Frequency characteristics of a stopped organ pipe

r»fciUE:cY c: cs

OF A

stepped oroam pipe

TQWi'ER IIARDT STEVEBS
B. S., Kansas State Teachers 1 College, Pittatottrg, 1 26

/. tiiec ::

eubiuitte I in partial fulfillment of the rsquireaents

for the degree of

1 OF SC

RJ*SAS STATE AGRICULTURAL COLLEGE

Da

fTCmS

-TH
I92S

5&

TABLE OF CONTENTS

Page
INTRODUCTION 1

APPARATUS: DESCRIPTION AND OPERATION B

REFERENCE TO LITERATURE . . 7

LENQTH-FRE:UKKJY CHARACTER ISTIC 10

TEMPERATURE-Fl JY CHARACTERISTIC 12

PRESSURE-FREIIUENCY CHARACTERISTIC 13

3 19

D: Frequency 290. 26 21

Dl 23

S: Frequency 325.75 * 25

F: Frequenoy 345.13 27

F#: Frequenoy 365.66 28

G: Frequency 387 .42 30

G^: Frequenoy 410.44 31

A: Frequency 434.85 32

k§ : Frequency 460 .70 o3

B: Frequency 488.11 34

C: Frequenoy 517.15 35

MOUTH CORRECTION 37

SUMMARY , . 4

IHTRODIJCTIOI

This research began with an attempt to calibrate a var-
iable r.ltch organ pipe for use as a source of sound vibra-
tions In the study of resonators. It was honed that by
means of such a pipe any vibration frequency between 256
v.p.s. and 520 v.p.s. might be obtained with a certainty of
+ .1 v. p. 8. As the work progressed It was found that sev-
eral factors affecting the frequency of such a source had to
be eliminated, or measured and corrected. These Included
length, temperature, pressure, and mouth opening. The first
was the means whereby the variations of frequency were to be
obtained; the last could be eliminated by making the mouth
of such size and shape that the fundamental tone would speak
over the octave range. Temperature and wressure, however,
seemed to be more difficult to control, consequently It was
decided to determine their effects both for this puroose,
and to check up on previous determinations of these factors.
As the work developed It was found that the correction to
length due to the mouth opening could be computed very eas-
ily from the same data.

Among those who have assisted in making this work suc-
cessful I am especially Indebted to professor |* V. Floyd,

who directed my research, and rendered me valuable assist-
ance in various ways. The use of the chromatic octave set
of tuning forks was made possible by the generosity of Dr.
F. E. tester of Kansas University. I aa also grateful to
i rofessor J. 0. Hamilton and to several other members of
the physics department for suggestions and assistance in
various matters.

apparatus; dsscbiptioi aid operation.

The apparatus and equioment used In this work, draw-
ings and diagrams of which apoear on subsequent pages , con-
sisted of a fan driven by an A.c. motor, an air pressure
regulating tank, an adjustable, stopped organ pipe, a pho-
nelescor>e, an adjustable resonator, and a chromatic octave
set of Koenig tuning forks. The forks gave vibration fre-
quencies corresponding to the Bach equi tempered scale, A-435
Ut 3 -258.57 at 21.5° C. , Ut 4 -517.15 at 21.5° c. The frequen-
cy of Ut 3 was determined by comparison with a fork of 256
v.p.s. which had reoently been calibrated by the Bureau of
Standards. A-435 was determined by comparison with an or-
chestral bell of 440 v.p.s.

The effective length of the pipe could be adjusted by
■sans of a threaded piston rod connected through suitable
mechanism to a hand wheel within easy reach of the operator.
The distance from base block to piston face was read from a
scale and vernier attached to the oipe. A thermometer was
kept with its bulb In the air stream below the mouth of the
pipe to indicate the temnerature. A manometer in communi-
cation with the same cavity indicated the pressure, and was
read by means of a cathetoaeter whose accuracy was checked

3A

FLOOR PLAN SKETCH OF
APPARATUS .

1. "otor.

2. Fan.

3. Pressure regulating tank.

4. Adjustable organ pipe.

5. Tuning fork.

6. Adjustable resonator.

7. phonelescope.

8. Light tube with ground glass screen at end near 9.

9. Operator's position.

10. Cathetoraeter.

11. Hand wheel for adjusting ipe.

12. Incandescent lamp.

13. Slit.

1. Air inlet.

2. !.:outh of pipe.

3. Piston.

4. Adjusting Rod.

5. Metric scale for indicating pipe length.

6. Vernier for reading scale.

7. Piston rod, threaded.

8. Rod for attaching vernier to piston.

9. Guide rods for upper end of piston rod.

10. Gears for moving piston.

11. Leveling screw.

12. Manometer.

13. Thermometer.

r

-J

■

■ ;

A <ij u $ t<*biz Pc*on<z te

Vernier P / pe.

FlG.Ul.To Follow P3B

I 1

X>

I-

..■9/

H

r/crv

as jk .05 mm. In the early part of the work a manometer
tube of about three mm. bore wag used , but it was found to
be unreliable due to capillary action and was finally re-
placed by a tube of about seven mm. diameter, and this in
turn by one of ten mm. diameter.

The adjustable resonator was made in the physics de-
partment shop by professor Floyd and myself. It consisted
of a piece of brass pipe two inches in diameter and about
ten inches in length, partially closed at one end by a
brass disc bored five-eights inch diameter. vithin the
pipe a piston moved on the end of a small pipe that served
to transmit the sound energy to the phonelescone and to car-
ry a rack, enabling it to be moved by means of a pinion.
When the forks were in use they were held in a lathe chuck
of about nine pounds weight.

