kolbrek2884 Horn Theory Part 1.pdf

Tube, Solid State,

Loudspeaker Technology

Article prepared for www.audioXpress.com

Horn Theory: An Introduction, Part 1

By Bjørn Kolbrek

This author presents a two-part introduction to horns—their definition,

features, types, and functions.

tical horns, as it applies to loudspeakers.

It reviews the basic assumptions behind

classical horn theory as it stands, presents

the different types of horns, and discusses their

properties. Directivity control, wave-front shapes,

and distortion are also discussed.

In this article, I try to keep the math

simple, and, where it is required, I explain

or illustrate the meaning of the equations.

The focus is on understanding what is

going on in a horn. The practical aspects

of horn design are not treated here.

ρ 0 : Density of air, 1.205 kg/m 3 .

f : Frequency, Hz.

ω: Angular frequency, radians/s, ω = 2πf.

k : Wave number or spatial frequency,

there will be a considerable mismatch

between the source and the load. The

result is that most of the energy put into

a direct radiating loudspeaker will not

reach the air, but will be converted to

heat in the voice coil and mechanical

resistances in the unit. The problem is

worse at low frequencies, where the size

of the source will be small compared to

a wavelength and the source will merely

push the medium away. At higher fre-

quencies, the radiation from the source

will be in the form of plane waves that

do not spread out. The load, as seen from

the driver, is at its highest, and the system

is as efficient as it can be.

If you could make the driver radiate

plane waves in its entire operating range,

efficient operation would be secured at all

frequencies. The driver would work into

a constant load, and if this load could be

made to match the impedances of the

driver, resonances would be suppressed.

This is because the driver is a mechani-

cal filter, which needs to be terminated

in its characteristic impedance, ideally a

pure resistance. If the driver is allowed

to radiate plane waves, resistive loading

is secured.

The easiest way to make the driver

radiate plane waves is to connect it to a

long, uniform tube. But the end of the

tube will still be small compared to a

wavelength at low frequencies. To avoid

reflections, the cross section of the tube

must be large compared to a wavelength,

but, at the same time, it must also be

radians/m, k = 2 f

c c

ω π

S : Area.

p : Pressure.

Z A : Acoustical impedance.

j : Imaginary operator, j = √-1.

THE PURPOSE OF A HORN

It can be useful to look at the purpose

of the horn before looking at the theory.

Where are horns used, and for what?

Throughout the history of electroa-

coustics, there have been two important

aspects:

• Loading of the driver

• Directivity control

You would also think that increasing the

output would be one aspect of horns, but

this is included in both. Increasing the

loading of the driver over that of free air

increases efficiency and hence the out-

put, and concentrating the sound into a

certain solid angle increases the output

further.

Loading of the Driver . The loudspeaker,

which is a generator of pressure, has an

internal source impedance and drives an

external load impedance. The air is the

ultimate load, and the impedance of air is

low, because of its low density.

The source impedance of any loud-

s p eaker, on the other hand, is high, so

TERMINOLOGY

The article includes the following termi-

nology:

Impedance : Quantity impeding or re-

ducing flow of energy. Can be electrical,

mechanical, or acoustical.

Acoustical Impedance : The ratio of

sound pressure to volume velocity of air.

In a horn, the acoustical impedance will

increase when the cross-section of the

horn decreases, as a decrease in cross sec-

tion will limit the flow of air at a certain

pressure.

Volume Velocity : Flow of air through a

surface in m 3 /s, equals particle velocity

times area.

Throat : The small end of the horn,

where the driver is attached.

Mouth : The far end of the horn, which

radiates into the air.

Driver : Loudspeaker unit used for driv-

ing the horn.

c : The speed of sound, 344m/s at 20° C.

audioXpress 2008 1

T his article deals with the theory of acous-

small to fit the driver and present the

required load. To solve this dilemma, you

need to taper the tube.

When you do this, you can take radia-

tion from the driver in the form of plane

waves and transform the high pressure,

low velocity vibrations at the throat into

low pressure, high velocity vibrations that

can efficiently be radiated into the air.

Depending on how the tube flares, it is

possible to present a load to the driver that

is constant over a large frequency range.

