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recorder, flute, organ pipe, edge tone, edgetone, acoustic oscillator,
transit time oscillator, acoustic feedback, frequency, wavelength,
boundary layer theory, turbulent jet, jet velocity, jet slowing, phase
velocity
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in tables be properly aligned.
The Theory of the Edgetone Oscillator
Wade Blocker
wblocker@nmol.com
Abstract
The resonant acoustic column of such musical instruments as the flute is not
necessary for a tone to be produced. A tone results when a jet of fluid is
blown against an edge. Past efforts to explain this edgetone oscillator have
been unsuccessful. The assumption that the edgetone oscillator is an
acoustic transit time oscillator closely analogous to electronic oscillators
leads to a theory in almost exact numerical agreement with all critical
experimental data.
Table of Contents
Introduction
Edgetone Phenomenology
Past Experiments and Theories
A Transit Time Oscillator, fT = k ?
The Predicted fT Product Sequence
The Transit Time of a Jet Particle
Comparison of the Present Theory and Experimental Data
The Jet Wave Phase Velocity
Comparison of the Present and Past Theories
Summary
References
Introduction
Although such musical instruments as the panpipe, recorder, flute, organ
flue pipe, and common whistle had been known for millennia, it was only
discovered in 1854 by Sondhaus (Ref. 1) that the resonant acoustic column or
cavity associated with these instruments is not necessary for a tone to be
produced. A tone is produced when a jet of fluid from an aperture is blown
against an edge. Since the discovery of these edgetones an extensive
literature has accumulated, but no satisfactory theory was developed. In
1940 Lenihan and Richardson (Ref. 2) wrote "The problem of edge tones is one
which continues to form a battle-ground for rival theories, though a
complete solution seems as far off as ever." This statement has remained a
challenge for theorists to this date. No prior theory of edgetones has been
able to predict the data of basic experiments. In this paper it is assumed
that the edgetone acoustic oscillator is just another feedback oscillator
that can be explained in a manner closely analogous to how ordinary
electronic feedback oscillators are explained. The assumption that the
edgetone oscillator is a transit time oscillator very much like electronic
transit time oscillators leads to a theory in almost exact numerical
agreement with all critical experimental data, with no empirical or ad hoc
elements introduced to force a fit to the data.
Edgetone Phenomenology
The basic phenomenology of edgetones seems simple. A tone is produced when a
jet of fluid from an aperture is blown against an edge. The jet and
aperture-edge system are immersed in a surrounding fluid. Oscillations occur
in both liquids and gases. We will consider the aperture-edge system to be
defined by the simple diagram shown here composed of an equal sign " = " and
a less-than sign " < "
= <
where the two parallel lines of the equal sign " = " define the slit through
which the jet of fluid is blown and the less-than sign " < " defines the
edge against which the jet is blown. In this diagram take the origin of
coordinates at the center of the aperture's exit, with the positive
direction of the x-axis extending to the right through the point of the
edge, the positive direction of the y-axis extending upward in the plane of
the drawing perpendicular to the x-axis, and the positive direction of the
z-axis extending outward perpendicular to both x and y axes toward the
reader. The aperture, edge, and the plane jet of fluid emitted from the
aperture in the positive x-direction will be considered infinite in both z-
directions, thereby reducing the problem to be analyzed to two dimensions.
The x-z plane divides space into two halfspaces, an upper and a lower.
The aperture slit width, which we will denote by b, is the distance between
the two parallel lines of the equal sign which define the slit through which
the jet particles are blown. We will denote the aperture to edge distance by
X, therefore the x-coordinate of the edge point is X. The velocity of a jet
particle in the aperture to edge gap at the distance x from the aperture
will be assumed a function of x only, therefore independent of any
displacement y of a jet particle from the x-z plane, and will be denoted by
v(x). Therefore the initial jet particle velocity at the jet aperture will
be denoted by v(0). We are assuming that any displacement y of a jet
particle from the x-z plane is a minor perturbation to the motion of the
jet. The success of our theory will justify the assumption.
The aperture slit width b, the aperture to edge gap distance X, and the
initial jet particle velocity v(0) fix the frequency f of oscillation, and
it is also necessary to specify the oscillation mode or stage number.
We will first discuss the experimentally observed effect of changing the
blowing velocity v(0) for fixed b and X. As the velocity increases from
zero, oscillations begin at some point and the edgetone is produced. The
oscillation frequency then increases linearly as a function of the
velocity v(0), along a line of constant slope. At some velocity the
frequency jumps upward and then increases linearly with velocity v(0) along
a second line of greater slope. With a further velocity increase the
frequency jumps again to a third line of still greater slope. As many as
five or six jumps have been seen. Beginning with the lowest line, these
modes of oscillation are called stage 1, stage 2, stage 3, etc. With a
decrease of the velocity v(0) the frequency decreases along one of the lines
and downward jumps back to stage 1 occur. The upward and downward jumps do
not necessarily occur at the same points so this oscillator shows hysteresis
just as most oscillators do. The lines are observed to be straight, and if
extended go through or very near the origin.
We will discuss next the experimentally observed effect of changing X for
fixed b and v(0). As the gap width from aperture to edge increases,
oscillations begin at some point. The frequency then drops as the gap width
increases. There are frequency jumps to higher stages, consequently to
higher frequencies, as the gap width is sufficiently increased, but in any
given stage the frequency decreases as the gap width increases. Higher
frequencies in a given stage and then jumps back to lower stages with
hysteresis effects are seen if the gap width is decreased. The stages
identified are the same as those seen when the velocity was varied. Most
experimenters have found the frequency in a fixed stage to vary inversely as
the first power of the gap width, but some experiments have been done in
which the frequency varied inversely as the three-halves power of the gap
width. These latter experiments have been mostly ignored by theorists
attempting to explain edgetones.
Lastly we discuss the experimentally observed effect of changing the slit
width b for fixed v(0) and X. As b increases, oscillations begin at some
point. For sufficiently small values of b, the frequency appears to increase
linearly with b as the slit width increases, then to increase more slowly
with b as the slit width increases further, and finally appears to approach
a limit with still further increases in slit width. Presumably jumps between
stages might occur but they have not been reported. This data too has been
mostly ignored by theorists.
The ratio of the two frequencies observed in two different stages for the
same parameters b, v(0), and X is a constant for these two stages,
independent of the particular parameters chosen. If one of the two
frequencies is known, the other is automatically determined.
Most writers on edgetones would add to the experimental facts just given
that the jet disturbance in the gap when oscillations are occurring is
characterized by a phase velocity which is about half the jet particle
velocity, both velocities being tacitly assumed constant across the aperture
to edge gap. This an inference from certain theoretical and experimental
work. The present paper shows that this inference is incorrect. Its general
acceptance by theorists is one of the major reasons for prior failures to
develop an adequate theory of edgetone production.
Past Experiments and Theories
The number of experimental and theoretical papers written on edgetones is
very large. Only a few of the most important will be reviewed briefly here.
For the first oscillation stage, Koenig in 1912 (Ref. 3) proposed the
empirical equation
f = v(0)/2X (1)
He did not explain why the factor 2 in the denominator was necessary to fit
the experimental data.
