Dept. for Speech, Music and Hearing
Quarterly Progress and
Status Report
On the dimensioning of
bass-reflex enclosures
Liljencrants, J.
journal:
volume:
number:
year:
pages:
STL-QPSR
7
3
1966
007-022
http://www.speech.kth.se/qpsr
A.
O N TH.E E ~ ~ v i E I ; T S , T C
O >L 3ASS
~
- P a F L E X ENC:LOSURES
J. L i l j e n c r a n t s
Abstract
---
A t h e c ~ e t i c 2 : a n a l y s i s shows that the b a s s - r e f l e x loudspeaker
s y s t e m exhibits t h e characterin",ics of a minimum-phase highpass f i l t e r .
It is t h u s possible t o apply the well-.known Butterworth and Chebyshev
approximations t o ideal filter:: on t l ~ elcudspzalcer s y s t c m dimensioning.
A number of such dimensionings z r c examined f o r t h e i r implications on
frequency and t r a n s i e n t Tesponoe, !.oudspeakcr diaphragm e x c u r s i o n and
e n c l o s u r e volume. A method of finding the phyoicnl conditions is given
t h a t will implement a n a r b i t r a r y s y s t e m behzvior s ~ e c i f i e di n t e r m s of
the poles of the t r a n s f e r function.
Introduction
--.-
-
--
Since the b a s s - r e f l e x e n c l o s u r e w a s invented by T h u r a s (5) i n 1930
a number ~f r e c o r n m e n d a t i o ~ shave clrcu!-ated i n the l i t e r a t u r e t o g e t h e r
with indications of pnssihie pe r i o r m a n c e improvements, A b a s s - r e f l e x
s y s t e m h a s a considerab?e xiumber of d e g r e e s of f r e e d o m , It is thus
n a t u r a l t h a t the a n a l y s i s mostly is confined t o s p e c i a l c a s e s , T h i s m a y
imply that i n t e r e s t i n g phe-omcna a r e overlooked, Often it i s a l s o difficult t o s e e the influence of different paraxlzeters on t h e s y s t e m p e r formance,
The p r o b l e m of s y s t e m optimization w a s t r e a t e d by Iieibs (2) who
a l s o gave a commented review of the l i t e r a t u r e , T3e optimal solution
d i s c u s s e d below is r c t v e r y different f r o m t52t of Keibs, but s o m e m o r e
of the p r a c t i c a l l y interesting p3rarneters a r e kept i n the t r e a t m e n t , We
will a l s o u s e the terrrinology of I-nnc?e;.n c i r c c i t t h e o r y a n d s e e how wellknown ideal f i l t e r approximztions m a y he applied t o t h e problem,
A s sumpti~xn~g
To,- the trttztmct?:: of tl1f.s probl-em we of c c u r s e r e s t r i c t o u r s e l v e s
t o the l o ~ ~ ~ - f r e q u eranee,
i ~ c y The loudspeaker cone is t h u s a s s u m e d t o
mp-re 2-s a y-igjd p i a t ~ nan4 the vlhnle s y s t e m is r e g a r d e d as a Feint s o u r c e
mech~.r.!.ca!3Smens,i.o,?r z;.e assumed t o Ee s m a f l
o r t i n ? i i e r wc=.c!n, f:l,.?,
compared t~ the sound .tvz7.~e
length,
F o r the analysis we u s e the impedance analogy of Fig. 111-A-1.
All the elements a r e t r a n s f e r r e d t o acoustical quantities. The detailed
derivation of the analogy is omitted since it is easily found i n the handbook literature, s e e e. g. Beranek (1)
.
The acoustical elements of the analogy have the following meaning:
-
e B1
(Rg+Re)sd
R1
--
+Rr
( ~ 1 ) ~+ R
is the total acoustical resistance of
s
the amplifier and the loudspeaker.
It consists of the reduced electrical
resistance, the acoustical resistance of the
diaphragm suspension Rs, and t h e radiation
resistance Rr f r o m both s i d e s of the diaphragm.
2
hil= m l / s d
2
C1 = clSd
is the acoustical m a s s of the loudspeaker
element including the air load, ml is the c o r r e sponding mechanical mass.
is the acoustical compliance of the diaphragm
suspension and cl is its mechanical compliance.
is the acoustical resistance of the box and
Rb
C2 =
is the p r e s s u r e f r o m the generator/amplifier
having a n electromotive force e and a n internal
electrical resistance R
B, 1, and Re a r e the
g'
field strength of the speaker magnet, and the
length and electrical resiotance of the voice
coil. S is the effective a r e a of the diaphragm.
d
v/yL
is its acoustical compliance. V is i t s volume
and P O and c a r e the density and speed of sound
of the a i r .
is the acoustical m a s s of the a i r in the bassreflex port, and finally
is the acoustical resistance of the port including
radiation.