Directivity Control . The directivity of

a cone or dome diaphragm is largely un-

controlled, dictated by the dimensions

of the diaphragm, and heavily depen-

dent on f requency, becoming sharper

and sharper as frequency increases. You

can solve this problem by using multiple

driving units and digital signal process-

ing, but a far simpler and cheaper way to

achieve predictable directivity control is

to use a horn. The walls of the horn will

restrict the spreading of the sound waves,

so that sound can be focused into the

areas where it is needed, and kept out of

areas where it is not.

Directivity control is most important

in sound reinforcement systems, where a

large audience should have the same dis-

tribution of low and high frequencies, and

where reverberation and reflections can be

a problem. In a studio or home environ-

ment, this is not as big a problem.

As the art and science of electroacous-

tics has developed, the focus has changed

from loading to directivity control. Most

modern horns offer directivity control at

the expense of driver loading, often pre-

senting the driver with a load full of reso-

nances and reflections. Figure 1 compares

the throat impedance of a typical constant

directivity horn (dashed line) with the

throat impedance of a tractrix horn (solid

line) 1 . The irregularities above 8kHz come

from higher order modes.

Initially, it is a three-dimensional prob-

lem, but solving the wave equation in 3D

is very complicated in all but the most

elementary cases. The wave equation for

three dimensions (in Cartesian coordi-

nates) looks like this 2

2 2

is insignificant.

2. The medium is considered to be a

uniform fluid. This is not the case with

air, but is permissible at the levels (see

1) and frequencies involved.

3. Viscosity and friction are neglected.

The equations involving these quanti-

ties are not easily solved in the case of

horns.

4. No external forces, such as gravity, act

on the medium.

5. The motion is assumed irrotational.

6. The walls of the horn are perfectly

rigid and smooth.

7. The pressure is uniform over the

wave-front. Webster originally consid-

ered tubes of infinitesimal cross-section,

and in this case propagation is by plane

waves. The horn equation does not re-

quire plane waves, as is often assumed.

But it requires the wave-front to be a

function of x alone. This, in turn, means

that the center of curvature of the wave-

fronts must not change. If this is the

case, the horn is said to admit one-pa-

rameter (1P) waves 6 , and according to

Putland 7 , the only horns that admit 1P

waves are the uniform tube, the para-

bolic horn with cylindrical wave-fronts,

and the conical horn. For other horns,

you need the horn radius to be small

compared to the wavelength.





(1)

2 2

∂ ∂ ∂ ∂



x y z



This equation describes how sound waves

of very small (infinitesimal) amplitudes

behave in a three-dimensional medium.

I will not discuss this equation, but only

note that it is not easily solved in the case

of horns.

In 1919, Webster 3 presented a solu-

tion to the problem by simplifying equa-

tion 1 from a three-dimensional to a

one-dimensional problem. He did this

by assuming that the sound energy was

uniformly distributed over a plane wave-

front perpendicular to the horn axis, and

considering only motion in the axial di-

rection. The result of these simplications

is the so-called “Webster’s Horn Equa-

tion,” which can be solved for a large

number of cases:

φ + −φ=

k 0

(2)

dx dx dx

π , the wave number or spatial fre-

quency (radians per meter),

φ is the velocity potential (see appendix

for explanation), and S is the cross-sec-

tional area of the horn as a function of x.

Because the horn equation is not able to

predict the interior and exterior sound

field for horns other than true 1P horns,

it has been much criticized. It has, how-

ever, been shown 8, 9 that the approxima-

tion is not as bad as you might think in

the first instance. Holland 10 has shown

that you can predict the performance of

horns of arbitrary shape by considering

the wave-front area expansion instead of

the physical cross-section of the horn. I

have also developed software based on

the same principles, and have been able

to predict the throat impedance of horns

with good accuracy.

The derivation of equation 2 is given

in the appendix. You can use this equa-

tion to predict what is going on inside

a horn, neglecting higher order effects,

but it can’t say anything about what is

going on outside the horn, so it can’t

predict directivity. Here are the assump-

tions equation 2 is based on 4,5 :

1. Infinitesimal amplitude: The sound

pressure amplitude is insignificant com-

pared to the steady air pressure. This

condition is easily satisfied for most

speech and music, but in high power

sound reinforcement, the sound pres-

sure at the throat of a horn can eas-

ily reach 150-170dB SPL. This article

takes a closer look at distortion in horns

due to the nonlinearity of air later, but

for now it is sufficient to note that the

distortion at home reproduction levels

FUNDAMENTAL THEORY

Horn theory, as it has been developed,

is based on a series of assumptions and

simplifications, but the resulting equa-

tions can still give useful information

about the behavior. I will review the as-

sumptions later, and discuss how well

they hold up in practice.