Schmidtke in 1915 (Ref. 4) thought that the different frequencies observed
in different stages for the same fixed set of parameters b, v(0), and X
should be harmonically related. Later more accurate experiments showed this
to be incorrect. Schmidtke extended Koenig's equation as
f = nv(0)/2X (2)
where n was the stage number.
Krueger in 1920 (Ref. 5) proposed that the factor 2 in the denominator of
equations 1 and 2 was properly associated with the jet particle velocity
v(0) rather than with the aperture to edge distance X and indicated that the
phase velocity of the jet disturbance in the aperture to edge gap was only
half the jet particle velocity. This proposal was later almost universally
accepted by those attempting to explain edgetones. It did give a plausible
reason for the otherwise unexplained factor 2. However, the proposal will
turn out to be wrong. His form of the frequency equation would be
f = n[v(0)/2]/X (3)
Carriere in 1925 (Ref. 6) published experimental results showing the
variation of the edgetone frequency with slit width b, with other parameters
fixed. This is the only data the author is familiar with that gives edgetone
frequency f as a function of slit width b, with other parameters fixed, and
is the data on which the discussion above of the variation of edgetone
frequency with slit width b is based. (The author is aware that Carriere's
name, properly spelled, has an accent mark. I apologize for its omission.)
Carriere also gave experimental results in which the frequency varied
inversely as the three-halves power of the aperture to edge distance X.
Carriere's data did not receive much attention from later writers on
edgetones. However, his data will be extremely important in verifying the
theory to be offered in this paper.
Brown in 1937 (Ref. 7) published an extensive and perhaps the best yet
collection of experimental data. His experimental data are excellent, and
are of primary importance in verifying the theory of the present paper.
Brown had made frequency measurements for more than one value of slit width
b and affirms Carriere's finding that the frequency is a function of the
slit width, but he gives detailed data in this paper for only one slit
width. He established that the frequencies in different stages are not
harmonically related. His data give the edgetone frequency for a single slit
width b as a function of initial jet velocity v(0) and gap width X. Although
not stated by Brown, as pointed out later in this paper it can be deduced
from Brown's data that for some conditions the edgetone frequency is
inversely proportional to the three-halves power of the gap width X, thus
confirming this finding also of Carriere's. Brown took photographs of smoke
filled jets on which he was able to measure wavelengths of the jet wave near
the edge. These wavelengths multiplied by the frequency gave a phase
velocity which was about one-half of the jet particle velocity at the
aperture. Brown interpreted this to mean that the phase velocity of the jet
wave was about one-half the jet particle velocity in general agreement with
the supposition of Krueger, tacitly assuming that each velocity had a single
unique velocity regardless of position in the gap. This interpretation by
Brown of his data is incorrect. He tacitly assumed that the jet particles
did not slow down, and that the wavelength and phase velocity at all points
in the gap were equal to the wavelength and phase velocity which he
determined just before the edge. He gave values for the different stages of
the gap width divided by his measured wavelengths, assuming this to be the
number of waves in the oscillation of the jet in the gap. This number turned
out to be approximately equal to or somewhat larger than the stage number,
leading to the conclusion that the number of wavelengths in the gap was
equal to or greater than the stage number. All of Brown's assumptions just
mentioned are wrong and every conclusion of Brown's based on these
assumptions is wrong. This will be discussed in detail later. Most later
theorists attempting to explain edgetones relied heavily on Brown's data and
his interpretation of that data. They overlooked the errors of
interpretation which Brown made which this paper points out. In checking a
theory against Brown's data, the excellent original data as recorded by
Brown should be used and not the data as interpreted by Brown. Brown's
purely empirical equation for the frequency, slightly simplified, is
f = n v(0) / 2X (4)
where n = 1.0, 2.3, 3.8, 5.4 for stages 1, 2, 3, and 4.
The papers by Carriere and Brown are presently the definitive papers that
together best set forth the experimental facts of edgetone oscillations that
were outlined above. These are the authors whose data every theorist on
edgetones should try to explain and predict. The most significant parts of
their data will be given later in this paper. Brown's paper has received the
preponderance of theoretical attention. The present author is not aware of
any prior theoretical paper giving attention to explaining Carriere's
results, although his results have not been challenged.
Jones in 1943 (Ref. 8) published remarks that confirmed Carriere's finding
that for some conditions the frequency varied inversely as the three-halves
power of the aperture to edge distance. Jones noticed that this occurred
when the gap was wide and the jet was turbulent. Jones suggested a mechanism
for the tone production similar to that adopted in the present paper.
However Jones did not go beyond suggesting this mechanism to analyze
quantitatively the consequences and did not produce a successful theory.
Jones' remarks received little attention, although they offered the key for
the solution of the edgetone problem.
Curle (Ref. 9) and Powell (Ref. 10) in 1953 both relied on the presentation
and interpretation Brown gave of his data, and each independently proposed
that there were one and a quarter, two and a quarter, etc., wavelengths of
the jet wave across the aperture to edge gap. They proposed the purely
empirical equation
f = [n + (1/4)]v(0)/2X (5)
where n is the stage number. This equation conformed to Brown's
interpretation of his data which gave the number of wavelengths of the jet
wave in the aperture to edge gap as equal to or greater than the stage
number of the oscillation occurring. It will be shown later that Brown's
interpretation is wrong, and that his experimental data, properly
interpreted, offer no support for this equation, which must be recognized
therefore as an unsupported conjecture. It cannot be used to predict the
data of Carriere and Brown.
The frequency equations proposed were empirical. These equations do not
give any attention to the inverse three-halves power dependency of frequency
on gap width X seen by both Carriere and Jones (and also deducible from
Brown's data); nor to the strong dependency of frequency on aperture slit
width b known to exist from the work of Carriere, and confirmed by Brown
although Brown gives no detailed data. The parameter b, the aperture slit
width, is not a parameter of these equations; in effect, they ignore the
dependence of edgetone frequency upon aperture slit width without offering
any explanation for that ignoration. Theorists continued to treat the jet
velocity as constant, despite the known fact that these jets slow down
rapidly.
A Transit Time Oscillator, fT = k ?
It will be assumed that the edgetone oscillator is an ordinary feedback
oscillator that can be explained in a manner exactly analogous to how
electronic feedback oscillators are explained. A plausible feedback
mechanism and feedback loop will be defined, and a search made for the phase
condition on the feedback loop that results in the positive feedback that is
necessary for continuous oscillations.
The critical factors that control the frequency of any feedback oscillator
are time relationships. The basic parameter is time. Other factors, such as
the geometry of the oscillator, are important only in that they control the
timing of events. With this viewpoint, an obvious possibility is that the
edgetone oscillator may be a transit time oscillator, whose frequency is
determined by the transit time of a jet particle from the jet aperture to
the edge. To check the assumption that the edgetone oscillator is a transit
time oscillator is simple, but it seems not to have been done previously.
The only requirement is that we predict the transit time, which is easily
accomplished. This oscillator has perhaps the simplest frequency equation
possible. For a transit time oscillator, the frequency equation is
fT = k (6)
where f is the frequency, T is the transit time, and k is a constant for a
given stage or oscillation mode. There is a sequence of values of k, with
each stage or oscillation mode having its own unique value of k. k is the
number of whole periods of the oscillation in the transit time T plus a
fraction which is any excess of fT over an integral number of periods. If T
is approximated by T = X/v(0), this equation becomes very similar to the
empirical equations adopted by previous theorists. The philosophies are very
different, however, since equation 6 demands a realistically determined
transit time taking account of jet slowing. The slowing of the jet turns out
to be critical. Previous theories tacitly assumed that the jet velocity
remained constant all across the gap and did not explicitly take account of
jet slowing, which is almost surely the major reason for previous failures
to develop a successful theory of edgetones.
