On the intensimetric analysis and monitoring of flue organ pipes. 1 Introduction

Transcription

1 On the intensimetric analysis and monitoring of flue organ ies Domenico Stanzial FSSG, National Research Council of Italy, Fondazione G. Cini, Isola di San Giorgio Maggiore, I-3014 Venezia, Italy, A short reiew of intensimetric analysis and monitoring of flue organ ies based on the new concets of radiating and oscillating sound intensities will be addressed in this aer. This kind of general analysis dates back to a decade ago when the concet of elocity of acoustic energy transort was rigorously introduced in In that aer it was clearly shown that the acoustic article elocity is always decomosable into a comonent in-hase with the sound ressure and another one orthogonal to it in the Hilbert sace. Roughly seaking this second comonent coincides with the hase-uadrature comonent when the field is a monochromatic one. From this general roerty it follows that acoustic energy can trael with an aerage elocity bounded in modulus between 0 and c (the sound seed). A further rogress was made some years later in 1996 and in 1997 when the olarization roerty of sound intensity was discoered and measured. The story culminates in 003 with a great synthesis when the four-dimensional treatment of the linear acoustic field was outlined in strict analogy with the electromagnetic field. The flue organ ie was a faithful witness all along this research ath and this aer will briefly summarize results obtained for sound intensity measurements erformed inside and outside the ie. There is no doubt about it: the flue organ ie is a real aradigm of acoustics. 1 Introduction Besides its musical nature and esecially due to its reliability the flue organ ie sound system has been widely used in the history of acoustics as a reference instrument. As shown in Figure 1, W.C. Sabine used it for exciting rooms [1] in its ioneering works on sound reerberation. For the same reason different ranks of organ ies are also emloyed as a didactical tool in many Physics classrooms for demonstrating the behaiour of the normal modes of ibrations of a column of air. In more recent years, starting from the seenties when the deeloment of sound intensimetric techniues reached its technological maturity [], the sound field of flue organ ies was monitored both outside [3] and inside the ie wall using standard as well as non-standard acoustic methodologies [4]. This aer just resents a short reiew of the nonstandard kind of intensimetric analysis deeloed for general sound fields by the author and its co-workers starting from 1994 [5,6,7] and which for the reasons gien aboe has been referentially alied to flue organ ies considered as a reference sounding object. This kind of analysis was first enisaged with the aim of giing a firm hysical interretation of exerimental data coming from standard sound intensimetry, but it become soon clear that a major reision of theoretical fundamentals of the concet of energy flux density in acoustics was necessary in order to score the lanned goal. In the following sections thus the state of the art of main ideas and theoretical discoeries which has been successiely erformed along this research ath are reorted and some measurement results obtained in the case study of the organ ie are interreted according to the new scheme of intensimetric analysis. Basic theoretical facts regarding energy roagation in sound fields and organ ies monitoring.1 Velocity decomosition, radiating and oscillating intensities Figure 1: Four small organs aaratus designed and used by W.C. Sabine in his ioneering exeriments on reerberation. Organs were fixed 5 meters aart. One of the most amazing theoretical conseuences which can be mathematically inferred from the definition of the standard actie intensity but haing a hysical meaning not yet clearly understood, is the general and uniue decomosition of the air article 641

2 Forum Acusticum 005 Budaest elocity field into one comonent ector always haing null stationary time-aerage = 0 and another one whose time-aerage is instead usually different from the null ector: 0. The mathematical reason for this strange behaior of the elocity field deends from the fact that sound ressure is defined in linear acoustics as the artial time deriatie of the real alued kinetic otential field φ ( x,t) i.e. = ρ0 φ t. Thus the roerty = 0 is forced to hold eerywhere within the sound field. This is not the case for the air article elocity field. In fact the elocity turns out to be defined as a sace-like uantity = φ [9]. Formally its intensimetric decomosition can be accomlished as follows. We are looking for an exression of the elocity as a sum = + where: i) is reuired to hae the same time deendence as the ressure at a gien oint x and ii) is reuired to hae zero time-aerage. Thus for any time-indeendent ector field M( x ) we can clearly define : = M( x ) in such a way to satisfy the first condition, but in order that = 0 is also satisfied, we find for M the uniue solution M = I where I= is the standard definition for actie intensity. In fact ( ) = = M = 0 I M = = I. sound field [8]. A smart obseration haing only a theoretical interest but which is worthwhile to be reorted here, is that sound energy density can be considered as really frozen at elocity nodes. In fact, for any fixed freuency at air elocity nodal oints, lines, or 3-D surfaces, sound energy density can exist only in its ure otential form. In this form it can certainly oscillates but only in the time dimension. One can deduce this imortant fact because at elocity nodal oints both the radiating and the oscillating intensities anish identically at each instant of time ( a r 0) so telling us that there energy cannot moe in sace at all. We hae thus found the following corresondences between acoustic air article elocities, intensities fields and their resectie stationary time-aerage alues. Table 1: Air article elocities & sound intensities = I Velocities Intensities 0 a = a = I = = 0 r = r = 0 The hysical interretation gien to intensity fields a and r is that the former reresents the intensity accounting for the transort of sound radiation (i.e. energy moing along actie intensity stream lines with the same seed c of the sound) while the latter accounts for normal modes energy locally oscillating back and forth through a gien satial oint of the Figure : To to bottom Intensimetric exerimental setu; SPL sectrum; Internal and external monitoring of the mean sound energy density along the ie axis (Kinetic energy is roortional to the triangle oints cure, Potential energy is roortional to the circle oints cure - last 5 oints are located outside the ie) 64

3 Forum Acusticum 005 Budaest. The continuity euation for sound energy density, intensimetric indicators and intensity olarization The relationshi between sound energy density w which is the sum of otential and kinetic energy 1 ( ) 1 1 w= w + wk = ρ0c + ρ0 and the intensity j = is well understood locally by means of the continuity euation w t + j = 0. Taking the timeaerage of this euation one get the fundamental roerty of the actie intensity I = 0 stating that, within a sound field where no acoustic ower sources are resent, radiation streamlines (also called ower streamlines) neer cross each other in sace. A failure in the exerimental erification of this roerty means that acoustic sources are indeed resent in the field. In the case of a flue organ ies internal 1-D sound field the aboe euation reduce simly to I x = 0 and its exerimental monitoring simly reduce in erifying that I ( x ) has or not a constant alue along the ie axis. This exeriment was actually run [4] and the obtained results (see Figure 3) clearly showed that sound ower sources are indeed resent and continuously distributed along the ie axis esecially in the region near the mouth. This sources are clearly identified with air whirlwinds due to the breaking of the air flow oer the labium edge. Table : Intensimetric indicators and their meaning η = z ξ = Definition z + z σ = z + Meaning Percentage of radiation referred to total sound energy density Pressure-elocity correlation indicator (Power factor) Balance between otential and kinetic sound energy density In the case of exerimental setu shown in Figure the measurement of the indicators η and ξ gae the results shown in Figure 4. Figure 4: The measurement of the ower factor ξ (circle oints) along the ie axis clearly indicates that the internal sound field owns a more resistie character in the zone corresonding to the maximum of otential energy. Een if the ercentage of radiation is ery low (η < 5%, triangle oints) the exerimental cure for η confirms (see Figure 3) a relatie maximum of radiation at about 0.4 m from the bottom of the ie. Figure 3: Exerimental eidence of the resence of acoustic ower sources along the ie axis. The comarison with the eta indicator reorted in Figure 4 confirms that sources are located mainly in the zone oer the mouth of the ie (x < 0.5 m). Other interesting energetic roerties of sound fields can be obtained by the measurement of intensimetric indicators. Our analysis has been focused on the hysical interretation and monitoring of the three indicators defined in Table where z = ρ0c is the characteristic imedance of air. The analysis associated with the indicator σ is more strictly related to the behaior of oscillating intensity and then to. In this case study we found the trend reorted in Figure 5 where it is eident that σ gets relatie maxima at locations where a erfect balance between kinetic and otential energies is achieed by the sound field (see Figure ). It is interesting to notice that this indicator is ractically unaffected by the drastic change in the energetic situation from inside to the outside arts of the organ ie as registered instead in Figure, 3, 4. This indicator is anyway strongly deendent from the sectral contents of the sound field 643