The problem of sound propagation in

horns is a complicated one, and has not

yet been rigorously solved analytically.

SOLUTIONS

This section presents the solution of

equation 2 for the most interesting horns,

and looks at the values for throat imped-

ance for the different types. You can cal-

culate this by solving the horn equation,

but this will not be done in full math-

ematical rigor in this article.

The solution of equation 2 can, in a

2 audioXpress 2008

www.audioXpress.com

 

∂φ ∂φ∂φ∂φ

− + + =

d d lnS d

where

k = 2 f

general way, be set up as a sum of two

functions u and v:

φ = Au + Bv (3)

where A and B represent the outgoing

(diverging) and reflected (converging)

wave, respectively, and u and v depend on

the specific type of horn.

In the case of an infinite horn, there is

no reflected wave, and B = 0. This article

first considers infinite horns, and pres-

ents the solutions for the most common

types 11 . The solutions are given in terms

of absolute acoustical impedance,

spherical coordinates. If you use spheri-

cal coordinates, the cross-sectional area

of the spherical wave-front in an axi-

symmetric conical horn is S = Ω(x+x 0 ) 2 ,

where x 0 is the distance from the vertex

to the throat, and Ω is the solid angle of

the cone. If you know the half angle θ

(wall tangent angle) of the cone,

Ω = 2π(1 - cosθ). (5)

In the case where you are interested in

calculating the plane cross-sectional area

at a distance x from the throat,

satisfying performance at low frequen-

cies. As such, the conical expansion is

not very useful in bass horns. Indeed,

the conical horn is not very useful at all

in applications requiring good loading

performance, but it has certain virtues in

directivity control.

EXPONENTIAL HORN

Imagine you have two pipes of unequal

cross-sectional areas S 0 and S 2 , joined by

a third segment of cross-sectional area

S 1 , as in Fig. 3 . At each of the disconti-

nuities, there will be reflections, and the

total reflection of a wave passing from S 0

to S 2 will depend on S 1 . It can be shown

that the condition for least reflection oc-

curs when

S 1 = √S 0 S 2 . (8)

This means that S 1 = S 0 k and S 2 = S 1 k,

thus S 2 = S 0 k 2 . Further expansion

along this line gives for the nth

segment, S n = S 0 k n , given that each

segment has the same length. If k

= e m , and n is replaced by x, you

have the exponential horn, where

the cross-sectional area of the

wave-fronts is given as S = S t e mx .

If you assume plane wave-fronts,

this is also the cross-sectional area

of the horn at a distance x from

the throat.

The exponential horn is not a

true 1P-horn, so you cannot ex-

actly predict its performance. But

much information can be gained

from the equations.

The throat impedance of an in-

finite exponential horn is

+ 

=   

x x

( )

;

S xS

(6)

you can find the specific throat imped-

ance (impedance per unit area) by mul-

tiplying by S t , the throat area, and the

normalized throat resistance by multiply-

ing by

The throat impedance of an infinite con-

ical horn is

c 2 2

 

ρ +

=   

+ 

k x jkx

(7)

S 1kx

2 2

THE PIPE AND THE

PARABOLIC HORN

Both these horns are true 1P

horns. The infinite pipe of uni-

form cross-section acts as a pure

resistance equal to

(4)

An infinite, uniform pipe does not

sound very useful. But a suitably

damped, long pipe (plane wave

tube) closely approximates the re-

sistive load impedance of an in-

finite pipe across a wide band of

frequencies, and is very valuable

for testing compression drivers 12,

13 . It presents a constant frequency

independent load, and as such acts like

the perfect horn.

The parabolic horn is a true 1P horn

if it is rectangular with two parallel sides,

the two other sides expanding linearly,

and the wave-fronts are concentric cylin-

ders. It has an area expansion given as

S = S t x. The expression for throat im-

pedance is very complicated, and will not

be given here.

The throat impedances for both the

uniform pipe and the parabolic horn are

given in Fig. 2 . Note that the pipe has the

best, and the parabolic horn the worst,

loading performance of all horns shown.

