Content-Type: multipart/mixed; boundary="-------------0002072203985" This is a multi-part message in MIME format. ---------------0002072203985 Content-Type: text/plain; name="00-66.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-66.comments" This paper should be displayed and printed as plain "typewriter type" text in a monospaced font such as Courier in order that tabular data in tables be properly aligned. Size of file about 76 KB. The author's e-mail address is wblocker@nmol.com. Comments will be welcomed. This paper in slightly revised form can also be found on the Internet at URL http://www.nmol.com/users/wblocker/index.htm along with two related papers. ---------------0002072203985 Content-Type: text/plain; name="00-66.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-66.keywords" 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 ---------------0002072203985 Content-Type: text/plain; name="edgetone.txt" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="edgetone.txt" NOTICE: This paper should be displayed and printed as plain "typewriter type" text in a monospaced font such as Courier in order that tabular data 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. References (1) C. Sondhaus, Ann. Phys. 91, 128 & 214 (1854) (2) J.M.A. Lenihan and E. G. Richardson, Phil. Mag. 29, 400 (1940) (3) W. Koenig, Phys. Z. 13, 1053 (1912) (4) E. Schmidtke, Ann. Phys. 60, 715 (1919) (5) F. Krueger, Ann. Phys. 62, 673 (1920) (6) Z. Carriere, J. de Phys. et Rad. 6, 52-64, (1925) (7) G.B. Brown, Proc. Phys. Soc. 49, 493 & 508 (1937) (8) A. T. Jones, J. Acoust. Soc. Am. 14, 127(A) (1942/1943) (9) N. Curle, Proc. Roy. Soc. A216, 412 (1953) (10) A. Powell, Acoustica 3, 233 (1953) (10a) A. Powell, J. Acoust. Soc. Am. 33, 395 (1961); 34, 163 (1962) (11) D. Marcuse, Engineering Quantum Electrodynamics; Harcourt, Brace and World, Inc.; New York, 1970, pp. 127 et seq. (See the first edition of this book, not the second.) (12) H. Schlichting, Boundary Layer Theory, McGraw-Hill Book Co., New York, 1960 (13) V. L. Streeter, Fluid Mechanics, McGraw-Hill Book Co., New York, 1960, p. 88 (14) J. R. D. Francis, A Textbook of Fluid Mechanics, Edward Arnold (Publishers) Ltd., London, 1958, Chapter 8; J. M. Kay, An Introduction to Fluid Mechanics and Heat Transfer, Cambridge University Press, Cambridge, 1963, Chapter 2; P. S. Barna, Fluid Mechanics for Engineers, third edition, Plenum Press New York, 1969, Chapter 6 (15) K. Karamcheti, A. B. Bauer, W. L. Shields, G. R. Stegen, and J. P. Woolley, National Aeronautics and Space Administration publication NASA SP- 207, pp. 275-304 (esp. pp. 288, 295), (1969) (16) K. Karamcheti and A. B. Bauer, SUDAER Rept. 162, Dept. of Aeronautics and Astronautics, Stanford Univ., (1963) (17) W. Blocker, J. Acoust. Soc. Am. 55, 458(A), (1974) (18) Young-Pil Kwon, J. Acoust. Soc. Am. 100 (5), November 1996, pp. 3028- 3032; J. Acoust. Soc. Am. 104 (4), October 1998, pp. 2084-2089 Copyright © 1999 by Wade Blocker ---------------0002072203985--