With the assumption that the edgetone oscillator is a transit time feedback
oscillator, analogous to electronic oscillators, explaining edgetones
theoretically is reduced to two independent problems, predicting the fT
product sequence, or values of k, as a function of stage number, and
predicting the transit time T of a jet particle from aperture to edge. In
making these predictions we will assume that the longitudinal motion (x-
motion) and transverse motion (y-motion) of the jet particles are
independent of each other. That is, we treat the oscillation as a
perturbation to the motion of the jet. This procedure turns out to be
successful.
It is possible to dispense with predicting the fT product sequence
theoretically, and to determine the sequence empirically by simply
calculating T using current theories of jet slowing, and then multiplying
the experimental values of f in each stage by their corresponding T values.
The empirical result is that indeed each oscillation mode or stage has its
own unique value fT. This will be shown later by our Table 1. The advantage
of the theoretical approach is that it also allows conclusions to be drawn
about the phase velocity of the jet wave in the aperture to edge gap, which
can be compared with Brown's data. The theoretical approach will be shown
first.
The Predicted fT Product Sequence
The plane through aperture and edge divides all space into two halfspaces,
an upper and a lower. Because of the mirror symmetry about the x-z plane, we
assume that this is a push-pull oscillator. Therefore the pressure
fluctuations on the upper and lower sides of the jet should be 180 degrees
out of phase when oscillations are occurring. The jet particles of necessity
respond to the pressure difference on the two sides of the jet. We postulate
that, driven by the alternating pressure difference between the two
halfspaces which acts on the jet particles, the jet particles are directed
in alternating puffs into the upper and lower halfspaces. This creates the
alternating pressure difference between the two halfspaces, and this
alternating pressure difference controls the y-component of the jet particle
motion, thereby completing the feedback loop required for continuous
oscillations to exist. This is exactly analogous to how the ordinary
electronic push-pull oscillator is explained, with pressure variations in
the acoustic oscillator being taken as the analog of voltage variations in
the electronic oscillator. A similar mechanism has been suggested before
[Jones (Ref. 8)], but was not followed to the point of producing a
successful theory. With the oscillator identified as an acoustic feedback
oscillator, and the feedback loop defined, we look for the phase condition
on the feedback loop that reinforces the excitation and makes the
oscillations self-sustaining.
Assume the pressure difference across the jet varies sinsusoidally with
time. Since the aperture to edge gap distance in the usual case is a small
fraction of the wavelength of the tone produced by the edgetone oscillator,
we assume that this pressure difference has the same value all across the
gap at any given instant of time. (The wavelength of the tone produced, c/f,
where c is the velocity of sound in the fluid, should not be confused with
the wavelength seen in the jet wave.) This sinusoidal pressure difference is
the dominant influence on the transverse or y-motion of the jet particles.
(Instead of using standard mathematical notation, I will write some
following expressions as they would appear if written and intended to be
executed in a Basic language program. In particular I will use * to indicate
multiplication, and ^ to indicate exponentiation, and the standard rules for
the order of evaluation are assumed.)
Define w as 2*pi*f , t as clock time, e as the time a particular fluid
particle is emitted from the aperture, and T as (t - e). T is thus the
length of time that a particle emitted at the time e has been traveling
since it was emitted from the aperture. For the usual Greek letter pi, I
have substituted the word pi, and for the usual Greek letter omega I have
substituted the letter w. The quantity f is the frequency of oscillation.
The differential equation for the motion of the jet particles in the y-
direction is therefore
y'' = sin(wt) (7)
with the initial conditions that y = y' = 0 when t = e or when T = 0. The
primes indicate differentiation with respect to time. Any constant amplitude
factor has been taken as unity in this equation since it would not affect
the argument that will be made.
Integrating this equation with the given initial conditions, we obtain the
solution
y(wt, we) = (wt - we)cos(we) + sin(we) - sin(wt) (8)
Again any constant factor common to all terms on the right side of the
equation has been ignored. From equation 8 we can derive other valid forms:
y(wT, we) = (wT)cos(we) + sin(we) - sin(wT + we) (9)
y(wt, wT) = (wT)cos(wt - wT) + sin(wt - wT) - sin(wt) (10)
y(wt, wT) = (wT)cos(wt - wT) - 2sin(wT/2)cos[wt - (wT/2)] (11)
Equation 9 gives us the path in the aperture to edge gap of a particle
identified by its time of emission e from the aperture. Equation 10 will
give us the values of k in equation 6 which define the expected edgetone
frequencies, and equation 11 will give us information about the phase
velocity of the jet wave in the aperture to edge gap. Equations 10 and 11,
coupled with an equation from standard fluid dynamics theory for the slowing
of a turbulent jet, will allow us to predict with remarkable accuracy the
frequencies and phase velocities that Brown observed in his experiments.
These predictions are made without introducing any empirical or ad hoc
factors into the theory to force a fit to the experimental data.
The kinetic energy lost by the jet is the energy source for the edgetone
oscillations. From a simple assumption concerning this energy loss a
frequency condition can be derived which turns out to be in agreement with
the experimental data. Assume that the instant of most rapid increase of
pressure in a halfspace coincides with the instant of the greatest rate of
loss of kinetic energy by the jet to that halfspace. The assumption will be
justified by the success of the resulting theory. For a sinusoidal pressure
variation, the instant of most rapid increase of pressure occurs as the
pressure changes from negative to positive values and goes through zero.
The instant of the greatest rate of loss of kinetic energy by the jet to a
halfspace should be the instant of greatest mass commitment to the
halfspace. The greatest mass commitment to a given halfspace should occur as
the jet particles switch simultaneously at the aperture and edge out of that
halfspace. Jet particles just out of the aperture always switch from one
halfspace to the other in phase with the driving pressure, that is, y(wt,
wT) for values of wT infinitesimally greater than zero changes from negative
to positive values as the driving pressure sin(wt) changes from negative to
positive values, so we require that the same be true at the edge if
oscillations are to occur. The driving pressure sin(wt) is zero and switches
from negative to positive values when wt equals zero or a multiple of 2*pi,
so y(wt, wT) at the edge must also be zero and switch from negative to
positive values when wt equals zero or a multiple of 2*pi. This reduces
equation 10 to an expression which we postulate is a required condition on
wT at the edge if oscillations are to occur.
wT = tan(wT) (12)
This equation is easily solved with a programmable electronic calculator or
computer, but it is also a well known equation whose solutions are tabulated
in many texts or collections of mathematical data. The solutions meeting
our criteria are the alternate zeroes of this equation. The solutions of
interest lead to
fT = 1.230, 2.239, 3.242, 4.244, 5.245, ... (13)
taking into account that wT = 2*pi*fT.
The product fT = 0.500 also meets our criterion of greatest mass commitment.
The jet particles at the aperture always switch from one halfspace to the
other in phase with the driving pressure and flow into a halfspace for a
halfperiod, so a maximum net mass commitment to a halfspace results if the
edge distance is such that jet particles reach this distance one halfperiod
after they are emitted from the aperture, even though no switching of the
jet at the edge from one halfspace to the other occurs in this case. This
additional term will be added to give the prediction that the observed
values of fT should occur in the sequence
fT = 0.500, 1.230, 2.239, 3.242, 4.244, 5.245, ... (14)
for the stage numbers 1, 2, 3, 4, 5, 6, ... .
The sequence 13 gives the frequencies of best operation of the ordinary
electronic transit time oscillator known as the klystron. This can be
deduced from the analysis of the klystron oscillator given by Marcuse (Ref.
11). Evidently, the klystron has no mode corresponding to stage 1 of the
edgetone oscillator.
Equation 14 immediately accounts for the numerical factor of 2 required in
the empirical equations 1 through 4 to fit the experimental data for stage
one. This is evident upon approximating T by X/v(0) for stage one. This is
not an adequate approximation for wide gaps, and consequently not usually
adequate for the higher stages of oscillation, and perhaps not always
adequate for stage one. By "wide gap" in the context of this paper is meant
that the gap width X is very much greater than the slit width b. The slowing
of the jet in crossing the gap is very important for wide gaps, amounting to
more than one-half in the usual case as will be shown later. Equation 14
automatically makes provision for the effects of jet slowing.
Equation 14 gives the basic prediction of the present theory. This
prediction is purely theoretical and has no empirical or ad hoc elements
introduced to force a fit to the experimental data. If this equation is in
agreement with the experimental data, then it is established that the
edgetone oscillator is a transit time oscillator. The transit time T is the
controlling parameter determining edgetone frequency. The system parameters
b, v(0), and X are of only incidental importance insofar as they determine
T. All sets of these parameters giving the same value of T will produce the
same frequencies f. That this equation is correct will be established by
comparing its predictions with Carriere's and Brown's experimental data.
The Transit Time of a Jet Particle
The transit time of a jet particle across the gap from aperture to edge when
there are no oscillations will be taken as a sufficient approximation to the
transit time when oscillations are occurring. The jet will be assumed
turbulent. For narrow gaps this assumption even if wrong will cause no
significant error in the calculated transit time and for wide gaps the jet
is almost surely turbulent, as observed by Jones. The transit time of the
non-oscillating turbulent jet can be calculated from the equations of jet
motion for a turbulent jet given in Schlichting's text on boundary layer
theory (Ref. 12). For the velocity v(x) of a jet particle at the distance x
from the jet aperture, Schlichting gives the equation
v(x) = v(0)/[1 + (x/x0)]^(1/2) (15)
x0 in equation 15 is defined by Schlichting. It is the distance before the
jet aperture of the hypothetical source point of the jet. From the general
equations given by Schlichting, it can be shown that
x0 = 3 s b/4 = 5.75 b (16)
where s (Schlichting uses the Greek letter small sigma) is an empirical
constant with the value 7.67 which is given by Schlichting (Ref. 12, pp.
605-607). This x0 is defined by the condition that the momentum flux for the
jet is b*v(0)^2. The jet behavior does not depend upon the density of the
fluid, or upon whether the fluid is a liquid or a gas.
From equation 15 the time increment dT required for a jet particle at
position x to travel the distance increment dx is
dT = [dx/v(0)]*[1 + (x/x0)]^(1/2) (17)
Integrating this equation with the initial condition that T = 0 when x = 0,
we obtain for the time T(x) required for a jet particle to reach the
distance x the result
T(x) = T0*{[1 + (x/x0)]^(3/2) - 1} (18)
where
T0 = (2/3)*[x0 / v(0)] (19)
The equation for T(x) can be solved for x to give
x(T) = x0 * {[1 + (T/T0)] ^ (2/3) - 1} (20)
where x(T) is the distance traversed by a jet particle in the time T.
The Comparison of the Present Theory and Experimental Data
It is immediately apparent without numerical calculation, by approximating
equation 18 for large and small values of b and x, that the predictions of
edgetone behavior from equations 18 and 14 are in qualitative agreement with
every aspect of the experimental behavior of edgetones pointed out above.
For fixed b and X, the frequency is predicted to vary directly as v(0). For
fixed b and v(0), the frequency is predicted to vary inversely as X for
small X, and inversely as the three-halves power of X for large values of X.
The fact that the jet is turbulent automatically introduces the three-halves
power of X necessary to fit the experimental data for wide gaps. For fixed
v(0) and X, the frequency is predicted to increase linearly as a function of
b for small values of b, and to approach a limit as b becomes large, in
agreement with Carriere's data. The parameter b although universally
neglected in previous theoretical attempts to explain edgetones is seen to
be as important as the parameter X in determining the edgetone frequency.
The constancy of the ratio of frequencies observed in different stages for
the same set of parameters b, v(0), and X, is predicted. The empirical
equations 1 through 4 for stage one are predicted by the first term of the
sequence 14. In every case the predicted behavior is in qualitative
agreement with the observed experimental facts.
It will now be shown that the predictions are in quantitative agreement with
the available experimental data. The best collections of experimental data
are those of Carriere and Brown. The critical test of any theory is to
predict the results of their experiments.
The most extensive compilation of experimental data is that of Brown. The
data of Brown's Table 1 will be exhibited in our Tables 2 and 3. For each
entry in Brown's Table 1, using Brown's b, v(0), and X as inputs to our
equation 18, the transit time T to the edge was calculated. This value of T
multiplied by the frequency f that Brown observed for that case gives the
results tabulated in our Table 1.
Table 1. Comparison of Theory with Experiment.
The fT Products for Brown's Table 1.
Stage Number 4 3 2 1
......................................................
f fT, Experimental
......... .....................................
20 -- 1.95 1.17 0.48
100 3.20 1.97 1.12 0.45
150 3.11 2.12 1.22 0.50
1200 3.31 2.24 1.24 0.47
2400 3.29 2.21 1.31 0.51
Average 3.23 2.10 1.21 0.48
......... ......................................
f fT, Theoretical
......... ......................................
All 3.242 2.239 1.230 0.500
.......................................................
The experimental values of fT in Table 1 are nearly constant for a given
stage and are in close agreement with the values our theory predicts. We
conclude that the edgetone oscillator is indeed a transit time oscillator.
Since the calculation of the values of T is independent of the theory that
predicted the theoretical fT sequence, the near constancy of the
experimental values of fT for a given stage establishes empirically,
independently of our theory for the fT sequence, that the edgetone
oscillator is a transit time oscillator.
For a further comparison with the experimental data of Brown's Table 1, the
gap width X for each entry in Brown's table was calculated as a function of
the stage number, b, v(0), and f. The transit time T was first found from
the sequence 14 using the stage number and observed frequency f. The gap
width X was then predicted using equation 20. The comparison of the
theoretical and experimental values of X is shown in our Table 2.
Table 2. Comparison of the theory with Brown's experimental data.
The slit width b is 0.1 cm in all cases. With the stage number, b,
v(0), and f as inputs to the theory, all gap widths X have been
predicted. The numbers appended to X in the column headings indicate
stage numbers. This table has the same arrangement of parameters as
Brown's Table 1.
...............................................................
f X4 X3 X2 X1 v(0)
(Hz) (cm) (cm) (cm) (cm) (cm/s)
...............................................................