The e l e c t r i c a l inductance of the voice coil may be neglected in
the frequency range of interest.
<
\A
\
-/
\
e
\
Generator
Loudspeaker element
Fig. 111-A-1.
>
C\
Box
Bass reflex port
Analogy of a bass reflex system. All elements are
transformed to acoustical quantities (MKSA).
Derivation of the fmquency response of the system
We make use of complex frequency
normalized to w o which is defined a s the geometric mean t o the
eigenfrequencie s of the loudspeaker and the box resonator:
Vle also use the following loss factors:
Loudspeaker
element:
a = l/Gl = w1R1C1 = R 1/ 9M 1
Box:
hp= R
Port:
v = 1
Total resonat o r :
b=
~ / M2
w ~
l/a2 = u + v
The acoustical resistances a r e not well behaved.
The value of
the resistance i o of importance mainly at frequencies in the vicinity of
the resonance of the pertinent system, ViTe assume these resonance
values of the resistances in the formulas a s constants, since the e r r o r s
at other frequencies a r e negligible.
To understand this one may recall
that a normally used loudspeaker element is m a s s controlled in the
frequency range just above the low-frequency cutoff.
Finally we introduce a couple of auxiliary quantities:
and
g tells us t o which extent the tuning of the loudspeaker is different
from that of the box resonator, and h will relate the box volume t o the
loudspeaker compliance.
With these normalized quantities we get the following expressions
for the impedances of the three branches of the analog diagram:
Under the point source assumption the sound pressure p a t the
distance r will be
The two f i r s t factors imply that the p r e s s u r e is proportional t o the time
derivative of the volume velocity (U ) out of the box while the r e s t of the
b
formula gives the distance dependence which is of no concern a t present.
The box volume velocity is easily obtained directly from the analog
diagram a s
The insertion of (8) and (6) into (7) will render the deoired e x p r e ~ o i o nof
sound p r e s s u r e as a function of frequency:
where P(x) stands for the fourth degree polynomial
4
P(x)= x
+ k 3x3 + k2x2 + klx + 1
with the coefficients
On t h e f i r s t p a r t of the problem t h e r e is room f o r a number of
subjective and objective viewpoints, and a c l o s e r discussion m a y be
justified.
VJe look a t the following:
1.
Frequency response. It is d e s i r a b l e with a n even shape of the
response curve. It i s a l s o well-known that the frequency range
of the s y s t e m m a y be extended below the loudspeaker resonance.
2.
Power handling capacity. The c l a s s i c a l advantage of the b a s s reflex enclosure over other s y s t e m s is that the diaphragm
excursions a r e diminished in the vicinity of the cutoff frequency.
T h i s will give a lower distortion at high output power,
3.
T r a n s i e n t response. The resonances of the s y s t e m should be
sufficiently damped. The c o r r e spondence of f IEquency and
t i m e responses conveyed by the Eaplace t r a n s f o r m should be
borne i n mind.
4.
Size. F r o m p r a c t i c a l r e a s o n s a s m a l l volume is advantageous,
Frequency response
The response (12) i s that of a minimum-phase cyotem, This m a k e s
it possible t o u s e the c l a s s i c a l approximations t o ideal f i l t e r s . Nearest
at hand a r e the Butterworth approximation (maximally flat) and the
Chebyshev (constant ripple i n the p a s s band).
O u r highpass function h a s a quadruple z e r o i n the origin and two
conjugate p a i r s of poles, the roots of P(x) = 0. One such p a i r will
together with a dual z e r o in the origin give r i s e t o a n elementary highp a s s resonance curve. A family of such c u r v e s is shown in Fig.
111-A-2.
T o go f r o m a known lowpass pole configuration t o the c o r r e
-
sponding dimensioning p a r a m e t e r s we d o a s follows. F i r s t the poles
a r e normalized.
This means that we introduce a scale f a c t o r r e n d e r -
ing the pole product a unity magnitu.de, The lowpass function will then
be
R is the magnitude of a pole in one of the conjugate p a i r s having the
l o s s f a c t o r d l , and 1/R and d2 are corre sponding f o r the o t h e r pair.
