Criterio de Bolt

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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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23:39:31



NORMAL FREQUENCIES FOR RECT,\NGttL-XR ROOh[S

131

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•o

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3o

-- lO

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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• •-•.


23:39:31



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.



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).


23:39:31

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