Experiment 20 -- 5.68 3.92 2.02 137
Theory 8.13 6.26 4.07 2.08
Experiment 100 3.48 2.41 1.57 0.75 212
Theory 3.51 2.67 1.69 0.82
Experiment 150 3.33 2.50 1.64 0.79 309
Theory 3.44 2.61 1.65 0.80
Experiment 1200 1.74 1.28 0.79 0.34 984
Theory 1.71 1.28 0.78 0.36
Experiment 2400 1.58 1.15 0.75 0.33 1750
Theory 1.56 1.16 0.71 0.32
...............................................................
The agreement of the theoretically calculated values of the gap width X with
Brown's experimental values is excellent, being exact or almost exact in
many cases. The calculation of Brown's Table 1 shown in our Table 2 does not
critically depend upon the theory used to predict the theoretical fT
sequence of equation 14. Our Table 1 justifies an empirical fT sequence fT =
0.50, 1.25, 2.25, 3.25, ..., which is so close to the theoretical sequence
of equation 14 as to make little difference in calculating Brown's Table 1
using equation 20, with T from this empirical sequence rather than from
equation 14. It is therefore established empirically, as well as
theoretically, that the edgetone oscillator is a transit time oscillator. To
challenge this conclusion is to challenge not just the theory behind the
predicted fT product sequence of equation 14, but to challenge either
Brown's experimental data or Schlichting's equation for the slowing of a
turbulent jet, since these two things alone suffice to give the fT product
sequence and to predict Brown's Table 1.
The theory can be compared with Brown's data in other ways. Table 3 shows
the comparison of the predicted values of the frequency with the
experimental values found by Brown. The transit time T was calculated for
each of Brown's cases using equation 18, and the frequency was then found
substituting this T into equation 14. The predicted frequencies are within a
few percent of the experimental values.
Table 3. Comparison of the theory with Brown's experimental data. The slit
width b is 0.1 cm in all cases. The first line in each group of three lists
the experimental values of gap width X and initial jet velocity v(0) that
give the experimentally observed frequency in the second line. The
corresponding theoretically calculated frequencies are given in the third
line.
.............................................................. ........
Stage Number 4 3 2 1 v(0)
(cm/s)
.............................................................. ........
Gap Width X, cm -- 5.68 3.92 2.02 137
f experimental, Hz -- 20 20 20
f theoretical, Hz -- 23 21 21
Gap Width X, cm 3.48 2.41 1.57 0.75 212
f experimental, Hz 100 100 100 100
f theoretical, Hz 101 114 110 111
Gap Width X, cm 3.33 2.50 1.64 0.79 309
f experimental, Hz 150 150 150 150
f theoretical, Hz 157 159 151 152
Gap Width X, cm 1.74 1.28 0.79 0.34 984
f experimental, Hz 1200 1200 1200 1200
f theoretical, Hz 1180 1200 1210 1270
Gap Width X, cm 1.58 1.15 0.75 0.33 1750
f experimental, Hz 2400 2400 2400 2400
f theoretical, Hz 2370 2440 2250 2340
.............................................................. ........
Our equations do fit and predict Brown's experimental data. The agreement of
the theoretical predictions with Brown's experimental data is excellent,
within a few percent. The theory has no empirical or ad hoc elements
introduced to force a fit to Brown's experimental data. Brown's data cover a
very wide range of parameters, a factor of 120 for frequency f, 12.8 for
initial jet velocity v(0), and 17.2 for gap distance X. The agreement of the
theoretical predictions with Brown's experimental data is too close to allow
a conclusion that the agreement is a fortuitous accident. It will be shown
later that Brown's data on the phase velocity of the jet wave are also
predicted.
Brown's data for the lower frequencies and wider gaps give values of the
ratio (X/x0) appreciably greater than one, so therefore our equations 14 and
18 predict that the edgetone frequency in these cases should be exhibiting
an inverse three-halves power dependence upon gap width X. That our
equations predict Brown's data shows that this is the case. Brown's data
therefore confirms what Carriere and Jones both observed, that for some
conditions the frequency varies inversely as the three-halves power of the
gap width X.
The theory will next be compared with Carriere's data giving the frequency f
as a function of slit width b with other parameters fixed. Unfortunately
Carriere gave the blowing pressure rather than the initial jet velocity so
we are faced with the problem of converting blowing pressure to initial jet
velocity. The conversion is not straightforward. For Bernoulli's equation to
apply it is necessary that the jet flow be steady, frictionless, and along a
streamline, and that fluid density be a function of pressure only (Ref. 13).
These conditions are not always, or even usually met. For a Borda tube
aperture it has been shown theoretically that the initial jet velocity is 50
percent of the value predicted by Bernoulli's equation, for a sharp edged
orifice the jet velocity is typically about 63 percent of the Bernoulli
value, and for a Venturi orifice it can be close to 100 percent (Refs. 14).
From the apparatus diagrams given by Carriere, conversion factors of about
50 to 63 percent seem appropriate, and conversion factors in this range give
good agreement with our theoretical predictions while higher values would
not. This conversion factor is an empirical factor we are forced to
introduce in order to compare the theory with Carriere's data.
Our Table 4 shows the comparison of theoretical predictions with part of
Carriere's data, using 63 percent as the conversion factor in determining
the jet velocity from Carriere's pressure values. The comparison was made in
two ways. First, the theoretical values of T(X) were calculated using
equation 18 and the resulting fT products were found for comparison with the
predicted values of fT. The agreement with the value 0.500 predicted for
stage 1 is almost exact, establishing stage 1 as the oscillation mode for
Carriere's data. All of Carriere's data appear to be for oscillations in
stage 1. Second, using the calculated values of T(X) and assuming stage 1
oscillations, the expected frequencies were predicted using equation 14. The
theory gives the right variation of frequency with slit width b for a 20 to
1 variation in b. Any error in the velocity conversion factor would appear
as a constant scaling factor for the experimental fT products and the
theoretically calculated frequencies.
Table 4. Comparison of theory with Carriere's data
giving edgetone frequency f as a function of slit
width b. The gap X was fixed at 13.40 cm. The blowing
pressure was 10 cm of water. The jet velocity is taken
as 2550 cm/sec, or 63 percent of the value from
Bernoulli's equation.
.........................................................
b fT f
(cm) (Hz)
.........................................................
Experiment Exp. Theory Exp. Theory
1.00 0.54 0.50 70.4 65.2
0.90 0.53 0.50 68.0 64.2
0.80 0.52 0.50 64.4 61.9
0.70 0.53 0.50 63.0 59.4
0.60 0.53 0.50 59.8 56.4
0.50 0.52 0.50 55.2 53.1
0.40 0.49 0.50 48.4 49.4
0.30 0.50 0.50 44.6 44.6
0.20 0.48 0.50 36.4 37.9
0.10 0.53 0.50 29.6 27.9
0.05 0.63 0.50 25.6 20.3
..........................................................
It is important to consider Carriere's experiments in which the frequency
varied inversely as the three-halves power of the aperture to edge distance.
This is what the present theory predicts for wide gaps and a turbulent jet.
Jones noticed that this variation occurred when the gap was wide and the jet
was turbulent. Table 5 shows the comparison with Carriere's data. T(X) was
calculated using equation 18 for each value of X given by Carriere. Then,
assuming stage 1 oscillations, the expected frequencies were predicted using
equation 14.