FIGURE 2: Throat acoustical resistance r A and reac-

tance x A as a function of frequency for different horn

types.



m m



You should note that equation 7 is iden-

tical to the expression for the radiation

impedance of a pulsating sphere of radius

x 0 .

= − +





(9)







, the throat

resistance becomes zero, and the horn

is said to cut off. Below this frequency,

The throat resistance of the conical

horn rises slowly ( Fig. 2 ). At what fre-

quency it reaches its asymptotic value de-

pends on the solid angle Ω, being lower

for smaller solid angle. This means that

for good loading at low frequencies, the

horn must open up slowly.

As you will see later, a certain mini-

mum mouth area is required to mini-

mize reflections at the open end. This

area is larger for horns intended for low

frequency use (it depends on the wave-

length), which means that a conical horn

would need to be very long to provide

CONICAL HORN

The conical horn is a true 1P horn in

FIGURE 3: Joined pipe segments.

audioXpress 2008 3

When m = 2k or f = mc

4 π

the throat impedance is entirely reactive

and is

Above cutoff, the throat impedance of

an infinite hyperbolic horn is

ance is purely reactive. But what happens

at this frequency? What separates the ex-

ponential and hyperbolic horns from the

conical horn that does not have a cutoff

frequency?

To understand this, you first must

look at the difference between plane and

spherical waves 10 . A plane wave propa-

gating in a uniform tube will not have

any expansion of the wave-front. The

normalized acoustical impedance is uni-

form and equal to unity through the en-

tire tube.

A propagating spherical wave, on the

other hand, has an acoustical impedance

that changes with frequency and distance

f rom the source. At low f requencies

and small radii, the acoustical imped-

ance is dominated by reactance. When

kr = 1—i.e., when the distance from the

source is



m m







z j

= − −





(10)

−





S 2k 4k











, (12)





1 T

− −

1 T

The throat impedance of an exponential

horn is shown in Fig. 2 . Above the cut-

off frequency, the throat resistance rises

quickly, and the horn starts to load the

driver at a much lower frequency than

the corresponding conical horn. In the

case shown, the exponential horn throat

resistance reaches 80% of its final value

at 270Hz, while the conical horn reaches

the same value at about 1200Hz.

An infinite horn will not transmit

anything below cutoff, but it’s a different

matter with a finite horn, as you will see

later.

You should note that for an exponen-

tial horn to be a real exponential horn,

the wave-front areas, not the cross-sec-

tional areas, should increase exponential-

ly. Because the wave-fronts are curved,

as will be shown later, the physical horn

contour must be corrected to account for

this.



− −









µ µ



and below cutoff, the throat impedance is

entirely reactive and is

 

 

µ µ

1 1

(13)

ρ  

z j

 

−  

1 T

−  

 

where

µ is the normalized frequency, µ=

The throat impedance of a hypex horn

with T = 0.5 is shown in Fig. 2 . The

throat impedance of a family of horns

with T ranging from 0 to 5 is shown in

Fig. 5 .

Exponential and hyperbolic horns have

much slower flare close to the throat than

the conical horn, and thus have much

better low frequency loading. When T <

1, the throat resistance of the hyperbolic

horn rises faster to its asymptotic value

than the exponential, and for T < √2 it

rises above this value right above cutoff.

The range 0.5 < T < 1 is most useful

when the purpose is to improve loading.

When T = 0, there

is no reactance com-

ponent above cutoff

for an infinite horn,

but the large peak in

the throat resistance

may cause peaks in

the SPL response of

a horn speaker.

Due to the slower

flaring close to the

throat, horns with

low values of T will

also have some-

what higher distor-

tion than horns with

higher T values.

—the reactive and resistive

parts of the impedance are equal, and

above this frequency, resistance domi-

nates.

The difference between the two cases

is that the air particles in the spherical

wave move apart as the wave propa-

gates; the wave-front becomes stretched.

This introduces reactance into the sys-

tem, because you have two components

in the propagating wave: the pressure

that propagates outward, and the pres-

HYPERBOLIC HORNS

The hyperbolic horns, also called hyper-

bolic-exponential or hypex horns, were

first presented by Salmon 14 , and is a gen-

eral family of horns given by the wave-

front area expansion





S S cosh Tsinh





(11)

T is a parameter that sets the shape of

the horn ( Fig. 4 ). For T = 1, the horn is

an exponential horn, and for T → ∝ the

horn becomes a conical horn.

x 0 is the reference distance given as x o =





2 π where f c is the cutoff frequency.