As stated in the introduction, the fir^t consideration
was that of determining the length-freauency characteristic
of the pipe. To determine the length of the pipe needed to
give a certain frequency, I set the proner fork directly in
front of the mouth of the resonator as shown in Figs. 1 and
2. Then I adjusted the resonator so that when the fork was
vibrating it gave a maximum displacement of the light on the
screen. Then, with the fork vibrating and the pipe sneaking
I adjusted the pine length by turning the hand wheel until

the phoneloacope showed no beats more raoid than one in ten
or more seconds. The frequency was found fro* a table of
frequencies for the Bach A-435 scale, and the corresponding
pipe length read from the scale and vernier to tenths of a
millimeter. This operation was repeated for each tone in
the chromatic octave, and the length-frequency characteris-
tic curve was clotted from the data so obtained. Through-
out this determination the teraoerature was kept between 21°
and 22°c. , and the oressure was nearly 4.52 cm. water as
could be read with the cathetometer.

Two of the above described determinations were run at
temoeraturea of 28°C. and 20°c. respectively in an effort to
obtain the temnerature-freoiiency coefficient of the pire.
They were run before the adoption of the water manometer and
cathetometer for measuring creesurea, so that they cannot be
taken as wholly reliable. Later, in another part of my work
I waa able to check up on this fnctor in another way which
I consider more accurate.

During the time that I was trying to make the above de-
scribed determinations, I noticed that fluctuations of pres-
sure due to faulty operation of the fan caused aporeciable
variations in the frequency of the pipe. In fact, this led
to the use of the cathetometer. I hapoened one day to hear
a closed tube blown as a whistle, and thought I detected a

;lon of pitch with increased wind force. At about the
ease time I read nayleigh^s comment to the effect that In-
creased wind pressure invariably raises the itch of an or-
gan ipe. As a result, I attached the problem of finding
out if there were any condition under which Increased r-res-
eure would result in a lowering of the nltch.

Tor this purpose I set the pipe in approximate unison
with each of the forks In succession, then varied the pres-
sure from the lows it that would cause the pipe to speak its
fundamental tone (lower critical oreesure) to the highest
pressure that could be obtained, or that would permit ths
fundamental to apeak (vmmmm orltlcal nressure). At various

eseures within this range I determined the frequency of
the pipers tone by counting the beats bstween it and ths
fork. In "his part of my work I found the visual asthod
of counting; bsats the better near unison, end the auditory
asthod better for more ranid beats. In so far as rosslble
T ussd the two methods eve a check upon each other.

From the data of the Immediately preceding determina-
tion It in possible to oompute the mouth correction for any
condition of pressurs and frequency contained therein, by
the equation,

c - V - L

T~

In which c is the correction in oentimeters of pips length;

V Is the velocity of sound in free air at the nrevailing
temperature; N is the frequency of vibration; and L is the
measured length of the r>lpe.

The pres-ure-freouency characteristic curves slotted
frooi the first list of data were very rough if made to pais
through the joints as read. I felt sure of the frequencies,
for they vere carefully checked to a much closer approxima-
tion than the distances of the ooints from the amroximation
curve indicate. The manometer then caoe up for criticism,
and one of the Isrger bore was substituted to see what its
effect upon the regularity of the curves would be. It was
an improvement, but cpplllarity seemed to have considerable
effect in it, so an even larger one was used in checking
some of the data.

RfTCPRWCSS TO LITEBATURK.

Previous to this time very little work seems to have
been done on the above mentioned subjects. wost text-books
of physics give the simole inverse ratio for the length-
frequency characteristic. **or example, *?eld and Palmer
(Text-book of Modern Physics, oage 362) say:

"Since the distance from a loop to a node is one-quar-
ter of a wave-length, the length of a closed organ pive la

one-quarter of the wave length of its fundamental tone;
hence the fundamental of a short pipe is of higher pitch
than the fundamental of a long one."

The mouth correction is discussed as follows by- Lord
Rayleigh (Theory of sound, p. 218, Vol. 2, Ed. 2.)

■We have seen how to take account of am upper open end,
but according to the rule of cavallle-Coll the whole addi-
tion which must be made to the measured length of an open
pipe in order to bring about agreement with the simple for-
mula, t* s L z r l, amounts to as much as 3 1/3 R, very much

a a
greater than the correction (1.2 F) necessary for a simple

tube of circular section open at both ends. This discre-
pancy is sometimes attributed to the blast, ^ut it must be
remembered that the lower end is very much less open than
the upoer end, and that if a sensible correction on account
of deficient openness is required of the latter, a much more
important correction will probably be necessary for the for-
mer. Observations by the author have shown this to be the

case. - - The considerable correction to length found by

Oavaille-Ooll is not attributable to the blast, but to the
contracted character of the lower end treated as open in the
elementary theory. The rine of oitch due to the

• t = time of vibration; L = wave length; 1 * oioe length;
» - velocity of sound.

9

wind increases with r>re3sure. Thus, in *he ease referred to
above, the pipe under a Treasure of 2.7 cm. of water gave a
note about 2 vibrations per second sharker than that of the
fork, but when the wind oressure was raised to 10.7 cm. the
excess was as much as 11 vibrations per second. Then the
pressure was raised much farther, the ipe was overblown and
gave the octave of its proper titch. "'his, of course, cor-
responds to another mode of vibration of the aerie 1 column."
e following is in regard to the effects of wind pres-
sure and temperature on organ pipe frequencies as given by
Cavaille-Coll (??elmholtz» , nensations of Tone, p. 89, Kd. 4)

■As to strength of wind, as pressure varies from 2 3/4
inches to 3 1/4 inches, the "itch number increases by about
1 in 300, but as ure varies from 3 1/4 inches to 4 in-
ches, the pitch number increases by about 1 in 440, the
whole increase of rresrure from 2 3/4 to 4 Inches increases
the pitch number by 1 in 130.