If we execute a lowpass/highpass t r a n s f o r m a t i o n by m e a n s of the
frequency substitution X = 1/x we get the analogous highpass function
Fig. 111-A-2.
Normalized highpase resonance curves.
The insert illustrates the definition of
Q and loss factor.
The only difference a s compared t o the lowpass function is the
quadruple z e r o and that the polynomial coefficients change order.
An identification with (1 1) will give
= kl = gb
+
a/g
+
ghv
The left m e m b e r s of the equations a r e thus given quantities and the
dimensioning p a r a m e t e r s a r e found by solving the system. This is
-
of course r a t h e r intricate t o d o i n a general manner.
that we have five independent variables (since b
u
VJe observe
+
v) and two
of them can thus be chosen with a certain degree of freedom.
T h e r e is a duality between some t e r m s in the left and right
m e m b e r s of (1 7), the similarity in position of the f a c t o r s K
dl
- b,
- g,
and d 2 - a being apparent. This might provoke a temptation
t o put some entirely unjustified equality signs h e r e and thus identify
a "speaker resonance" and a "box resonance" in the compound system.
In reality such a n identification would not give any non-trivial solution t o (1 7).
Table 111-A-1 l i s t s the c h a r a c t e r i ~ t i c oof some different c a s e s
t o compare. The f i r s t alternatives a r e the Butterworth and a
Chebyshev (see a l s o Fig. 111-A-3) and the case quoted by Keibs
a s optimal.
T h e r e is a l s o one case where one of the pole p a i r s h a s
been given a v e r y l a r g e l o s s factor making the system resistance
controlled in the low frequency range.
then have a
+ 6 d ~ / o c t a v eslope.
The frequency response will
F o r a wider comparison we finally
examine the special c a s e s with a n infinite baffle and a closed box.
In these c a s e s the loudspeaker resonance frequency is used f o r the
normalizing, g = 1, and the l o s s factor is s e t t o unity giving a
reasonably flat response.
The acoustical compliance of the closed
box is chosen equal t o that sf the loudspeaker element, h = 1.
1
2
Bass reflex
Chebyshev
B a s s reflex
Butterworth
0.131
4
0.061
0.678
0.681
0.180
110. 893
0.317
0.604
0.680
1.300
1.469
0.383
0.925
0.925
0.383
1
0. 383
0.925
1
0.766
Bass reflex
Optimal acc.
t o Keibs
0.417
0.909
1
1.2
0.417
0.909
1
1.2
Bass reflex
Highly d a m p e d
:1
1.469
3
0. 34
3
0. 59
0.681
1
0. 383
1
.400
.50
5
Infinite baffle
Loss factor = 1
0.5
0.866
1
1
1
6
Closed box
Loss factor = 1
h = 1
0.707
1.225
1.414
1
1.414
A
T a b l e 111-A-1.
2.783
1.29
0.31
1.15
0
0
1.365
0.148
1.02
0.05
0.05
1.41
2.62
0
0
1
0.94
2.31
0. 1
0. 1
1
1.44
2.40
0
0
1
1.01
2.1
0.1
0.1
1.21
2.33
4.24
0
0
1.23
1.69
3.75
0.1
0.1
-
-
0
1
-
-
-
-
1
1
-
1.482
0.925
2.613
0.925
3
1.462
3.414
3.440
5.52
-
2.613
.
2.400
5.09
i
C h a r a c t e r i s t i c s of t h e d i f f e r e n t s y s t e m s considered.
.
Fig. 111-A-3. a. The poles of a lowpass Butterworth s y s t e m a r e equally
spaced on a s e m i c i r c l e . If the r e a l p a r t s of the poles
a r e scaled down with the f a c t o r aT/cug the poles will l i e
on a n e l l i p s e and the s y s t e m will get a Chebyshev behavior.
b. A normalizing of the pole product will give a root locus
f o r the lowpass Chebyshev c a s e ( - - - - - - - - ). A reflexion
i n the unit c i r c l e will r e n d e r the corresponding locus f o r
the highpass c a s e (
). The LP/HP t r a n s f o r m a t i o n
a l s o gives a quadruple z e r o i n the origin.
The scaling p a r a m e t e r is a r b i t r a r i l y chosen f r o m p r a c t i c a l
reasons. Two c a s e s of Table 111-A-1 a r e indicated with dots.
In analogy with (12 ) and (14) the closed box will have
with the i d i n i t e baffle h
04
Fig, 111-A-3 shows the loci of the poles for normalized
~ h e b ~ s h econfigurationst
v
It is seen here that the cutoff frequency
is lowered when the loss factoi. is decreasedr This means that a
m o r e extreme Chebyshev design should provide a n extended frequency
range.
If one on top of this can find a design with g > 1 the gain
will be even l a r g e r with given cul.
The pole configurations and frequency response curves for
the different alternatives of Table III-A-1 a r e shown in Figs.
1114-4 and III-A-5.
In Figs. III-A-5
- III-k-7
the frequency
scale has been r e -normalized t o the resonance frequency of the
loudspeaker. This will make the curves bring out the performance
of the different box design types with reepect t o one and the same
loudspeaker element. To make the re-scaling possible the value of
g must be known which implies that the Eqs (17) have t o be solved
first. The actual solutions used a r e indicated in Table III-A-1 and
could be regarded a s near optimal.
If some special frequency response is wanted it can be constructed by adding two template curves of Fig. III-A-2.
The nor-
malizing will then involve that the unit frequency of these templates
should be held with geometric symmetry around the normalizing
frequency. The displacement will then determine R in Eq (1 6 ) .
When appropriate values of R and the 10s s factors have been
chosen they can be inserted into the left member of Eq
(17). It
should then be remembered that the poles of magnitude R have t h
in the high-pass case, Eq (16), a s opposite t o the
2
low-paee caee of Eq (1 5.j.
l o s s factor d
Fig. 111-A-4.
~ o l e / z e r od i a g r a m s f o r the different s y s t e m t y p e s of
Table 111-A-1.
~
Frequency responses for the alternatives of
i 111-A-5.
~ .
Table 111-A-1. The frequency scale i s normalized to the resonance frequency iol of the loudspeaker element.
Fig. 111-A-6.
-
Diaphragm excursions for the alternatives of Table 111-A 1 .
Fig. 111-A-7.
Step responses for the alternatives of Table 111-A-1.
The time scale i s normalized t o the resonance
frequency of the loudspeaker element.
Diaphragm excursion
The variation of diaphragm excursions with frequency differs from
the sound p r e s s u r e response in a 12 d ~ / o c t a v eslope and an antiresonance.
This is directly seen from a comparison between Eqs (12) and (14). It
is also seen that the frequency and loss factor of the antiresonance depend
only of the properties of the box and port.
The diaphragm excursions
for the different alternatives a r e shown in Fig. 111-A-6.
It is now directly seen that the combined losses in the box and port
have t o be rather small if the antiresonance is t o have any significance
justifies the assumption above that
for the diaphragm excursions. T h i ~
v / ~may be neglected in the numerator of Eq (9) which will give r i s e t o
Eq (121,
The electrical input impedance of the system is the sum of the
essentially constant (at low frequencies) blocked impedance and the
motional impedance. The variation with frequency of the motional
impedance is identical in shape with the diaphragm velocity which in
t u r n is obtained from the curves of Fig. 111-A-6 by superimposing a
tilt of t 6 d ~ / o c t a v e . It i s now easily understood that the two peaks in
the impedance curve d o not give a very useful information on the system.
The interesting thing is rather the dip between them that tells the resonance frequency w
2
and loss factor of the box resonator.
Transient properties
A complete analysis of the transient behavior of the system i s
rather involved both theoretically ancl practically. The following discussion will thus only briefly review the general method and we then
proceed t o some experimental measurements.
In general, when a system with the t r a n s f e r function H(G) is excited
with a signal having the Laplace transform G(s), then the transform of
the output will be
The response in the time domain will then be the inverse transformation
of this, o r
To study the transient behavior of the system only, one should use a
signal with a transform G(s) a s simple as possible. The natural choice
would then be the unit impulse having the transform 1. The impulse
response is then
A practical problem when this is applied t o a highpass system a s our
loudspeakers is that the infinitely high unit impulse is directly t r a n s
mitted and then followed by the interesting part of the response.
-
The
unit step has the transform G(s) = l/s, o r with normalized frequency
G(x) = l/x.
If we disregard a proportionality constant the transform of
the 8tep response of our system will be
An inverse transformation will give the general form of the step response
as
o and w a r e the real and imaginary p a r t s of the poles, and A and cp a r e
the magnitudes and phases of the residues
. The impulse response
d i f f e r s from this in the values of the residues and a n additional t e r m
c?-(t ), the directly transmitted input pulse.
F o r a closer treatment of the Laplace transform and especially
the residue calculus the reader is referred t o the abundant handbook
literature on the topic (see for instance (3)).