Table 5. Comparison of the theory with Carriere's data for which
the frequency varied inversely as the three-halves power of the
gap width. The oscillations are identified as stage 1. The
blowing pressure was 16 cm of water. The jet velocity is taken
as 2455 cm/sec, or 48 percent of the value from Bernoulli's
equation. The slit width is 0.25 cm. All numbers are Carriere's
except the theoretical values for f in the third column.
................................................................
Experiment Experiment Theory Experiment
X f f f*[X^(3/2)]
(cm) (Hz) (Hz)
................................................................
16.7 28.2 29.2 1920
15.9 30.0 31.3 1900
14.9 33.3 34.3 1910
13.9 35.2 37.8 1820
12.9 42.6 42.0 1970
11.9 47.6 47.0 1960
10.9 54.0 53.1 1940
9.9 62.0 60.6 1960
8.9 74.0 70.1 1940
8.1 80.0 79.6 1860
.................................................................
The agreement shown is excellent. The success of the present theory in
predicting the results of critical edgetone experiments confirms this
theory, and offers confirming evidence for the theory of jet motion
presented in Schlichting's book. It suggests a new technique to determine
the velocities of jet particles as a function of slit width b, initial jet
velocity v(0), and distance X from the jet aperture.
The Jet-Wave Phase Velocity
The present theory predicts that the phase velocity of the jet wave is not a
fixed quantity across the gap as tacitly assumed by Brown and accepted by
later theorists, but is a function of position in the gap. Equation 11
demonstrates this most conveniently. For values of wT small enough that
sin(wT) can be approximated by wT, equation 11 becomes
y(wt, wT) = wT [cos(wt - wT) - cos(wt - wT/2)] (21)
which in turn with the same approximation for sin(wT) reduces to
y(wt, wT) = [(wT)^2] * sin(wt - 3wT/4) (22)
The last equation indicates that the phase velocity at any point very near
the aperture is four-thirds times the jet particle velocity at that point.
This is perhaps more obvious if T in the sine term of equation 22 is
replaced by its equivalent as a function of X given by equation 18.
For values of wT greater than about 2, or for values of fT greater than
about 0.318, the first term of equation 11 dominates its behavior and the
second term can be neglected. Equation 11 is then approximated by
y(wt, wT) = (wT)cos(wt - wT) (23)
This first term taken alone indicates a phase velocity at a point equal to
the jet particle velocity at that point. Therefore for values of fT greater
than about 0.318, the phase velocity at a point is almost equal to the jet
particle velocity at that point. The phase velocity at the position x(T) in
the gap should then be closely approximated by v(x) where v(x) is given by
equation 15. Both jet and phase velocities are functions of position in the
gap.
These conclusions about the phase velocity of the jet wave are very
different from those of previous theorists, who neglected the slowing of the
jet particles and tacitly took the jet velocity anywhere in the gap as
always equal to the jet velocity at the aperture. The phase velocity of the
jet wave was believed to be about half that value anywhere in the gap, as
exemplified by the common interpretation of equations 3 and 4, Brown's
interpretation of his experimental data, and Curle's and Powell's
interpretation of equation 5.
Comparison of the Present and Past Theories
No past theory could even attempt to predict all the basic phenomenology of
edgetones outlined above, and certainly not the detailed experimental data
of Carriere and Brown, so the only detailed comparison with past theories
that is possible is to discuss their inferences about the phase velocity of
the jet wave in the gap. It will be shown that Brown's data on the phase
velocity of the jet wave have been misinterpreted by Brown and others, and
that Brown's data are predicted by the present theory.
Karamcheti, Bauer, Shields, Stegen, and Woolley in 1969 (Ref. 15) discussed
briefly the availability of experimental evidence on the phase velocity of
the jet wave in the aperture to edge gap. They state "the only experimental
information on the phase criterion is that indicated by Brown's measurements
of the wavelength from smoke pictures of the oscillating jet." Brown
injected smoke into the jet and photographed the jet disturbance in the gap.
There was not enough detail in the photographs to allow the number of
wavelengths in the jet disturbance in the gap to be counted. He was able to
define and measure a wavelength in the jet disturbance in these photographs
only near the edge and only for the higher stages. He states that
photographs of stage one were such that no wavelength could be defined. For
the other stages he could only make measurements near the edge since
insufficient details of structure were discernible in earlier parts of the
jet path. Great accuracy should not be expected in these measurements. The
measured wavelength multiplied by the frequency gave a phase velocity which
was about one-fourth to one-half of the velocity Brown attributed to the
jet, which was the velocity of the jet at the aperture. Brown did not
tabulate the phase velocities themselves but instead gave the ratios of the
phase velocities he determined to the initial jet velocity at the aperture.
It should be noted that Brown had no experimentally determined values for
phase velocity in stage one since he could not determine wavelengths in
stage one. However he did guess a value of wavelength for stage one and,
using this guessed value, calculated a phase velocity for stage one.
Inexplicably, he entered these guessed values into his table of what were
otherwise experimentally determined numbers. This invalid entry in Brown's
table of data seems not to have been noticed by most theorists, for example,
Curle and Powell, who in trying to explain Brown's results seem to have
regarded this entry as valid experimental data. The ratios Brown determined
experimentally are valid numbers, but Brown misinterpreted their
significance. Brown's values for the wavelength are valid only immediately
before the edge. There are the implicit assumptions in Brown's treatment of
his data that the jet particles do not slow down and that the wavelength in
the jet disturbance in the gap is the same at all positions in the gap, and
therefore that both the jet particle velocity and the jet wave phase
velocity are constant independent of position in the aperture to edge gap.
Brown concluded then that the phase velocity (assumed constant) was a small
fraction of the jet particle velocity (also assumed constant). There is no
basis in theory or experiment for Brown's assumptions. It is known that
these jets do slow down rapidly. However later theorists neglected the known
slowing of the jet and uncritically accepted Brown's interpretation of his
data. Karamcheti and Bauer (Ref. 16; also see Ref. 15, page 295) have noted
that assuming a single disturbance wavelength and propagation velocity
throughout the edgetone jet is not correct, but their remark seems to have
passed unnoticed by most theorists.
Brown's procedure gives the phase velocity not at a point but averaged over
one wavelength. Although Brown did not state it this way, obviously what
Brown determined was the ratio of the phase velocity of the jet wave
averaged over one wavelength immediately adjacent to the edge to the jet
particle velocity at the aperture, and stated this way Brown's
experimentally determined ratios are valid numbers. For the higher stages,
for which the wavelength is much shorter since there are more wavelengths in
the gap and the jet has slowed appreciably, Brown's average is taken over a
short distance just adjacent to the edge. The present theory predicts that,
except very near the jet aperture, the phase velocity of the jet wave at a
point is very nearly equal to the jet particle velocity at that point.