For temperature, I found by numberous observations at
very different temperatures that the following practical
rule is sufficient for reducing the * itch number observed
at a Riven temperature, to that due to another. It is not
quite accurate, for the air blown from the bellows is often
lower than the external tem-erature. Let P be the oitch
number observed at a given temoerature, and d the difference

10

of temperature, in degrees F. Inen the pitch number is
P(l + .00104 d) according as the temperature is higher or
lower. The practical operation is as follows: supposing
P = 528 and d » 14 increase of temperature. To 528 add 4
in 100 or 21.12, giving 549.12. Divide by 1000 to two pla-
oes of decimals, giving .55. Multiply by d ■ 14, giving
7.70. Adding this to 528, we get 555.7 for the pitch num-
ber at the new temperature."

LEHGTH-FRE^UKKCY CHARACTER I5TIC.

The accompanying data and the graph thereof show the
true relationship between the length of a stopped organ
pipe, at constant temperature and pressure, and the fre-
quency of its prime tone. The important points to be ob-
served are, (1) that the relationship is not linear, but
approximates in the graph, an arc of a conic, perhaps an
hyperbola. (2) The length of the pipe needed for C 4 is less
than half that for G3. As will be shown later in this work,
the mouth correction for the shorter position is greater
than for the longer. Hence, a shorter actual length is re-
quired, as the addition for the total effective length
brings it to the proper value. For example, the measured
length for G3 is 30.87 cm.; the mouth correction is 2.45 cm.

11

effective length 33.32 cm. For C4 the e

ffective length is

13.9

- 2.94 | 10.84. 2 X 16.84 = 00.68,

which gives the

length for the lower tone within 2.5 v.p

.8. This includes

temperature change and experimental error, and is of the

order of 1$ of the frequency range over

which the calcula-

tion

Is made.

Calibration Data.

Tuning fork.

Pipe length,
21.5°C.

Tone

Frequency, 21. 5° j.

08

258.57

31.40 OB.

*

275.94

29.35 "

D

290.23

27.49 "

D,

307.49

25.74 "

1

325.75

24.00 M

F

345.13

22.38 "

n

365.68

20.90 ■

G

387.42

19.55 w

og

410.44

18.20 "

A

434.85

16.74 n

A#

460.70

15.60 M

B

488.11

14.49 »

C

517.15

lo.47 ■

The graphioal representation of these points appears

on the next page.

•

stuo 't/j. 6u9~j dd ij

m

UNIVERSAL CROSS SECTION PAPER

I I I I - - 1 - I -

12

TEMPERATURE-FREQUENCY COEFFICIENT.

In order to determine the relation

between temperature

and frequency of the

pipe, I made two length- frequency de-

terminations at temperatures of approximately 28°C. and

20°c. respectively, keeping the pressure

i practically oon-

stent in both cases.

This was done befc

>re the adoption of

the oathetometer for reading pressures,

so that its value

is only approximate and it serves to give by itB average,

an indication of the

true coefficient.

The following data

shows the method of arriving at the probable value for the

temp erature- frequency

• coefficient,
itata.

o o
«1 " «2 Ll -

Lg V.p.8. per

i'one 1 .41

■a. length .

n Ae

7.6°C. .41

cm. .7

C3 .00146

7.6 " .26

.75

9j .00936

7.6 M .26

" .9

D .00106

7. " .17

" 1.0

Dff .00789

5.6 - . .31

1.1

E .00187

.3 ■ .23

1.25

I .00157

6.6 " .23

" 1.45

If .00138

7.7 " .24

1.65

.001^2

7.2 n .24

" 1.75

Gfi .00141

7.4 " .21

1.95

A .00127

6.6 " .25

M 2.2o

A| .00184

6.2 ■ .23

" 2.7

B .00205

— — — — —

2.8

G

. .0027

Cavaille-Coll'

s velue .00188

-

13

Gj? and D# are of entirely different order from the remain-
der of the values, hence are probably errors. Leaving them
out, the iaebn of the other ten is .00153.

Typical Calculation.

i A« « 1Q ( L i - 1 2 ) v.p.s. per mm.

% ^ G k(»i -eg)

For C3 this becomes,

4.1 x .7 a .00146
7.6 x 25Q.6

In this 0B8e,.AN represents a difference of one vibra-
tion per second in frequency, and^e a temperature change of
one degree J. .-.eferenoe to the pressure- frequency curves
taken at various temperatures will serve as a check upon the
work, and will be discussed later.

pressurk-frehUlkjy ooimoiv*.

Many things may be considered under this head. The
rate of change of frequency with change of pressure is the
outstanding feature. Pressure above which the tome is un-
certain and not good for musical purposes is perhaps of
chief importance to musicians and instrument makers. Upper
and lower critioal pressures for the fundamental tone of the
pipe is also of interest, x'hese will now be discussed in

14
the order given above.

As nay be seen by examining the accompanying curves,
the frequency rises rapidly with increase of pressure over
about the lover one-half or one- third of the total pressure
range. It then reaches a noint at which the frequency is
practically constant over a considerable increase of pres-
sure, then in a few instances decreases slightly, to rise
again before changing over to the upper partial. This is in
contradistinction to the statements of Payleigh and Cavallle
Joll that frequency rises with pressure.

Due to the lack of facilities for maintaining constant
pressure above 10 centi leters of water, all but two or three
of the curves are incomplete, and I have indicated by dot-
ted lines the curves as they will probably be found when
this work is supplemented and completed. My prediction is
that all of the curves will be found to have the general
shape of those given for and C5, the difference being that
the bend of the curve will come at a higher pressure for
each successive semitone In the octave, and the pressure
range of slight frequency variation rill increase for the
higher notes of the octave. I also expect to find that each
of the partials follows a similar type of variation in this
respect.

