criterio de bolt
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Ili1' J()I•I,•NAi, Of •llIP
Note on Normal Frequency Statistics for Rectangular Rooms
R. [|. B()LT
.[tOltslit;.• aborat.ry, M•a.•-sachusellsn.•tilute • 'l'echnology, ambr•d•e,Ma.ssachuscfl•
(Rcccive{I May 21, 1946)
{ I.lX)\VI(;t )rclin•i•aryet)(•rt
requent3 spacing statistics, some of the
n•athemati('al results have been reduced to
charts. A detailed paper on this subject is in
preparation,but certain resultsare presentedat
this time with the thought that they may be of
assistance o others ('()neemealwith properties of
normal frequencies.
Figure 1 may be used o detern•ine he average
number N of normal frequencies, up to a fre-
quency u in a rectangular room of volume V
having dimensionsL, pL, qL. The length L is
arbitrarily specified s the longestdimensionof
the room. The dimension atios p and q are equal
to or less than unity, both being equal to unity
for a (-ube.
()he first uses he insert chart in the upper left
corner to obtain the room pr(•t•(•rti(m factor P
fr()m 1t•(' rati()s p and q. 'Fht,n one selects,
interpolates, the appropriate curve
('orrest)ondingo the volume of the room' this
curve is lhen used to transform from a frequency
v in c.p.s., read along the right-hand vertical
scale, to the corresponding dimensionless re-
quencyparameteru read along he bottom scale.
Next one selects he appropriate curve
correspon(tingo the value of the roomproportion
factor P obtained above. This curve is used to
lra•sform from the previouslydetcn•ine(t value
{)f • 1() ]•(' correspondingalue ()f N ()n the right-
l•a•(1 s('ale, which gives (lircclly the average
t•t •l•l)cr )f •ormal frequenciesu ) lo lhe
.ks an example, consider a ro()m I()X 15 X30
f('ct, a•(l let us find the average number of
•orn•al frequencies up to v=100 c.p.s. Here
p-0.5 and q=0.33, which yield a value of
P=3.3 from the insert chart. This room has a
v()lume V=4500 cubic feet' and from the
interpolated V curve we find a value of •=1.5
c(•rrcsI)on(ls o v=100 c.p.s. An intcrI)olatcd
curve for N with P=3.3 intersects •=l.5 at
• R. H. Bolt, "Influence of room proportions on normal
fn'(l•Cn('y spacing,"J..\cous. Soc..\n•., 17, 101.\ (1945).
N=28 wl•icl• is the al•l)r(•xi•ttc •t•l•cr
normal freqt•encies.
Figure 1 may also be used to (tetermi•e t •o
average spacing between adjacent normal fre-
quencies in a rectangular room at a given
frequency. he aboveprocedures used o obtain
P and •. Then the curve •=f(•) for the proper
value of P is used o transform rom u to •, read
along he eft-hand ertical cale. hisvalueof •
is the average frequency spacing n terms of u.
This value must be transformed from the u scale
to the v scale through the appropriate V curve
for the given room. However, n doing his it will
usuallybe necessaryo divide the u and v scales
by a common divisor; factors of 10 are most
convenient for this purpose.
To contissuehe ab()ve example, et us find the
average requ(m('y pacingai 100 cycles n the
10X15X30 rr•on•. The value of u is again 1.5;
this intersects h(. •:f(•) curve for P=3.3 at
•=0.022. We now divide the u and v scales by
100, take u = 2.2 through the curve V=4500, and
find approximately = 160: whence,dividingby
100, we obtain (dv?•,=l.6 c.p.s. which is the
average frequency spacing.
Figure2 gives plot of the ow requency pacing
index • as a function of rectangular room pro-
portions. he spacingndex hasbeendefinedas
follows:
in whichb s • •wtu•tln(,-ma reqtie•cy' t)ace,
is the avcrag('sl)a('( (as (•btaincd rom Fig. for
example)at the mcan normal frequencyof the
space ;and/• and •6 are the limiting requencies
(dimensionlessarametersas used in Fig. 1)
between which the summation in the above
equation is taken.