A representative selection of hypex con-

tours is shown in Fig. 4 .

FIGURE 4: A family of representative

hypex contours, T = 0 (lower curve), 0.5,

1, 2, 5, and infinite (upper curve).

WHAT IS CUTOFF?

Both exponential

and hyperbolic horns

have a property called

cutoff. Below this

frequency, the horn

transmits nothing,

and its throat imped-

FIGURE 5: Throat impedance of a family of hypex horns.

4 audioXpress 2008

www.audioXpress.com

= +

sure that stretches the wave-front. The

propagating pressure is the same as in

the non-expanding plane wave, and gives

the resistive component of the imped-

ance. The stretching pressure

steals energy from the propagat-

ing wave and stores it, introduc-

ing a reactive component where

no power is dissipated. You can

say that below kr = 1, there is re-

actively dominated propagation,

and above kr = 1 there is resistive

dominated propagation.

If you apply this concept to the

conical and exponential horns by

looking at how the wave-fronts

expand in these two horns, you

will see why the cutoff phenom-

enon occurs in the exponential

horn. You must consider the flare

rate of the horn, which is defined

as (rate of change of wave-front

area with distance)/(wave-front

area).

In a conical horn, the flare rate

changes throughout the horn,

and the point where propagation

changes from reactive to resistive

changes with frequency through-

out the horn.

In an exponential horn, the

flare rate is constant. Here the

transition from reactive to resis-

tive wave propagation happens at

the same frequency throughout

the entire horn. This is the cutoff

frequency. There is no gradual

transition, no frequency depen-

dent change in propagation type,

and that’s why the change is so

abrupt.

−

= − (16)

where Z m is the terminating impedance

at the mouth.

gZ b

The expressions for a, b, f, and g are

quite complicated, and are given by

Stewart 15 for the uniform tube, the coni-

cal, and the exponential horn.

You see that the value of mouth

impedance will dictate the value

of the throat impedance. As ex-

plained previously, there will usu-

ally be reflections at the mouth,

and depending on the phase and

magnitude of the reflected wave,

it may increase or decrease the

throat impedance. A horn with

strong reflections will have large

variations in throat impedance.

Reflections also imply stand-

ing waves and resonance. To avoid

this, it is important to terminate

the horn correctly, so that reflec-

tions are minimized. This will be

discussed in the next section.

It’s interesting to see what ef-

fect the length has on the perfor-

mance of a horn. Figure 6 shows

the throat impedance of 75Hz

exponential horns of different

lengths, but the same mouth size.

As the horn length increases, the

throat resistance rises faster to a

useful value, and the peaks in the

throat impedance become more

closely spaced.

Finite horns will transmit

sound below their cutoff fre-

quency. This can be explained as

follows: the horn is an acousti-

cal transformer, transforming the

high impedance at the throat to

a low impedance at the mouth.

But this applies only above cutoff.

Below cutoff there is no trans-

former action, and the horn only

adds a mass reactance.

An infinite exponential horn

can be viewed as a finite exponen-

tial horn terminated by an infinite

one with the same cutoff. As you

have seen, the throat resistance

of an infinite exponential horn is

zero below cutoff, and the throat

resistance of the finite horn will

thus be zero. But if the imped-

ance present at the mouth has a

non-zero resistance below cutoff,

a resistance will be present at the

throat. This is illustrated in Fig. 7 ,

where a small exponential horn

with a mouth three times larger

a fZ

FIGURE 6: The effect of increasing the length of a 75Hz

exponential horn with krm = 0.5. The lengths are (top to

bottom) 50, 100, and 200cm.

FINITE HORNS

For a finite horn, you must con-

sider both parts of equation 3.

By solving the horn equation this

way 3, 15 , you get the following

results for pressure and volume

velocity at the ends of a horn:

p m = ap t + bU t (14)

U m = fp t + gU t (15)

where p and U denote the pres-

sure and volume velocity, respec-

tively, and the subscripts denote

the throat and mouth of the horn.

You can now find the impedance

at the throat of a horn, given that

you know a, b, f, and g:

FIGURE 7: Finite exponential horn terminated by an

infinite pipe.

audioXpress 2008 5

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: