边棱音产生机制__1937

The mechanism of edge-tone production This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1937 Proc. Phys. Soc. 49 508 (http://iopscience.iop.org/0959-5309/49/5/307) Download details: IP Address: 202.117.145.242 The article was downloaded on 18/04/2011 at 14:33 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience THE MECHANISM OF EDGE-TONE PRODUCTION BY G. BURNISTON BROWN, M.Sc., PH.D., Lecturer in Physics, University of London, University College Received I O April 1937. Read 25 June 1937 ABSTRACT. Former theories of the production of edge tones are reviewed and a new theory is put forward which accounts in a qualitative manner for the phenomena described in the preceding paper. The effect of the wake of a disturbance on the portion of the jet immediately following it is considered, and it is concluded that the compressibility and structure of the surrounding medium must be taken into account, and that, for low velocities, wake disturbance is the controlling factor, whilst for higher velocities the effect of the sound emitted must be taken into account. An attempt is made also to account for the differential sensitivity to frequency found in sensitive jets in a former research. I. FORMER T H E O R I E S L T H O U G H edge tones had been discovered for some fifteen years, Helm- h o l t ~ ( * ~ ) did not discuss them except in connexion with an organ pipe. I n A this connexion he refers to his earlier paper on “ The discontinuous motions of fluids ”(‘@and states that the bounding surfaces of the streammust be looked upon as vortical surfaces, i.e. surfaces “faced with a continuous stratum of vortical filaments or threadlike eddies. Such surfaces have a very unstable equilibrium.” An example of this instability is the phenomenon of sensitive flames, and he makes a reference to Tyndall’s experiments on gas jets. He continues : “ This revolution into vortices takes place in the blade of air at the mouth of the pipe, where it strikes against the lip. From this place on it is resolved into vortices, and thus mixes with the surrounding oscillating air of the pipe, and accordingly as it streams inwards or outwards, it reinforces its inward or outward velocity, and hence acts as an accelera- ting force with a periodically alternating direction, which turns from one side to the other with great rapidity. Such a blade of air follows the transversal oscillations of the surrounding mass of air without sensible resistance.”* The view that the vortices form only at the edge is not borne out by e~periment‘~’. Helmholtz also refers to Schneebeli’s(”) mechanical explanation, which was that the air stream took the place of a solid reed. This view was also held by Cavaille C~ll‘’~’, Sonreck(19) and Hermann Smith @O). It is clear, however, that the tone-production is only explained if a pipe or other resonator is present, the oscillations in which cause the transverse motion of the air stream. Lord Rayleigh in his Theory of Sound, 2nd edition (1896), does not attempt any explanation, but in connexion with the bird-call suggests that “when a symmetrical excrescence reaches the second plate, it is unable to pass the hole with freedom, and the disturbance is thrown back, probably with the velocity of sound, to the first * Reference (IS), p. 395. The mechanism of edge-tone produclion 509 plate, where it gives rise to a further disturbance, to grow in its turn during the progress of the jet" (p. 412). The first attempt to account for the production of tones in the absence of any resonator was made by Wachsmuth('). According to him, with a blunt edge no sound occurs until the distance of the edge from the orifice is such that the stream can spread on to both sides : the sharper the edge, the smaller this distance, but he does not explain why a minimum distance exists however sharp the wedge. When the stream spreads over the edge it must pass up one side first, and thus an asymmetry occurs which produces a periodicity in the velocity, through the alternate freeing and checking of the stream, and this must cause a pulsation in the air supply. That the stream starts to bend before the edge is reached is due to its behaving like a flexible bar, which buckles in the middle when it first strikes the edge. The bent portion travels up without returning to the vertical, since it is not a rigid body and the backwash (Riickstau) from it maintains the stream in a disturbed condition at a point below the edge. Wachsmuth adds that his researches did not decide whether the actual tone-production was due to the to-and-fro motion of the lower straight portion of the stream or to vibration of the wedge itself: he preferred the latter theory since, when h is large, the tone is heard near the wedge and not near the orifice, With regard to the latter question, it can easily be shown that vibration of the wedge is not the cause of the tone, by holding the edge with the fingers, which, as. Benton" has remarked, provide very efficient damping. No difference in tone- production occurs. Wachsmuth did not justify his assumption that an air stream can be treated as a flexible rod and that it would break in the middle on striking an obstacle. He gave no physical explanation of the jumps in tone, and there is no evidence for the pulsation in the air supply. The problem was next examined by Konig(9), who developed a hypothesis similar to that made by Lord Rayleigh in connexion with bird-calls. He pointed out that the jet could not be treated as a body with periods of its own, so that a simple resonance principle cannot be invoked, and he added that neither considerations of friction along the wedge nor of pendulation of the stream threw much light on the mechanism of tone-production. He assumed that when the edge was struck a sudden stoppage in the stream occurred and a compression wave travelled out in all direc- tions with the velocity of sound. This would produce a disturbance at the orifice which would travel upwards with the velocity of the stream and create a new stop- page. If the velocity of sound in air is c, the period would be h (c+ U)/cU, and with c very much greater than U the frequency n would be given by U n = X * obtained I U But this gives double the observed frequency. Konig therefore was forced to suppose that somehow the first compression produces a rarefaction at the orifice and he then n=- -. 2 h * Reference (6), p . 126. 510 G. Burniston Brown He had to admit, however, that Goller’s results(“’, which he was using, showed that the factor was not 4 but between 0.4 and 0.5, and that for other intervals it became 1.0 and 1.5, and that his theory gave no account of this, neither did it explain why the tones do not commence with the highest audible note. He concludes by sug- gesting that since the velocity of sound is (c- U ) in the stream and c outside, this may cause pressure differences which might affect the tone. S~hmidtke(~) begins by asserting that the results of Wachsmuth and Goller show that n is proportional to U and inversely proportional to h and that the jumps in tone are octaves. This being so, Konig’s formula can be written where J = I for the first interval, 2 for the second and so on. (The last column in table 2 on p. 501 of this volume shows the value of ahn/JU.) Schmidtke points out that Konig’s derivation of the formula is unsatisfactory, and then attempts to make use of Kiirmdn’s vortex street. Such a street, he suggests, is set up along the sides of the wedge in stage I and between the orifice and wedge in later stages. Since he held that the frequency of the tones was equal to the number of vortices set free per second, it followed that in a KBrmiin street the same frequency could be produced again by moving the wedge a distance of h/2; i.e. for the same frequency and velocity the difference between the wedge-distances must be constant, Schmidtke could not prove this with the results he had obtained for water jets since he did not succeed in getting more than one jump. He quotes, therefore, some values from a research by Rieth(’”) with air jets, which support this conclusion, A table similar to that drawn up by Schmidtke but covering a much greater range of velocities has been given in table I of the preceding paper, and it can be seen that (l~,+~-h,,,) is approximately a constant, but it should be noted that the constant is not AI2 but more nearly A. A glance at figure 3 of that paper shows that although with respect to the vortex motion the wedge has moved only A/2 each time, with respect to the orifice it has moved a distance A, the vortex motion having moved away from the orifice by a distance A/2. Further, on Schmidtke’s theory, the product of n by the mean of (h,+, - h,) ought to be constant and he had to admit that it was not. Schmidtke, however, concluded that a case had been made out for a KBrmdn street between the orifice and edge, but he did not attempt an explanation of its origin and did not provide any direct evidence of its existence. Kriiger(26) was the first to realize that a disturbance travelling up the jet would have a velocity U different from the stream-velocity U. He made the further sug- gestion that the jet has a natural wave-length & corresponding to its own tone (“jet tone”) in the absence of the wedge. The oscillation of the stream would there- fore be greatest when the edge-tone wave-length and jet-tone wave-length were the same. As h increases, the difference between it and A,, becomes greater, and the amplitude of the forced oscillations consequently smaller. When h approaches the value 2h, , the wave-length jumps back to h,, Corresponding to an octave jump in tone, and resonance again occurs. Kruger did not