15

an the C„ graph, (next page) I have plotted three ourves

■

as follors: (1) fron data taken at 21°C. using the original
•■all bore raenometer; (2) from data taken at 24.2°C. with a
manometer of 7 rat. bore; (3) from data taken at 20.5°C. with
a 1 cm. diameter manometer.

It may be seen that the critioal pressures correspond
Tery closely in the three determinations, i>ue to the method
used for finding them, the values given by (3) are perhaps
the more reliable. Also, there is little doubt that curve
(3) represents more accurately than the others the behavior
of the pipe in this range.

Examination of (3) shows that near the lower critioal
pressure the frequency rises rather slowly, but between 3
and 4 cms. of water, the frequency rises very rapidly, be-
tween 4 and 5 cms. it increases at the rate of aoout one
vibration per second per am. water pressure.

To compute the pressure- frequency coefficient rithin

any range one may use the expression

Coefficient « A 1%

K .AP

where N is the frequency, A V is the frequency change for 1

cm. change in pressure, andAP is 1 cm. of water change in

pressure. In the above case it becomes

1 * .00386

258.6 X 1

16
That is, between 4 and 5 cms. pressure the frequency changes
at the average rate of about .00366 tines itself per one em.
change in pressure. If the pressure- frequency characteris-
tic were a straight line with the slope 4J£, the frequency at
any pressure would be

N ■ 258.6(1 + .00386(4. 5£ Pg)
where 4.5 is the pressure giving 258.6 and P g is any other
pressure. In view of the fact that the curve Is not a
straight line, the above cannot be used, but instead the co-
efficient must be calculated, if desired, at the point under
consideration on the curve.

The vertical lines at the rifht and left extremities of
the curve are the upper and lower critical pressure lines,
respectively. That is, the fundamental tone of the nipe
speaks only at pressures indicated between them.

The vertical broken line at 5 cm. pressure is perhaps
of as much interest and more value than the critical pres-
sure lines. Near that part of the curve, the tone of the
pipe gave strong evidence of the presence of upper partials,
and above thet point the tone became unsteady, or what musi-
cians term a split tone. The actual frequency varied within
a few seconds between .5 or 1 v.p.s. above the fork, to as
ouch as 3 or 4 v.p.s. above. This would indicate that it
would be inadvisable for several reasons to try to use this

17

pipe above 5 em. pressure as a musical instrument, especial-
ly in a sustained chord. A serious difficulty that arises
in this connection is due to the fact that the loudness of a
wind instrument tone is changed by changing the air pressure
with which the pipe is blown. Unless all the instruments of
the (supposedly) same pitch have the same oressure- frequency
coefficient, serious dissonances are sure to arise in sus-
tained chords unless some means is employed to compensate
this change in frequency, livery wind Instrument except the
organ may be thus compensated, so the organ must be equipped
with separate pipes voiceu for the various pressures.

I wish to call attention to t e minimum point on the
curve at about 7 cm. pressure. So far as I have been able
to discover, this has not been detected previous to this
ti.e, and adds a special qualification to the work of Ray-
leigh and Cavaille-Coll on this subject.

Data.

Pipe length, 30.87 cm.
Pressure Beats Seconds

1.32 cm.

80

20

4

1.95 "

100

38

2.6

2.70 "

80

31.2

2.5-

Small nanometer. 21 C.
Beats/ Second li

254.6
256.0
256.1

Cont'd on next page

UNIVERSAL CROSS SbSTION PAPER

3.43 on.

4.20 "

4.34 "

5.06 "

6.78 M

8.18 n
9.18

50 45
30 50

Unison
15 42
30 55
80 27.6
Second nartial

1.1

257.5

•6

258.0

258.6

.37

258.97

.85

259.45

2.9

261.50

774.

18

Data.

Pipe length, 30.87 cm.
Pressure Beats Seconds

1.40

era.

60

20

2.10

»»

20

12.8

3.04

n

Unison

4.29

■

30

12

5.20

ft

30

15

6.46

n

30

12

7.13

tt

32

10

7.83

it

30

6

8.74

n

30

5.2

at s/ sec.

1

S

255.57

1.56

257.01

258.57

2.5

261.07

2

260.570

2.5

261.07

3

261.57

5

263.57

6

264.5

Data.

Pipe length, 30.95 on.
Pressure Beats Seoonds

1.56 era.

40

7

3.23 "

30

9

3.56 "

30

20.6

4.53

Unison

5.27

12

24

6.21

12

6.62

5

21

Beats/sec.

K

5.7

252.87

4.44

254.13

1.45

257.12

258.57

.5

259.07

.48

259.05

• 24

258.81

Cont'd on next page

19

7.15 oau
7.76 *
8.66 "
8.83 w
8.99 "

20
30
30
30

21
14
12
10

Second partial

1

259.57

.14

260.71

2.5

261.07

3

261.57

C*3

The data taken at 21.4°c. with the snail manometer made
a emooth curve of the general shape of the others in the
lower part of the octave, and was considered as sufficiently
accurate for the purpose. Because of limited time I did not
repeat this data with the larger manometer.

It may be seen that this curve corresponds in general
to that for G . The pipe began to speak its fundamental
tone at 1.6 cms. water pressure. The frecjiency rose rapidly
to about 5 cms. pressure, then decreased slightly at 7 cms.
and Increased again before the fundamental gave place to the
next partial, the third harmonic, at 9.7 cms. pressure.

Comparing with the preceding curve, it is evident that
the "breaking point" or the point where the split tone is
notio e able ia at a slightly higher pressure than in the
former curve, lietween 4 and 5 cms. the coefficient is
.00329.

