End Correction at a Flue Pipe Mouth
by Johan
Liljencrants
Abstract
A simple notion is that the fundamental resonance of a pipe
occurs when the sound wavelength is half or a quarter of the resonator
length. It is however well recognized that the practical frequency
comes out lower than this, you have to apply an end correction, the
pipe appears to be acoustically somewhat longer than its physical
length. A formula for the basic mechanism behind this is theoretically
derived, then expanded into the case where the open end area is made
smaller than the pipe cross section.
The end correction was experimentally determined for several pipes with
mouths extending 360 and 90 degrees of the circumference. Formulas are
given to compute the end correction, using optimal coefficients found
from these measurements.
Introduction
The pitch f of
flue instruments like the flute, organ
pipe, or whistle is predominantly controlled by a resonator length L.
This length is closely connected to the wavelength  , where c is the
speed of sound. Examples
of the most common
cases are open and stoppered organ pipes, commonly characterized by
their working length as being half and quarter wavelength,
respectively. When such pipes oscillate at their fundamental pitch
their internal standing wave patterns of acoustic pressure and flow are
classically illustrated this simplified way. |
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Let us initially inspect a quarter wavelength
resonator tube of length L
and area A, closed at one
end. This tube has a total acoustical
mass 
and acoustical capacitance  ,
where 
is density of the air
and c is its speed of sound.
The effective resonating mass and capacitance
are less than these total
ones. They are reduced by the factor 
due to the sinusoidal distribution within the tube of flow
and pressure. (This factor is the area of a sine quadrant, divided by
the area of its circumscribed rectangle). Thus the effective mass is  and effective
capacitance  .
Inserting these into the resonance formula  and cleaning up, leads into the well known
quarter wave expression  for
the fundamental
resonance.
The flue end of a pipe where the air jet
blows across
the mouth is an open end of the resonator, having an acoustic flow
maximum. The acoustic pressure at this place
in principle is minimum. Still it is far from
zero, it is the sound pressure you can hear here. Inside the resonator the sound pressure is
considerably higher, its magnitude approaches the blowing pressure.
The air
immediately
outside the end of the pipe takes part in the acoustic oscillation.
This air makes the pipe appear to be acoustically somewhat longer
than its physical length. This apparent length
increase  is
called the end
correction. To compute the resonance frequency
the length measure one should use is the sum of physical length plus
the end correction. |
Theory
A theoretical basis for
computation of the end correction is the 'radiation acoustic impedance
of a circular
piston', reproduced here. This impedance tells the ratio of
acoustic pressure at the piston, divided by the flow rate induced by
it. The piston does not physically exist, it is an abstract
theoretical vehicle to state that one assumes the air speed to be the
same at all places across the tube end. This is a good
approximation, but not exactly true in reality, since air viscosity
reduces the flow rate in the boundary layer very close to the tube
surface.
Two different cases are illustrated here. In blue for a free tube end
and in
red for a baffled tube, i.e. when a wall limits the external sound to
spread only into a half space rather than all around.
There is a critical frequency, typically taken as kr=1, which implies that wavelength
equals the circumference of the piston/tube. At higher frequencies, or
greater r, the tube end tends
to
impedance match the ambience such that the tube does not act as a
resonator, but just a transmission line. The scale of whistles and
organ pipes is always such that kr
is substantially less than unity, so what applies to our
problem is the
left half of the diagram.
The impedance Z is composed
from two parts, the real resistance R
and the imaginary reactance X.
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R tells the
in-phase
component of the pressure to flow ratio. A given flow U will develop a power W=RU2
that is lost from the resonator and is radiated into the ambient space
to
become a useful sound. Normally this 'radiation resistance' is a
major determinant of the resonator Q value.
The reactance  is the
quadrature
component. The nice thing is that X turns out proportional to
frequency for kr<1, hence
its corresponding acoustical mass M
is constant within this kr<1 range. This
M
is the effective measure of the co-oscillating air outside the tube
opening. From the upward sloping blue X contour in the diagram, leaving
out trivial intermediate formula manipulations, we arrive at the most
commonly cited expression
 ,
where the equivalent tube radius is  .
Since it is rather immaterial whether the tube is
circular,
quadratic, or even moderately oblong rectangular, let us here stick to
the correction expressed in terms of the tube area, such that it
amounts to
For a baffled tube end we similarly find a slightly larger
value, namely 0.48 times the square root of tube area. |
End correction for a mouth
Now let us narrow the open end of the tube
with an aperture, into a mouth area B,
less than or equal to
the tube area A. This means
that we effectively remove the classical end correction and instead add
the acoustic mass
of a tube of area B.
The physical length of this tube is essentially
zero, but we have to add a new end correction for this one, indeed for
both sides of its
aperture. On the outside we can simply assume the same formula as
before. Similarly might apply also to the inside
of the aperture, but
only when B is appreciably smaller than the tube A. With a
growing aperture area such an internal end correction decreases to
ultimately vanish when B
reaches A. There are
plausible models for
this, but for the present we neglect any
internal correction.
There is a paradox
in that the basic correction formula for
the B aperture alone gives a
length
that is smaller than for the original A.
This might suggest the resonance frequency to rise because of the
constriction, contrary to all observation. To resolve this we must go
back to the circuit elements of the resonator. The capacitive element C (the elastic action of the
resonator interior) is like before, unaffected by the aperture.