Therefore, Brown's experimentally determined ratios, for the higher stages,
of the phase velocity near the edge to the jet particle velocity at the
aperture, should differ only slightly if at all from the theoretically
calculated ratio v(X)/v(0) at the edge. And this is indeed the case. For
every entry in Brown's Tables 2 and 3 our equation 15 gives v(X)/v(0) = 0.38
at the edge. Of the nineteen valid entries in Brown's Table 2 for the ratio
of phase velocity immediately before the edge to jet particle velocity at
the aperture, eighteen are in the range 0.36 to 0.43, in close agreement
with our prediction. One entry gives 0.47 for the ratio. (There are 20
entries in the table but one has to be discarded because it is for stage 1
and is not experimental data but is a guess by Brown.) For the highest stage
in Brown's Table 3, the ratio values are 0.40, 0.41, and 0.38, again in
agreement with our prediction for what Brown would see for the higher
stages. Schlichting's equation for the slowing of a turbulent jet (our
equation 15) gives an accurate prediction of the phase velocities Brown
observed. If Brown or others had calculated the slowing of the jet
particles, they would have recognized these phase velocities as being for
practical purposes just the velocities to which the jet particles had
slowed. Brown's experimental data are entirely consistent with the
conclusions of the present paper about the phase velocity, in fact, the
present theory predicts the phase velocities which Brown found. We are
forced to the conclusion that while Brown's experimental data are excellent,
serious errors have been made in the interpretation of these data. These
errors of interpretation contributed to previous failures to develop a
satisfactory theory.
It will be noted that the conclusion of this paper that at points not very
near the aperture the phase velocity of the jet wave at those points is
essentially equal to the jet velocity at those points is now an empirically
established fact, not dependent upon the theory of this paper that led to
equations 8 through 11, and to equation 14. If Brown's basic data are
correct and Schlichting's equations for the slowing of the jet are correct,
then this conclusion is correct. The conclusion follows from Brown's data
and Schlichting's equations, neither of which depends upon anything in this
paper.
There are major problems with Brown's presentation and interpretation of his
experimental data. Brown could not experimentally define a wavelength for
stage one oscillations. However he assumed that for stage one the wavelength
was just the aperture to edge distance, and he calculated values for phase
velocity in stage one on the basis of this assumption. In his text he stated
that he had done this. Unfortunately, he included these assumed and
calculated values for stage one in his table of real experimentally
determined values for the other stages, without an explicit warning in the
table that for stage one the values given had not been experimentally
determined but were in effect guesses. This is the entry in Brown's Table 2
that was rejected from consideration in the paragraph above. Brown had no
experimental value of wavelength or phase velocity for stage one. Strangely,
the existence of this invalid entry in Brown's table of data has not been
previously pointed out. It is an unfortunate fact that past theorists,
subsequent to Brown, using these data overlooked Brown's statement of what
he had done and treated these particular values for stage one as valid
experimental data. Curle and Powell both made this error. There is certainly
no basis in Brown's data for Curle's and Powell's application of their
equation (our equation 5 above) to stage one. (The next paragraph will show
that Brown's data does not support this equation applied to any stage.) This
inattention to what Brown wrote and the resulting confusion about Brown's
data have undoubtedly contributed to the failures to develop an adequate
theory.
The lack of detail except near the edge in Brown's photographs made it
impossible to count the number of wavelengths in the oscillation of the jet
in the gap. The number assigned by Brown for stage one and entered into his
table of what was otherwise numbers derived from experimental data, one
complete wavelength, was a pure guess as we saw in the paragraph immediately
preceding. This entry must be rejected. Except for stage one, the numbers
assigned were obtained by dividing the gap width by the wavelengths measured
just before the edge, which was the only place with sufficient detail to
allow the wavelengths to be defined and the measurements made. This
procedure tacitly assumed that the wavelength was constant and independent
of position in the gap. The measured wavelength values of course are valid
but must be properly interpreted. There is neither experimental nor
theoretical justification for Brown's assumption that the wavelength
measured just before the edge was the wavelength at all positions in the
gap. This gave much too large a result for the number of wavelengths in the
gap because the wavelength in the jet gets smaller as the edge is approached
since the jet is slowing down. The integer part of every value that Brown
gives for the number of wavelengths in the gap is one unit too large and the
fractional part above an integer has no validity either. Every entry Brown
gives for the number of wavelengths in the gap must be rejected. Theorists
making uncritical use of Brown's data who accepted his interpretation of
that data were led to incorrect conclusions. For every stage Brown, Curle,
and Powell each assigned one complete wavelength too many for the integer
part of the number of wavelengths in the jet disturbance in the gap. It is
evident that the values Brown tabulates for the number of wavelengths in the
oscillation of the jet in the gap have no validity whatever, and offer no
confirmation or support to the equation (equation 5 above) that Curle and
Powell proposed for the edgetone frequency. That equation is completely
devoid of experimental support by Brown's data and is therefore an
nsupported conjecture. All subsequent theoretical papers (and there are
many) following Powell's lead and adopting this equation are based on the
false premise that this equation is supported by Brown's experimental
data. Therefore these papers must be rejected or at least critically
reexamined. Brown's and later theorists' interpretation and treatment of
Brown's excellent experimental data were unchallenged until the present
paper.
Powell has probably been the most influential writer on edgetones since
Brown, and certainly the most prolific. Powell has continued to publish in
various journals numerous articles and letters on edgetones (too many to
reference all here, but see for example Ref. 10a) offering elaborations of
the viewpoint and equation he first presented in 1953. The influence of
Powell upon other theorists has been very great, which in many respects is
unfortunate. The continued acceptance of Brown's assumptions and
interpretation of his data by Powell vitiates most of Powell's work on
edgetones. Powell's approach to the problem of edgetones and his equation
(equation 5 above) have been adopted as a starting point by many later
theorists attempting to explain edgetones. It will be apparent that the
present paper completely invalidates Powell's and his followers' approach to
the problem of edgetones. (Note added in 1999: Powell's and his followers'
efforts to explain edgetones, continuing to depend upon Brown's assumptions
and interpretation of his data, now cover a period of almost fifty years,
and have yet to produce a theory or procedure that can predict the
experimental data of Carriere and Brown.)
It truly seems that the major reason for the long failure to have a
sucessful theory of edgetones has been the uncritical acceptance by
substantially all theorists after Brown of the faulty interpretation Brown
offered of his data. Brown tacitly assumed that the jet particles do not
slow down, and that the wavelength measured immediately before the edge was
the wavelength at all positions in the gap. These assumptions are critical
features of Brown's interpretation of his data. Powell accepted the
treatment and interpretation Brown gave of his data. Powell's great
influence on later theorists, a result of his 1953 paper and his persistent
espousal since of the viewpoint he presented then, may be in large part
responsible for the failure to have previously identified the shortcomings
in Brown's paper. Theorists just did not consider that Brown might have been
seriously wrong in his treatment of his data.
Theorists attempting to explain edgetones have also neglected large parts of
the known phenomenology of edgetones. They have not given attention to the
finding of Carriere, confirmed by Jones and also deducible from Brown's
data, that for some conditions the edgetone frequency varies inversely as
the three-halves power of the gap width X, or to the finding of Carriere,
confirmed by Brown, that the edgetone frequency depends upon the aperture
slit width b. These dependencies are too strong to be ignored in an adequate
theory of the edgetone oscillator. No prior theoretical paper known to the
author gives serious consideration to Carriere's work, acknowledges that a
three-halves power dependence of frequency on gap width X exists which must
be explained, or makes the slit width b an important parameter of the theory
proposed. None takes account of the possible effects of jet slowing upon
edgetone production.