Data.

20

Forte C3#

ipe length,

.01

Pressure

Beats

Seconds

Beats/sec,

■

X • O

50

10

5

268,

,94

.-j •

50

16.2

3

270,

► 94

4.03

20

31

.66

273,

,21

4.73

Unison

273,

5.50

20

36.4

.55

274,

,-■:.

6.06

20

31.8

.64

274,

r HI

7.

20

35

.57

274,

, 1

8.57

30

n

1.3

275,

.31

50

20

2.5

276,

AA

9.i

Second

partial

821.

,82

UNIVERSAL CROSS SECTION PAPER

21

D: Frequency 2-0.23

In this curve also the downward trend In the region of
the split tone is apparent, but due to the lac: of suffic-
ient air pressure the full curve between critical pressures
could not be determined and I have indicated by a dotted
l.-'ne the probable course of the curve in the unknown re-
gion. This is done on the strength of the evidence of the
two preceding curves. The breaking point is at about 5.
cms. pressure and the subsequent "flat" part of the curve
is broader than in either of the preceding curves.

Both the upper and lower critical pressures are some-
what higher than those of the two preceding curves, and
these points will be seen to rise on through the octave, the
upper one the nore rapidly.

The coefficient at 4 to 5 cms. pressure is .00605.

Ofttft.

Temperature 21.0°C.
^essure Beats Seconds

1.91
3.42 »

20
60

3
19

Pipe at 27.35 cm.

Beats/sec. N

6- 284.

3.4 286.83

Cont'd on next page

4,65 oma.

8

24

• 33

289,90

5.43 "

10

23

• 43

2-.0.64

7.1;; »

20

30

.66

290.33

7.34 "

SO

27

.74

290. 7

8.04 "

30

30

1

291.23

10.74 «

Second

i-tial

870.69

22

670

O

riG.vw

Pressure -Frequency
Characteristic
of-
D

Follow*. P 21.

~7 8 T

Pressure - Cms. af Water

~7Q 7T

UNIVERSAL CROSS SECTION PAPER

23

N

Here again the curve Lao the same general shape up to
.5 era. pressure as the former curvce have to the point of
negative coefficient. As in the curvo for »n», T have
dotted the probable course of the curvo on up to the upper
cr tlcr.l j ressure. rote that the brea^In^ point is at
si ghtly higher pressure, and the flat part of the curve
broader than in any of those previously considered. ~ne
to the fact that the 4 to 5 en. pressure range is on the
steeply rising part of the curve, the coefficient is some-
what greater, via. .00752. This curve as dete rained by
means of the email manoueter was considered fairly reli-
able, and «as not repeated.

Data.

ure, 21.5°C.

Pipe length, 25.55 en,

essuro

Beats Seconds

Beats/Sec.

■

2.18

30 6

5

V

3.96

50 15

2

305.49

4.73

Unison

307.49

5.71

10 3.5

.28

307.77

6.57

50 22.5

2.08

309.57

Cont'd on next page

<?Zl

? 3 Of

^ 3 03

v.

s

■ 3 07
3 OS
3 OS
3 04
3 03
3 02

<t>

s

ricix

Pressure -Frequency
Characteristic

Foil* m/j £.23

3 7 5 6 7 5 T~

Pressure cms. of water

~n> — n 73:

UNIVERSAL CROSS SECTION PAPER

24

7.86

50 24.2

2.06

30.. 55

8.99

50 25

2

309.49

9.47

50 32

1.56

30.. 05

12.00

Second partial

-22.47

25

E: Frequency 325.75

The first determination of points for this curve gave
a rather rough line, curve 1), so I repeated the work with
the second manoneter* (Results shown by curve 2.) Here
again have had to supply the probable continuation of the
curve ~bove 0.5 era. pressure. The critical pressures are
seen to be somewhat higher, and the breaking point is aore
difficult to determine, but is between 6 and 7 oas pressure.
The coefficient for (2) between 4 and 5 is .0076.

Data.

Teoperature ,1) 21.6°C; .2) 23.6°c. Pipe length 24 cm.
Pressure Beats Seconds Beats/Sec. N

2.10

20

3

6.6

313.15

4.17

30

10

3

322.75

5.00

20

35

.57

325.18

5.11

Unison

325.75

6.23

30

IS

1.58

327.33

7.35

10

17

2,35

328.10

8.07

50

17

3

328.75

9.28

50

16

3.1

328.85

9.43

50

16

3.1

328.85

1-

Second

arti al

977.25

T6

E: Temperature 23.6°C.

os; ure

Beats

Seconds

Beata/Sec.

N

1.81

40

7

5.7

320.05

2.46

40

10

4

321.75

3.13

,0

11.5

3.48

322.27

3.76

30

18

1.66

324.09

4.55

Unison

325.75

5.26

30

16

1.87

327.62

5.97

40

15.5

2.66

328.41

Pipe at 24.

.2;

add 2.3 v,

>t) .8.

7.82

30

1

4.44

330.19

8.82

30

14-

4.5

330.25

The last tr/o readings were made by using the length-
frequency character "etc to deter. ine the change in fre-
quency due to the greater pipe length. This added to the
actual difference between pipe and fork gives the differ-
ence as it would be with the pipe at its original length.

177

— r | - 1 : | - - - : 1 1 -_ — I .. |. ;|,..|., ; : |.'