The in(lex ½ has the l)roperties of a mean
square(t atio of a('tt•al t() averagespacing ver
the band /• to /•. This quantity is directly a
measure of statistical fluctuation in frequency
spacing.Thus, •k= 1 if every actual spaceequals
130
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NORMAL FREQUENCIES FOR RECT,\NGttL-XR ROOh[S
131
O0
JIi Ir
/
II I
?•
ioo
)o
•o
;o
3o
-- lO
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s
3
o. ooi
•.1 •.$ I.o l.$
FIo. 1.
the correspondingverage pace;=2 if the
actual normal frequencies re distributedat
random Poissonistribution).he larger he
valueof •, the largerwill be the fluctuationn
frequencypacingnd subjecto certain ualifi-
cations)he argerwillbe hesoundransmission
irregularity r responseoughnessf the room
over hespecifiedrequencyange. he root-mean-
squarepacingf actual ormalrequenciesas
distinct from the smoothaveragespacinggiven
in Fig.1) canalso ecalculatedrom as ollows'
•rm• •-•.
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132 R. H. B O 1, T
.I
I.O
q
ß .• .4 .5 .6 .? .8 .9
• --I.8
--• N..,, : ,,.• ..< _-- • _•' ,
•,•,.o / •.• ? •
-,., 7 ///
N• x• • • / •ntour.f for
. Roomdimensions : pL=q L
//
•/ FREQUENCYPACINGNDEX
/
7
.6
.5
.4
.2
FIG. 2.
2.6
2.4
2.2
Y
2.0
1.8
1.6
1.4
1.2
1.0
Curve enclosesdimension otios giving
smoothest requency response ot low
frequencies n smoll recfangulor rooms.
Dimension rofios= I: X:Y
. /
,
1.2 1.4 1.6 1.8 2D 0 4000 8000 12,000 16,000 20,000
X VOLUMECU. T.
FIG. 3. Room proportion criterion.
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NORMAL FREQUENCIES FOR RECTANGULAR ROOMS 133
The index •k•plotted in Fig. 2 is calculated over
the interval u = 0.5 to u = 1.5. As can be seen from
Fig. 1, this interval contains approximately the
25 lowest normal frequencies in the room. The
width of this band is chosen to include about
enough normal frequencies to make the calcula-
tion statistically significant. The quantity •k• is
then a measure of response luctuation for the
lowest normal frequencies n a rectangular room.
Figure 2 displays quantitatively the intuitive
expectation that response rregularity is great in
a cube (•k•= 3.5); in a 1 X 1 X 2 room (•kl= 3.3);
in a 1 X 2 X 2 room (•k•= 3); and in other com-
mensurate cases. Values of •k• are lower for
incommensurate ratios and they have a broad
minimum range i,n the neighborhoodof p=0.8
and q = 0.6.
The significance of •k• as a measure of room
performance must be studied further in con-
junction with experimental tests. Tentatively,
however, one may deduce from the results of
Fig. 2 a room proportion criterion. For reasons to
be discussed n a subsequentpaper, it appears
reasonable o select •k•= 1.5 as a rough limiting
value belowwhich the responsewill be relatively
smooth. n actual cases, he influenceof damping
must be incorporated to obtain a true measure of
response rregularity.
The room proportion criterion obtained in this
way is presented in Fig. 3. Here the dimension
ratios of the rectangular room are given as
1 X' Y. It is important to recognize he range of
validity of this criterion, that is, the band of
frequencies over which •k• is calculated. Ac-
cordingly, the range of validity is also given in
Fig. 2 as a function of room volume and fre-
quency. The lower boundary of the region of
validity corresponds pproximately to the lowest
normal frequency in the room. This will be ex-
actly the lowest normal frequency in a cube, but
in rooms with other proportions there may be a
few lower normal frequencies.
This tentative criterion is consistent with room
proportion rules which have been employed in
practice.2, Thus the 2:3: 5 and the 1: 4/2:4/4
proportions lie within the smooth responsearea
in Fig. 2. The present criterion, however, indi-
cates that a relatively broad area of X' Y values
may be acceptable.
2 . E. Volkmann, "Polycylindrical diffusers in room
acoustical design," J. Acous. Soc. Am. 13, 234 (1942).
a C. P. Boner, "Performance of broadcast studios de-
signed with convex surfacesof plywood," J. Acous. Soc.
Am. 13, 244 ( 942).
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