But the inertive element M of
the tube is
augmented with the 
M that comes in the aperture. When
we compare the two as given above we get
The paradox is resolved by the fact, that when the
area B is made smaller, then
the additive M becomes larger. Recognizing that
for small deviations from 1, the square
root in the frequency formula functions like 
, we
then finally arrive at
We would have appreciated when the coefficient in
this result had rather been 0.34 , such that it complied with the
original formula when B=A.
But when we assume the mouth aperture to be partially baffled, that
would
correspond to a credible increase in the coefficient.
The importance of this theoretical derivation is to
suggest that the end correction is basically proportional to tube area over square root of mouth
area. It remains to find practical values for the proportionality
factor, and how to specify resonator physical length.
The basic
relation was used as a
direct value for the end correction by Ising (1969), apparently adopted
from Ingerslev and Frobenius (1947).
Experimental measurements
A number of
tubes with one end closed were examined for their passive fundamental
resonances. The open ends were adjustable for variation in cutup H.
The system was
excited by an external loudspeaker, driven from a tone generator, and
frequency was monitored
with a counter to 0.1 Hz accuracy. A
midget microphone was placed inside the resonator such that the
resonance peak could be located by adjusting frequency. Having found
the -3 dB points fa
and fb of the
resonance
peak, then the resonance is found as f1=(fa+fb)/2 while the quality
factor Q=f1/(fb-fa).
Knowing f1
and sound speed (ambient temperature should be accounted for) the
actual quarter wavelength is computed. This then equals the physical
resonator
length L plus a sought,
experimentally determined end correction. This is compared to the
theoretical model, the right member of the expression
The various experimental values were collected in an Excel spreadsheet.
A final step is to find values for the coefficients 
, by trial and error, such
that the model values (computed from A, B, and H) come maximally close
to the experimental ones.
Cylindrical whistle, 360 degree mouth
A
set of cylindrical tubes were used, where length L from stopper to end was in all
cases set to 187 mm, corresponding to resonance in the 400 Hz range.
The internal diameters were 13.4, 24.5, 46.0, and 69.5 mm, wall
thickness between 1.5 and 2 mm. For each, a block of same outer
diameter was placed at variable
distances H from the open end
in order to simulate the pipe languid and foot.
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The two coefficients for
H
and shown here are
empirically
adjusted for a best match between theory and experiment, using
the same values for all tube
diameters. The data points line up reasonably well along
the
diagonal
in this plot of measured vs. predicted end corrections.
The RMS error between data and model is 1.2 mm. Optimizing for a single
tube diameter, the
match may improve using slightly different coefficients. But including
the tube diameter as another parameter in the optimization did not
improve
the match substantially.
The small
colored figures tell values of H.
Here resonator length is taken as the
tube proper, such that the case of infinite cutup can be included. But
common practise is otherwise to define the resonator length as
the sum of L and H.
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is interesting to note the additional term with the cutup H.
This apparently accounts for an extra distance for the sound wave to
travel, from the end of the tube to an effective center of the mouth
opening. This may come from that the mouth is located at the side of
the tube rather than centrally at its end, as was postulated in the
theoretical model. |
Square organ pipe, mouth on one side
| The
following data were obtained from two regular square
stoppered wooden organ pipes where the cutup could be varied by moving
the
labium. This means that the front plate length L differed between data points,
while T was constant. |
Here the common way, defining the resonator length as T = L+H is shown in the lower
formula and the graph, while the upper formula uses the front plate
length L. The two formula alternatives render identical results for the
effective length Le,
which is the quarter wavelength at fundamental resonance.
These two pipes render the same optimal coefficients once the length T of the bigger one was
artificially increased by 8 mm. This extra correction is the thickness
of its pipe walls, apparently introducing an extra length or a baffling
effect. Here the RMS error between data and model is
1.7 mm. |
Because
this graph and formula is based on the total length T, the end
correction becomes
negative at sufficiently large H.
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The
most obvious difference against the previous 360 degree mouth is that
the coefficient for the term
is now much bigger, supposedly because the external sides of the pipe
act as baffles.
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Conclusions
The foundation for the mouth end correction was theoretically
established as having a factor where A and B are
the areas of resonator and mouth.
This was verified by resonance measurement on a number of different
pipes. In matching the end correction model to the experimental data it
was found that the weight of this factor differs very much depending on
the rotational extent of the mouth opening, here 90 and 360 degrees
were examined. For a mouth at the side of the resonator tube also a
fraction of the cutup height should be added to the end correction.
It must be noted that the fundamental speaking frequency of the
pipe when blown is not precisely the same as the resonance frequency
studied here.
When blown, the pipe adjusts to whatever phase angle is imposed by the
flue exciting mechanism, most probably characterized by its Ising
intonation number. This, in turn, is determined from cut up, flue
airband thickness, and blowing pressure.
References
Beranek L L (1954): Acoustics.
McGraw-Hill.
Ingerslev F, Frobenius W (1947): Trans. of the Danish Acad. of
Technical Science, No 1.
Ising H (1969): Über die Klangerzeugung in Orgelpfeifen.
Diss.,Technisches Universität Berlin.
2006-09-30
JLs