W

-*-*- 1

y

F1G.X

Pressure -Frequency
Characteristic

o r
E

Follows P 26

~J 3 5 7* 5 7 73 7? 73~

Pressure- crr\5.oi water

UNIVERSAL CROSS SECTION PAPER

27

F: Frequency 345.13

n thlB curve the brea ing point is at about 8 cu.
pressure, and the final point on the curve is apparently
at the niaxinwa point just before the downward turn of the
split tone* I have not attempted a continuation of ti.is
curve, but would be inclined to make it similar to the ; re-
ceding ones. The coefficient between 4 and 5 cm. pressure,
which is dec dedly on the straight part of the curve, is
.0065.

at,.,.

Temperature, 22.3°C.
rossure Beats Seconds

1.94

30

4.1

3.72

20

3.5

3.95

50

16

.00

20

17

.19

unison

. a

20

20

6.00

30

18

6.41

30

16

7.70

30

11

8.74

30

8

.66

30

8

15

ocond

partial

Pipe length,

22.47 c :.

Beats/Sec.

N

7.13

338.

5.7

339.43

3.1

342.03

1.17

343.

34C.13

1

346.13

1.66

346.7

1.87

347.00

2.7

347.83

3.75

348.88

3.75

348.88

1035

Id 35

349

FIG.X I

Pressure-Frequency
Characteristic

of-

r.

FoVo ws P 21

TS

UNIVERSAL CROSS SECTION PAPER

aTer

r : i v i '•--! ' ! I

28

T§: Frequency 365.68

Here again have plotted two curves. :1) b fro.
data taken with the s.;all r.ianoaeter, (2) using the large
bore 1 cm. ) manometer. Both are at practically 21 C C. n
(2) there is evidence of the breaking point at about 3 c .
pressure. Y&en the pipe was Bounding at this point the
8 lit tone was quite evident. This is the highest point in
the octave at which detected the split tone: hence the
subsequent curves will represent only the lower straight

rts of the curves found for the lower tones. Each will be
respectively a smaller fraction of the entire curve between
the critical pressures.

• or this tone on curve (2), between 4 and | c . pres-
sure, the coefficient Is .0046.

Data.

Curve 1) Temperature 22.3°C.
Pressure Beatc seconds

2.17
3.68
4.36

30 7.5
20 32
Unison

Pipe length 20.84 em.
Beats/Sec.
I

.62
Cont'd on next page

365.06
. 68

£9

5. 30 16

5.85 30 10.

6.90 20 3

Pipe at 21.31; add 7.. 8
lo. o: 50 17

Total

1.87

2,8

6.66

2.94
7.
10.

367.55
368.53
372.34

376.60

Data.

F#« Laf£

pipe length, 20.81

Pr ©scare

Be ate

second*

Beate/Seo.

1

2.46

10

4

361.68

3.3o

30

18

1.66

364.02

4.26

15

27

.55

365.13

4.55

Unison

365.68

4.85

20

32.8

.61

366.:

5.

30

15

2

367.68

7.12

40

9.5

4.21

36 J. 89

8.42

.0

8.5

4.7

370,38

9.26

6.5

6.16

371.54

—

T~

1018
3 76
3 7J
37*
373
372
37 1

370

369

368

367

366

365

3 6*

L_

363
3 62
3JLL

3 =F

/
x A

FIG XII
ure -Frequency
\Qtacter\stic
or
F~
Follows P.zq

UNIVERSAL CROSS SECTION PAPER

7 1-~l 7 -" :

~ru 7T

TT

30

G: Frequency 387. 42

Although I tooli data for this tone with eaci, .t the
three sizes of raanoueter, I have plotted only t*at for the
largest, because I consider it the more significant of the
three. The points are so placed that one might draw a
straicht line through then with only a small de rture fro.,
any one point. However, It may be that the tone of the pipe
actually follows such a curve as have drawn, in all cases,
in the lower part of its pressure range. Certainly there is
no suff cient reason for so far disregarding n:y readings as
to draw a straight line ayproxi mating the curve. The co-
efficient at 4 to 5 m* s .00387.

Data.

Temperature

21

3C.

P

ipe length 19

,43 era.

Pressure

Beats

Seconds

Beats/Sec.

N

2.74

60

20

3

384.42

3.45

20

15

1.33

386.09

4.52

Unison

387.42

4.88

10

16

.63

388.05

5.31

20

1

1.05

388.47

6.54

40

16

2.

389.92

8.56

40

8.5

4.7

3i2.12

.55

.0

5

8

395.42

18.50

Sec

ond

P rtial

1162.26

1 'MM 1

| ! i:. | I millllllll 1 | | | 1 ! |

II OZ-

*

393

39Z

39/

390

J. 367

^ 368

A 1 W V

1 o 3a7

| g see

®/

3&S

FIG.XU1

389

P

ressure-Freai/enc\

f

r

Characteristic

Oh

3 33

/

G

36Z

)

^ollowt P 30

3 9- 5 6 7 « Y to If 18

UNIVERSAL CROS

Pressure- Cms of water

i — i — i — i — l_ i r. i. : i : i ; n i fii i -■- x \

S SECTION PAPER

Gif: Frequency 410.44

Here again I have graphed only the final data fro.i
readings of the large manometer* The - ortion of the curve
thua obtained is evidently a straight line, the variation
of the points from it being so small t^at it may be due to
add t ion of various experimental errors, such as the effect
of capillar' ty. The 4 to 5 c ;. coefficient is .0016.

Dat .

Temperature 2l.5°C. Pi • length, 18.23 c .

Pressure Beats seconds Beats/Sec. N

3.09
3*84
4*52
4.8.
6.63
7.86
8.69
20.00

iats

Seconds

.0

15

30

28.6

Unison

18

23.6

40

10

40

7.8

40

5.8

Second partial

2.66

407.78

1.05

40- .3.

.10.44

.63

411.07

4

414.44

5.12

415.56

7

417.44

1231.32

r — i..„i-'.i.