Summary
This theory was developed in the years 1971 through 1973. A brief account of
the work was presented at the Los Angeles meeting of the Acoustical Society
of America, 30 October-2 November, 1973; and an abstract (Ref. 17) of that
presentation was published in 1974. (The abstract's statement that the phase
velocity of the jet wave in stage one is twice the jet particle velocity
there is incorrect. The error arose from the hasty assumption that only the
second term of equation 11 was important for very small values of wT.)
Except for increased emphasis on the discussion of past theories and of
phase velocity, and for minor changes elsewhere, this paper is as initially
written in 1974. The literature since early 1974 has not been reviewed. A
detailed account of the theory has not been previously published.
The most important conclusion of the present paper is that the edgetone
oscillator is an acoustic transit time oscillator, not different in
principle from many electronic oscillators. Perhaps equally or even more
important is that erroneous assumptions in Brown's presentation and
interpretation of his excellent experimental data are identified. These
assumptions tacitly adopted by Brown and unquestioningly accepted by later
theorists relying upon Brown's paper have had the most serious consequences
upon efforts to develop an understanding of the edgetone oscillator.
(Note added in 1999: One recent theorist on edgetones, Young-Pil Kwon (Ref.
18), in two interesting papers has independently noted in referring to
Brown's paper that "the jet velocity decreases with distance along the jet
axis" and that "the wavelength measured near the edge tip may be shorter
than the average wavelength along the stand-off distance" in consonance with
this paper. Although not greatly emphasized by him, this amounts to a denial
of Brown's assumptions about the jet and phase velocities. He did adopt
Powell's equation (our equation 5) as a starting point for his discussion of
the edgetone problem, but recognized that as a guide to edgetone oscillator
behavior this equation would require major modifications.)
The edgetone oscillator is not a complex oscillator. The crucial facts
necessary to understand its operation are few. Because of mirror symmetry
about the plane through aperture and edge, the edgetone oscillator is
recognized as a push-pull oscillator, which implies that the pressure
variations on the two sides of that plane are 180 degrees out of phase. The
jet particles necessarily respond to the pressure difference on the two
sides of the jet. An equation for the jet wave in the aperture to edge gap
results. This equation leads to the conclusion that the edgetone oscillator
is a transit time oscillator; to a frequency equation for the oscillator, fT
= k, where f is the oscillator frequency, T is the transit time of a jet
particle from the jet aperture to the edge, and k is a constant which
depends upon which oscillation mode is excited; and leads to the value of k
for each oscillation mode. At this point we have to recognize that jet
particles in traversing the aperture to edge gap slow down markedly, and
that at least for wide gaps the jet is turbulent. Applying standard
available results from fluid dynamics theory for turbulent jets gives us an
equation for the transit time T of the jet which takes account of jet
slowing. Using this value of T with the equation, fT = k, predicts all the
available experimental data on edgetones, without the introduction of any
empirical or ad hoc factors to force a fit to that data. It is thus
established empirically as well as theoretically that the edgetone
oscillator is a transit time oscillator. Conclusions about the phase
velocity of the jet wave follow from this theory which, once recognized, can
be seen as established empirically by the experimental data without
dependence upon the theory which initially led to the conclusions.
Although usually unstated, implicit in most previous theories of edgetones
(and also of the organ flue pipe) are the tacit assumptions that the jet
velocity remains constant at its value at the aperture, and that the jet
wave is characterized by a constant phase velocity which is a small fraction
of that assumed constant jet velocity. The first assumption is contradicted
by the known facts about jet behavior, and the second is a pure assumption
without support in experiment or theory. No theory based on these
assumptions has had success in predicting the data from basic edgetone
experiments, despite a plethora of theoretical efforts extending over
most of a century. None gives even a qualitative prediction of all the basic
phenomenology of edgetones outlined above. None of these theories
gives attention to or attempts to explain either qualitatively or
quantitatively the three-halves power variation of frequency with gap width
seen in some experiments, or the strong variation of frequency with slit
width which is known to exist and for which quantitative data is available.
The slit width is not even a parameter of these theories. Despite claims of
validity that have been made for these theories, none has been successfully
applied to predict the data of Carriere and Brown. These data exist, their
validity has not been challenged, and a successful theory must explain them.
The author regards the predictions of these data as the basic tests of any
edgetone theory. The theory of the present paper is the only theory of
edgetones yet developed that predicts and explains the results of these
basic experiments.
The theoretical situation is further complicated by the fact that a clear
distinction is not always made by theorists between the theory of edgetone
oscillations and the theory of flute or organ pipe oscillations. An adequate
theory of the organ flue pipe will have little application to the edgetone
oscillator. The impact of the Q-factor of the flute's or organ pipe's
resonant air column upon the oscillation frequency of the musical instrument
is so strong that a clear distinction between the two theories should be
made. Any theory of edgetones has direct bearing upon the theory of flutes
or organ pipes since the edgetone appears to be the exciting agent of
the flute or organ flue pipe. However the Q-factor of the musical instrument
so modifies the behavior of the edgetone oscillator as to make separate
discussions of the two oscillators at least desirable if not strictly
necessary. (See the discussion of Lord Rayleigh's data on the variation of
frequency of an organ flue pipe with changes of blowing pressure, which can
be found at http://www.nmol.com/users/wblocker/index.htm where the present
paper in a slightly modified form is also found. Lord Rayleigh's data shows
that changes of blowing pressure, and consequently of blowing velocity, that
would more than triple the frequency of an edgetone oscillator make a change
of only a few percent in the frequency of an organ flue pipe oscillator.
This stabilization of the frequency is attributable to the Q-factor of the
organ flue pipe. Even modest values of the Q-factor have very great effects
in stabilizing the frequency sounded by the pipe. Any parameter having an
influence this great upon the frequency sounded by the flue pipe should
receive great attention in any theory of the flue pipe. Most theories of the
organ flue pipe oscillator fail to mention the pipe's Q-factor and the few
that do give scant attention to its importance.)
The assumption that the edgetone oscillator is a transit time oscillator
leads immediately to a theory that predicts all the basic phenomenology of
edgetones and gives numerical predictions in almost exact agreement with all
critical experimental data. This is the first theory that can predict in
detail the experimental results of Brown and Carriere. But the theory is not
essential. Every important conclusion of this paper follows from Brown's
experimental data and Schlichting's equations for the the slowing of a jet.
These alone suffice to establish empirically the fT product sequence, thus
to establish empirically that the edgetone oscillator is a transit time
oscillator, and to establish empirically that, except possibly very near the
aperture, the phase velocity of the jet wave at a point is equal for
practical purposes to the jet particle velocity at that point. These things
are now empirically established facts, and are independent of the theory
that led to them.
The major factor leading to this theory was the conviction that the edgetone
oscillator must be an ordinary feedback oscillator which could be explained
in the same fashion that all other feedback oscillators are explained. With
this conviction, the feedback loop is easily identified and analyzed. In
fact Jones had already identified the feedback loop and the feedback
mechanism. The edgetone oscillator is an acoustic feedback oscillator, more
specifically a transit time oscillator. The calculation of the transit time
must take account of the slowing of the jet particles and take account that
the jet is turbulent. The same principles that explain the operation of
ordinary electronic feedback oscillators suffice to explain the operation of
this oscillator.
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Copyright © 1999 by Wade Blocker
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