substantiate this hypothesis by The mechanism of edge-tone production 511 comparison with experimental results. If this is done, the present research shows that the heights at which jumps in tone take place are not simple multiples of a fixed length, and further, the suggestion that the oscillations of the stream will be- come smaller as h approaches 2A,, is the reverse of what actually occurs : the oscilla- tions increase in amplitude as h increases in stage I until the jump occurs to stage 11, and then the amplitude of stage I1 increases until the jump to stage I11 and so on. The use of a resonance principle by Kruger does not, therefore, seem to be justified, and there is no explanation of A,,. Benton@) is another investigator who has claimed to show that a KBrmhn vortex system exists. He assumed that if the wedge were moved across the stream the point at which the tone disappeared would be a point on the path of the vortex centre. He measured the width b of this tonal boundary at different heights of the wedge and further assumed that h was equal to A. He found that blh fell from 0.5 to 0.28 as the slit-width increased from 0-1 mm. to 1.88 mm. Instead of proceeding with wider slits, he assumed that this was a limiting value independent of the slit- width, and so concluded that for "very wide slits" blh approaches the KBrmBn value of 0.28. It happens that for d= 2 mm. this ratio has a value close to KBrmBn's but for other widths it differs widely." So that, even if Benton's assumptions are admitted as correct, he does not succeed in presenting any evidence for the KBrmPn vortex street, and, further, he gives no theory of its origin or of the jumps, since he has denied the occurrence of the latter (cf. 5 3). Richardson@) while favouring the KhrmBn street theory, although providing no direct evidence for it, describes an experiment which indicates that the theory of a compression wave travelling backwards from the edge to the orifice with the velocity of sound is not correct. A second jet near the first one, but without a wedge, should have been affected by the sound wave, but no disturbance of any kind was observed. This experiment was made with water. 5 2. T H E KARMAN VORTEX STREET In view of the assumption by many writers that the conception of the KBrmBn street is applicable to fluid jets, an assumption which has led to the superficial claim that the phenomenon of edge tones and jet tones is thereby explained, and to the neglect of further research, it is important to consider this theory in greater detail. Von KBrmBn considered the case of two parallel rows, of indefinite extent, of isolated, equal, point vortices in a fluid which was incompressible and without viscosity. He was then able to show that there was instability for infinitesimal disturbances unless alh = 0.28, where a is the distance between the rows. He con- cluded that this configuration was definitely stable, but, as Lamb(") remarks, he did not prove this, and lately Schmieden(") has shown that for certain finite disturbances it is certainly unstable. It is clear that any application of this result to physics would have to be made with caution, and the experimental verification has never been satisfactory, even in * Reference ( z ) , table 3. 512 G. Burniston Brown the case of vortices at some distance in the wake of an obstacle(23). When gaseous jets are considered, the vortex rows cannot be of indefinite extent nor can the vortices be equal, and further, the fluid is not incompressible or free from viscosity. A priori, therefore, there is no reason for expecting to observe a constant ratio of a/h and certainly no reason to suppose that this will be 0.28. The lack of evidence for the assumption of the KBrmAn street in the theory of edge tones has already been commented upon in 0 I. In the case of jet tones which are produced in the absence of the wedge, Kruger and S~hmidtke('~) claimed to have found the exact KBrmBn ratio using water. Four observations only are given, viz. 0~30,0~27,0.28 and 0.29 from which a triumphant mean of 0.285 is obtained. It may be concluded, therefore, that there is no reason for supposing that the constant KBrman ratio would occur in jets and that, in fact, it has never been observed." Indeed it is remarkable that a mathematical theorem which neglects the compres- sibility of the medium should ever have been expected to account for the alternate compressions and rarefactions of which edge tones consist. 