/23t

i....i i i i I ■ II I — r~

■ I i i

4/7

4/6

4/f

4/4

4/3

4/X

9 4/0

^ 409
M

^ 40&

407
406

4 04
40 3

V. T

^

T-T-l

O

FIG.XN

Pressure-Frequency
Characteristic
o/-
G*

Folio WJ /'.J/

~K> 7T

^3T&

I 1 ! -I

Pressure - cmsofwate-r

■•■■<■ i i ■ '■ -i i i •' ^ I — i — i — i— j — -i-

UNIVERSAI, CROSS SECTION PAPER

32

A: Frequency 434.85

Only the final determination ia graphed here. An idea
of the email part of the total characteristic curve for this
tone that ia ahown hare may be . bt: ine . from a conaideraii-n
of the follov/ing:

Total prcasura range between critical preasurea - 20 am
Preeeure range represented by curve 5

Part of total characteristic re:>resenied - 2 ;'
Conaiderinc the fact that about O ' of the ch rrctor-
etic was needed in the former curves to determine even the
brea :ing point, it is evident that this and the remaining
curves tell very little about the actual operation of the
pipe at the important pressures. The 4 t. 5 cm. coefficient
ia .003. .

Data.

Temperature 2l.8°C.
Pressure Beata

Bacendi

Pipe length, 17.03 cm,
ata/sec . N

3.30

30

8.

5

3,53

451,32

3.64

30

16

1.9

432.95

4.50

Unison

434.85

4.90

20

22

• 91

435.76

6,92

40

11

3.63

438.48

8.62

40

7,

,2

5.35

440.40

13.00

Second Ml

rtirl

1304.55

r

/305

41$

442

441

440

457

j&M.

It

ricxv

Pressure-Frequency
Characteristic

of
A

7 7d 77 23"

* -cms. of water

K

UNIVERSAL CROSS SECTION PAPER

23

: rro.uoncy 460.70

Here the fraction of the total ch rretoristic repre-
resaited s about 23 . The 4 to 5 era. coefficient ia ,0039.

Data.

Temperature -

°1.9°C.

Pipe length,

15,89

c:.i

Pressure

•eats

Seconds

Beats/Sec,

I!

2.68

ti

15

2.66

458,

,04

4.15

30

28.8

1.04

459,

,66

4. CI

unison

460.70

4.. .9

30

27.6

1.06

461.

,76

6.48

40

10

4

464,

,70

8.76

40

6

6.66

467,

,36

26.00

partial

1382,

,10

UNIVERSAL CROSS SECTION PAPER

:;4

B: Frequency 438.11

This curve is fron data taken with the snail manometer.
t was sufficiently near a straight line that I did not
feel justified in repeat ng it due to lad: of t- c. The
part of the total character st'c shown is 18. Tg. The 4 to
5 cm. pressure coefficient is .0055.

Data.

orature 21.2°C.

r on sure

Beats

Seconds

Pipe length 14.87 en.
Beats/ Sec. N

3.75

20

3

6.66

431.45

4.53

30

13

.3

%81

5.23

20

31

.64

487.47

5.52

Un'son

488.11

6.76

30

15

2

490.11

7.88

30

7

4.28

492.39

3.94

20

4

5

493.11

9.64

20

3

6.66

494.77

32

Second partial

1464.33

I

/l&t

~T~\ ~T I i | . .J.... {.!:::) ;:

195
191
193

192

19/

IfO

189

o las

<U 167
466

48*

163
182.
4#/,

Q

FIC.KVU

Pressure-Frequency
Characteristic

°r
3

Folio wsp.3l

7D 7T

-J?r

UNIVERSAL CROSS SECTION PAPER

Pressure - cms. of water

35

C: Frequency 517.15

This curve s froi data ta :en with the second :ano-
eter. The part of the total curve represented is 8.3#.
The 4 to en. coefficient can not be obtained due to the
fact that 4.5 is the lower cr tical pressure. The 4.5 to
5.5 value is .0067.

Data.

Temperature 2l.6°C. Pipe length 13.9 cb.

Pressure Beats seconds Beats/Sec. N.

4.29 20 3.2

5.11 20 7.4

5.32 20 11.5

5.77 10 28

6.15 Unison

6.36 10 15

7.27 20 8

7.83 20 3.8

9.74 30 5

10.20 30 4.2

42 second partial

6.25

10.90

2.7

514.41

1.74

515.41

.36

516.79

517.15

.66

517.81

2.5

519.65

5.26

3.41

6

523.15

7.14

524.29

1551.

155/
5Z5

S3.3
52.2

5/9

»s/r

r-

1

J 5/6

£

5/5

5/3
S/2
S//
5/C

? I

~z\ — I — r

o

FIGXVlll.

Press ure- Freque ncy
Character istic

w

catty

ie

UNIVERSAL CROSS SECTION PAPER

e-cms oi wotier

^2T

CRITICAL PRBSSU

The relation of the lower critics! pressure to fre-
f— i cy, or to posit on of the tone in the ootava, is not
definite but in general it increases as the frequency
rises, and the length of the pipe decreases. The pressure
necessary to bring in the fundamental at any point varies
considerably with different trials. In general, the second
partial is the first sound besides the wind- rush, and is
suddenly replaced at about the presaures given, by the
fundamental.

The upper critical pressure, on the other hand, fol-
lows very closoly the arc of a parabola. As the frequency
of vibration rises, with decreasing length, a point is
approached at which a great, or practically infinite pres-
sure would be ecessary to elicit the upper part'al.
doubt further work on noutii corrections to length will be
necessary before this oat tor MB be developed further.

