of a radial horn are shown in Fig. 36 .

The horizontal expansion is conical, and

defines the horizontal coverage angle of

the horn. The vertical expansion is de-

signed to keep an exponential expansion

of the wave-front, which is assumed to

be curved in the horizontal plane. Direc-

tivity control in the horizontal plane is

fairly good, but has the same midrange

narrowing as the multicellular horn. In

4 audioXpress 2008

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α β

vertical direction

only, then the di-

rection of expan-

sion is changed.

The wave-front

expansion is re-

stricted vertically,

and is released

horizontally. The

result is that the

horizontal pres-

sure that builds up

in the first part of

the horn causes

the wave-front to

expand more as it

reaches the sec-

ond part. That it

is restricted in the vertical plane helps

further.

Because the wave-front expansion is

to be exponential all the way, the dis-

continuity at the flare reversal point

(where the expansion changes direction)

is small. In addition, the change of cur-

vature at the flare reversal point is made

smoother in practical horns than what is

shown in the figure.

quency, k c =

2 f

The problem of midrange narrowing

was solved by having a more rapid flare

close to the mouth of the conical part of

the horn. Good results were obtained by

doubling the included angle in the last

third of the conical part. This decreases

the acoustical source size in the frequen-

cy range of midrange narrowing, causing

the beam width to widen, and removing

the narrowing. The result is a horn with

good directivity control down to the fre-

FIGURE 35: Beam width of a multicellular horn constructed as

shown in the insert 34 .

FIGURE 36: Profile of a radial horn 42 .

ce hornS

In the early 1970s, Keele, then working

for Electro-Voice, supplied an answer to

the problems associated with multicel-

lular and radial horns by introducing a

completely new class of horns that pro-

vided both good loading for the driver

and excellent directivity control 42 .

The principle is based on joining an

exponential or hyperbolic throat seg-

ment for driver loading with two conical

mouth segments for directivity control.

The exponential and conical segments

are joined at a point where the conical

horn of the chosen solid angle is an op-

timum termination for the exponential

horn. Keele defines this as the point

where the radius of the exponential horn

addition, there is almost no directivi-

ty control in the vertical plane, and the

beam width is constantly narrowing with

increasing frequency.

FIGURE 37: Wave-front expansion in

reversed flare horns 41 .

reverSed Flare hornS

The reversed flare horn can be con-

sidered to be a “soft diffraction horn,”

contrary to Manta-Ray horns and other

modern constant directivity designs that

rely on hard diffraction for directivity

control. This class of horns was patented

for directivity control by Sidney E. Levy

and Abraham B. Cohen at University

Loudspeakers in the early 1950s 40, 41 .

The same geometry appeared in many

Western Electric horns back in the early

1920s, but the purpose does not seem to

be that of directivity control 17 .

The principle for a horn with good

horizontal dispersion is illustrated in Fig.

37 . The wave is allowed to expand in the

0.95sin

(27)

where

r is the radius at the junction point,

θ is the half angle of the cone with solid

angle Ω,

θ = cos -1 (1 -

Ω

), and

FIGURE 38: Example of the Electro-

Voice CE constant directivity horns. This

horn covers 40 ° by 20 ° 42 .

k c is the wave number at the cutoff fre-

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π .

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