5 3. T H E P H Y S I C A L THEORY O F EDGE-TONE P R O D U C T I O N As was mentioned earlier, the present writer, after giving a theory of the action of sound on fluid jets, favoured the hypothesis that edge tones are maintained by the sounds travelling back to the orifice and producing a vortex formation simiiar to that in sensitive jets, and that, in some way, the edge stabilizes the motion. The sound has been shown to arise in the region where thevortex first begins to throw out a filament; this filament as it advances from 8=o to 8'371. (cf. g 4 of the preceding paper) entrains air, and it is the sudden pressure changes in this region? that result in the propagation of sound waves. Although this hypothesis is partly correct it fails to account for the following facts: (a) In some of the cases where it was found that vortex motion which was apparently identical with that due to the wedge could be produced by the loudspeaker alone, it was observed that the intensity of the sound from the loudspeaker had to be easily audible, whereas no audible edge tone occurred. (Cf. 5 IO of the preceding paper.) The wedge therefore must do something more than just stabilize sound-produced vortex motion. (b ) In a case such as that illustrated in figure g of the plate facing p. 498 i t is impossible that an edge-tone vortex-formation with the wave-length shown could originate in disturbance at the orifice and not be affected by the superposition of vortices developed to the extent of those in figure 9. The only observable effect is the widening of the stream: the frequency is not altered. (The superimposed frequency was made to be a simple multiple of the edge tone in figure 9 merely to obtain a stationary stroboscopic view.) (c) Similarly it is difficult to see how a jet can give out two nearly pure tones simultaneously, if each is produced by vortices developed from a periodic disturb- ance propagated from the orifice. In consequence of these facts it is necessary to examine the effect of interposing an obstacle in a jet of fluid. The easiest case to consider first is that in which the angle t Reference (z) , p. 730. * Cf. reference (z) , table 3. The mechanism of edge-tone production 513 of the wedge is increased until it is ISOO, i.e. the jet impinges normally on a flat plate (see figure I a). In this case it is evident that if the jet is deviated towards the right side, the surrounding fluid in the neighbourhood of A is displaced. If this displacement takes place sufficiently rapidly a local increase of pressure occurs at A, and this in dissipating itself reacts on the jet, deviating it towards the left; this deviation, in its turn, produces an excess pressure driving the jet back towards the right, and so an oscillatory motion is set up. Such motion is easily observed with a stroboscope : no developed vortex formation is seen, but undulations such as are shown in the illustration" are observed. The nature of this type of disturbance of the surrounding air is a problem which has not yet been considered quantitatively. It is well known that if an obstacle is vibrated with very low frequency, there is a mass motion of the air from the front to the back and vice versa, and that pressure changes are negligible and can only be X -Y (e) _**.---__ JA 'T ::$;, Z F (4 ( b ) Figure I. observed in the vicinity of the obstacle. At higher frequencies the reverse is the case: the mass motion becomes negligible and a pressure wave spreads to great distances with the velocity of sound. It is between these limits that the types of disturbance that are responsible for the oscillations producing edge tones occur. Turning now to the wedge, it was found that with the velocity far below that at which oscillation commences, e.g. 82 cm./sec., it was possible to produce a vortex motion corresponding to that found in edge tones by moving the wedge slowly from side to side with an amplitude of about I nun. and a frequency of I or 2 per second. In this case the effect of such a slow motion is to cause the air to pass round the edge, from the side which the wedge is approaching to the opposite side. While this is done the stream is displaced : in figure I b for instance, if the wedge is moved to the left the air at B flows round the edge and moves the jet to the right ; if this movement is large enough the portion of jet lower down at D may move to the left. The first appearance of the vortex filament may now occur at the exposed portion F, instead of at E. If now the wedge is considered to be stationary, as it is in practice, the air at B can be displaced by the stream instead of the wedge. In the non-disturbed condition The fact that oscillation can be set up when a jet impinges on a flat plate or on a plate with a hole in it, as in the bird-call, or on other forms of obstacle, makes the effect of introducing an obstruction sideways towards the stream, as was done by Klug(Zs), very difficult to interpret since it may itself produce oscillation. PHYS. SOC. XLlX, 5 34 514 G. Burniston Brown the stream bifurcates at the edge and passes half up one side of the wedge and half up the other. Now it has been shown that the whole of the edge-tone phenomena occur within the sound-sensitive range of the