PRESSURE FRE.UE.CY COEFFICIENT

The rolati-n of pressure -frequency coefficient to fre-
quency, or to pipe length is not easily deducible frcw the

—

"1

T

■■■■\- l ::: l \ ili-;-|^:|; i I i ' -I::. 1

UNIVERSAL CROSS SECTION PAPER

" M g/

T" 5 J© * V

X

J

n

$

**

*>

*

">

fc
n

<&

Bj

*-

V

>^

O

; )

1)
v

.. _

37

data contained herein. In view of the fact thrt the coeffi-
cient at 4.5 en. pressure Is for a different RELATIVE part
of 8 ccesc've curves, it hardly see s to me to have any
particular significance except as It affects the calibra-
tion of such a pip* as I used in this experiment. If it
were possible to do so, it see.. 8 to me that a dot .ation
of the pressure- ipe-length ch-r-cteristic at constant co-
efficient, perhaps at the breaking point of each curve in
the above list., would be of considerable interest and value.
A» stated previously , the breaking points of only a few of
the lower tones of the octave could be found, due to lack
of constant air pressure above that whieh will sustain a
colu-an of water ten centimeters In height.

UTi; ORRECTION.

The work done to date on the question of nouth cor-
rections is, so far as I can f ind ; suransd up In the quot-
ations froo Rayle'gh that are given in the first part of
this wor . yleigh states that for an open organ pipe,
the total add t' on to t:.o length necessary to Give the
acoustic length of the pipe 'e 3.3 R . Subtracting .6 R
for the open end, which of course s ot present in a
closed pipe, there is left 2.7 tines the radius of the

38

circular part of the pipe for the nouth alone, such is hie

treatment of the question. t s given without any con-

eideration whatever for length of pipe, temperature, or

wind prescuro.

Froa the data given above it is a c :. lc nsAtsr to

calculate the mouth correction at any point on any of the

pressure -frequency curves by the formula

c • V3 - L
— Jr —

where c is the correction in centimeters length, V- is the
velocity of sound in free a r at the prevailing ten. erature ,
N is the frequency of vibratin, and L s the measured
length of the pipe. The following table shows the correc-
tion for each tone of the octave at 4.5 eau pressure and
8 cm. pressure reepectively.

Data*

Tone. Pipe -Length. 8°C.

C3

D
DJ

£

30.87

CD.

1

30.87

»

24.2

29.01

■i

21.5

27.35

ti

.1.8

5.55

n

21.8

24.00

|

21.6

n

23.8

Correct

ion

g%

pressure

of

4.5 on.

8

2.45 era.

2.17

era.

2.32 ■

. 7 "

2.36

M

2.42 «

2.28

It

2.50

2.31

W

2.61 "

2.23

H

2.57

Cont*d on next

•6

F 22.47 cm. 22.2° 2.68 cm. 1.507 cm*

F 20,84 ■ 22.3 2.74 " 2.24 "

G 13.64 ■ 22. .76 ■ 2.44 "

B# 18.33 M 19.9 2.78 ■ 2.53 "

A 16.88 ■ 20.5 .84 " 2.59 "

15.94 " 21. 2.865 - 2.60 »

14.07 ..3 8,87 2.63 "

13.9 " 21.3 2. 4 » 2,63 "

24.8 ... " .63 "

Without plotting curves, it may b© 8©©n that the mouth

correction is not constant, as it would be if it were a

definite function of the diameter. Instead, it is seen to

increase as the pipe length decreases. It decreases as the

pressure io increased an& also as the temperature increases.

f t were possible to vary the area or shape of the mouth

opening, some relation between correction and size or shape

of opening might appear, in fact, when I waa getting the

pipe in shape for this experiment, I found that the size of

the mouth opening had a greet deal to do with the pressure

at which the fundamental tone of the pipe would speak.

Since the presoure appears to influence the correction, it

seems reasonable to conclude that the size of mouth would

also have m effect.

n so far as my work can be used as a guide, it see .b

that the mouth correction varies with tern era turc length,

and pressure so that the frequency of the pipe follows the

simple rule,

H? V
4L

40
wherein N is the frequency, L the acoustic length, and V is

the velocity of sound in free air.

As a chec upon the accuracy of oy aeasurenents, I cal-
culated the acoustic lengths for 258 v.p.s. end 516 v. p. a.
and found the inverse octave relationship within •!£•

Referring again to the table of data for the mouth cor-
rection, t nay be seen that readings were ta en in a few
instances at various temperatures with identical condit ons
of a r pressure and pipe length., see also curves for C 3 >
page 15. Evidently the temperature frequency coefficient
raay be calculated from these readings to eheci: the previous
method. age 13). For C3 find, by this means, that the
coefficient is .002^; for E It is .0024; and for C^, .002*
Evidently, further wor.. is necessary before anything like an
accurate value for the ten erature- frequency coefficient can
be found,

SUGARY

n actual practice, the frequency of a stopped pipe i«
not inversely proportional to the length of the pipe itself.
The variation from such a relationship is caused by varia-
tion in the uouth correction at various lengths of the pipe.

J

41

The correction to length, necosr Vted by lac.: of
openness of the mouth, is not a function of the pipe d: a-
neter ae Rayleigh, llelniholta, and others ave assumed* It
is Inversely proportional to the length of pipe, pressure
of wind with wh'ch the pipe is blown, and the ten erature
■A which the pipe is operated. It is probable that the
shape or area of the r.outh opening also has an influence
upon the correction.

ncroasing the pressure of the wind with which a
stopped pipe is blown does not always increase the vibra-
tion frequency, although in w eneral it does* This is
another f enture which is not considered in any treatise on
the subject of sound*

ncreatse of temperature results in tn Increase of fre-
quency* The exact value of the temperature- frequency co-
efficient is iOwn,