In practice the residues a r e preferably calculated by graphical
methods from a plot of the poles in the frequency plane. An alternative
and often e a s i e r method is however t o build an electrical analog t o the
system and directly record its performance. Fig. 111-A-7 shows the
results of such a simulation of the system t r a n s f e r functions described
in Table 1114-1. (See
7'k
next page. ) In the same diagram there is a
time scale valid f o r the fairly representative case with wl = 2n
. 50
radians/second.
The most striking example is the Chebyshev case which now clearly
shows the price of the extended frequency range. The system has a very
weakly damped resonance near the cutoff frequency which accounts for a
long transient time.
If we look at the Butterworth case and the system
It i s usual that the transient properties of a system is tested
with tone b u r s t s a s the excitation signal. The output will then consist
of a stationary part being the input modified t o phase and amplitude
by the t r a n s f e r function H(s). It a l s o contains a transient part a t the
beginning and end of the bone burst. If we assume a sinusoidal input
of frequency w
g
the t r a n s f o r m G(s) in Eq (20) will be
The frequencies and l o s s factors of the components in the transient
response a r e the s a m e a s in the s t e p response (24) but the residues
get other values since (25) will enter a s a factor in the calculation of
these. The transient response contains no components of frequency
These a r e instead restricted t o the stationary response. The difg'
ferent appearances of the transient response when w i s varied h a s
g
nothing t o do with the system but instead reflects the influence of Eq
w
(25) on the residues.
The main advantage of the tone burst method is r a t h e r that it is
possible t o excite parasitic resonances m o r e o r l e s s selectively. T o
make the picture complete however the tone burst response f o r a
Butterworth system i s shown in Fig. 111-A-8.
Resonator volume
The second Eq (1 7) shows that maximal values of h should be
expected with u = v = b = 0, The corresponding values of the other
p a r a m e t e r s a r e given in Table 111-A-1. In practice it i s however
impossible t o have the resonator completely undamped, it is not even
desirable with respect t o the detrimental effects of standing waves.
The
table therefore a l s o l i s t s the m o r e practical solutions having u = v = 0.1.
The Chebyshev case i s a n exception in that it h a s no physical solution
for this magnitude of the l o s s factors. Instead a solution is given with
u = v = 0.05,
F o r a given loudspeaker element the box volume is inversely
proportional t o h. r"+ study of Table 111-A-1 reveals then that the case
with a heavily damped s y s t e m can give a box volume that is about 60 O/o
of that f o r the Butterworth c a s e . R e a contrast the minimum practical
volume f o r the exemplified Chebyshev design is about 7 t i m e s a s l a r g e
as for Butterworth,
Fig. 111-A-8.
T i m e r e s p o n s e of a fourth d e g r e e Butterworth s y s t e m
t o excitation with a sinusoid and a cosinusoid of frequency
WO.
The s t a t i o n a r y p a r t of the r e s p o n s e i s attenuated
3 dB and phase shifted 180 d e g r e e s . The high frequency
components of the s t e p in the cosinusoid a r e however
t r a n s m i t t e d without attenuation and phase shift.
Fig. 111-A-9.
Time response of a fourth d e g r e e Butterworth s y s t e m
t o excitation with a sinusoid of varying phase. The
excitation frequencies a r e iuo and 2 w 0 respectively. Only
the envelopes of the responses a r e shown.
Conclusions
The Butterworth behavior could be deemed a s the optimum in the
light of the previous discussion. The frequency range is not extended
a s compared t o the closed box of equal volume, but the diaphragm
stiffening effect i s fully utilized in the same time as the transient
time as a rule i s sufficiently short.
Apart from this the heavily damped system is Prominent mostly
because of its small volume, but its frequency response has t o be
corrected somewhere else in the total s y s t e m
Design
When system behavior has been decided on the left hand members
of (17) a r e given. The problem is then t o solve this equation system t o
find the parameter values.
Fig. 111-A-1 0 shows the result of a n i t e r a -
tive solution process that was executed on a PDP-7 computer. The loss
factors a r e given as functions of the box volume parameter h and the
tuning parameter g. The diagram directly indicates within which limits
of the various parameters the desired behavior may be realized. It i s
f o r instance seen that a s m a l l e r box necessitates smaller loss factors
in the box and port in the same time a s the demand on proper tuning of
the resonator becomes more proxxinent. It is thus not practical t o make
the box a s small a s the theoretical minimum.
On the other hand, if the
box is too large the resonator losses have t o be s o high that the antiresonance in the diaphragm excursion characteristic i s flattened out.
The essential point i s thus that for a given loudspeaker element
and system behavior the r e i s a definite optimal box volume, namely
the smallest practically possible,
When the parameters of Fig. 111-A-10 have been chosen we can
proceed t o find the physical dimensions of the resonator with the aid
of some properties of the speaker element.
The ones most easily
measured a r e w l , m l , and Sd. The resonance frequency i s determined
f r o m the maximum of the electrical impedance. The loudspeaker should
be baffled during this measurement s o that the a i r load is included in
the oscillatory system. After this the diaphragm is loaded with a known
and a new resonance frequency w is found. W e then get the
3
3
mechanical m a s s a s
mass m
The effective diaphragm a r e a i s approximately
is the mean diameter of the compliant edge suspension of the
d
diaphragm,
where d
After some manipulating with the definition equations we a r r i v e
a t the box volume
In passing we note that for a given resonance frequency the box will
be s m a l l e r f o r a smaller diaphragm a r e a and a l a r g e r mass.
These
quantities also have a direct influence on the loudspeaker efficiency.
If the bass -reflex port is of cylindrical shape the classical
theory yields
biz = p 0 l 0 / s
(29)
P
where S
r , and 1 ' a r e the a r e a , radius, and effective length of
P' P
the port, The interdependence of a r e a and radius makes
which shows that we cannot control the acoustical resistance of the
port merely by giving it certain dimensions, After appropriate substitutions we find the loss factor of the port as
-,
In general this value i s r a t h e r small. If one for some reason wants
a higher loss factor this has t o be accomplished with some kind of
damping m a t e r i a l in the port.
The conjunction of effective length and a r e a is given by E q (29).
The physical length of the tunnel is obtained by subtraction of the end
corrections. If we a s s u m e both ends t o be baffled the physical length
will be
1 = 1'
-
2
0.85
dp/2
(33)
which eventually will give the final formula
which i s shown graphically i n Fig. 111-A-1 1.
The a r e a of the port is i r r e l e v a n t t o the function of the s y s t e m
as long as i t s length i s c o r r e c t l y chosen i n correspondence t o the a r e a .
The most suitable p r a c t i c a l solution will thus be t o choose a tunnel
length equal t o the wall thickness of the box and then match the a r e a
t o this.
The following is a possible way of determining the l o s s f a c t o r s of
the box and the port.
First a microphone is mounted inside the box and
then the mounting hole f o r the loudspeaker is closed tightly. If the
r e s o n a t o r is now excited with a n external loudspeaker the microphone
output will give information on the resonance frequency w2 a s well a s
t h e 3 d B relative bandwidth which i s u
+ v'.
F i r s t the port a r e a is
adjusted t o give the c o r r e c t resonance and then the box is filled with
damping m a t e r i a l until the appropriate value of u is obtained.
If nec-
e s s a r y the port is then damped t o give a relative resonance bandwidth
u
+ v.
If the resonance is badly pronounced before the box is damped
t h i s is a good indicator that the wall construction is too weak o r a l ternatively that t h e r e a r e leaks in the box.
Finally the loudspeaker element is mounted and the l o s s f a c t o r a
is adjusted with damping m a t e r i a l around the loudspeaker basket The
c o r r e c t value of a is m o s t easily found by checking that the frequency
response is the desired.
Example
On a s m a l l loudspeaker element the following values w e r e d e t e rmined:
Fig. 111-A-11. Dimensioning diagram for the bass reflex port.
The port i s assumed t o be circular of diameter d
P'
and axial length 1. The curves show 1 a s
area S
?'
a function of dp. The parameter of the curves i s
1
-
Po
hg
4
in MKSA units.
Sd
Fig. 111-A-12. a. Step r e s p o n s e f o r a n e x p e r i m e n t a l b a s s reflex cabinet
designed f o r Butterworth behavior. The fine s t r u c t u r e
i s due t o p a r a s i t i c r e s o n a n c e s and reflexions.
b. F r e q u e n c y r e s p o n s e of the s a m e s y s t e m m e a s u r e d i n
a n anechoic c h a m b e r a t two different angles t o the
loudspeaker axis.
In the low frequency range it i s s e e n t h a t the a s s u m p t i o n
of a point s o u r c e i s justified.
The r e s p o n s e a t frequencies above 2.000 c / s i s due t o a
p a i r of t r e b l e loudspeakers mounted closely t o the b a s s
loudspeaker.