4 Forum Acusticum 005 Budaest and may hae scarce significance as local satial indicator in the case of continuous sectra. distance till crossing the alues of the actie intensity (noted as A in the figure). This critical distance can be called no return distance as sound radiation once treassed this crossing oint can no more come back to the internal field of the ie and is definitiely ejected in the outer enironment. Figure 5: Exerimental trend of σ indicator. The relatie maxima indicate a erfect euiartition of sound energy between kinetic and otential forms. Let s now come to the interesting subject of intensity olarization and its monitoring. Differently from light the word olarization referred to sound has nothing to do with the satial orientation of sound waes. These are longitudinal waes showing no olarization with resect to the direction of roagation of sound. The sound olarization refers to the intensity and secifically to the oscillating intensity. As reorted in Table 1, oscillating intensity is defined in such a way that r : = 0. This means that we hae to measure the time-deiation from its time-aerage in order to get a suitable exerimental uantity able to reresent it efficaciously. Roughly seaking this is the usual situation when measuring acoustic ressure but here, the ector nature of the sound intensity makes things mathematically more comlicated. The roblem anyway was rigorously soled in [6] and the first sectacular measurements of 3-D intensity olarization was reorted in [8]. In the case of organ ies anyway interesting exerimental results can be achieed by simly monitoring the effectie alue of oscillating intensity as reorted in Figure 6 and 7. One can see that the effectie alue R of the oscillating intensity is obtained from the full definition R = r r of the intensity olarization tensor gien in [6] as R = r. This coincides with the norm of the intensity olarization tensor and it is clearly a scalar uantity. Comaring Figure 5 and 6 we can notice that een though the trend of σ and R looks similar inside the ie, they hae a significant different behaior outside it. R in fact no more reaches a maximum alue showing instead a monotone decrement. Its alues decrease raidly with Figure 6: Exerimental comarison of actie (A) and oscillating (R) intensities inside and outside the to end of the ie along its axis. The no return oint has been found at 85 cm from the to end. The exerimental setu was the same as in Figure 1. Figure 7: The same comarison as in Figure 6. Here anyway actie (radiating) and oscillating intensities were measured along an axis orthogonal to the mouth of the ie (dashed line). The no return distance was found here at 4 cm from the mouth. The comarison of Figure 6 and 7 shows that the flue organ ie radiates sound more from the mouth than from the oen to. Aart from the absolute alue of actie intensity in fact the shorter distance of no return oint located near the mouth, just indicates that sound radiation escaes more easily from the mouth than from the to. 644

5 Forum Acusticum 005 Budaest.3 Four-dimensional treatment of sound and radiation ressure The exlicit introduction of the Minkowskian metric in acoustics in strict analogy with the electromagnetic field for treating in a rigorous mathematical form the concet of wae-momentum flux density has been recently done in [9]. There the concets of acoustic Lorentz transformations, sound eents and sound-cone related to the causality rincile has been gien and the conseuences of this generalization on the theoretical fundamentals of sound intensimetry has been sketched. The starting oint for understanding the hysical need of a 4-D formalization of general linear acoustics is the subtle distinction existing between the energy flux density (or sound intensity) j = and momentum flux density = j c. One easily understands that the difference between this two kind of fluxes is merely a concetual one gien that from the intensimetric oint of iew one can easily get once measured j. In fact, thanks to the four-dimensional treatment of acoustics it becomes clear that the sound radiation ressure is a matter of conseration laws and secifically regards the conseration law of linear momentum. In ractice the acoustic ector field j must aear at the same time in two continuity euations: once in the euation for local energy conseration and the second time in the continuity euation for conseration of sound field momentum. In fact, this latter euation gies rise just to sound radiation ressure which turns out to be a second order acoustical uantity whose magnitude deends from the orientation of a non-material target with resect to the direction of the air article elocity. This target is identified with the geometric surface, not necessarily coinciding with a field boundary, oer which the force of radiation is acting. In the case study of the organ ie two directions are of secial interest for the analysis: the direction of the longitudinal axis and the orthogonal one (transersal axis). This means that we hae to choose a target oriented in such a way that its normal direction is arallel to the longitudinal axis (for instance a target coinciding with the ie section: angle between the normal and the elocity directions θ = 0 ) and a target erendicular to it (for instance an area on the internal surface of the ie wall θ = π ). It can be seen from [9] that the ressure of radiation through any section of the ie is always negatie S = wk + w (that is, it is always comressing the sound radiation itself along the ie axis so constraining its motion) while the radiation ressure acting oer the internal ie wall is S = wk w and, being always orthogonal to the ower streamlines, it cannot do work on the radiation. As seen from the aboe exressions, in the first case the magnitude of the radiation force er unit area (just another name for the radiation ressure) amounts to the sound energy density, haing of course an always ositie alue. It amounts instead to the Lagragian density of sound, then haing a ositie or negatie alue, in the second case. Measures of S and S done in the flue ie are reorted in Figure 8. Figure 8: Measures of time-aerage alues of radiation ressure along the ie axis. Asterisks: orthogonal comonent of the radiation force S (Lagragian density), suares: comonent of radiation force along ower streamlines S (energy density). It is clear from Fig. 8 that the Lagrangian density of sound changes its sign at oints where wk = w (see Figure for comarison) while w doesn t show significant ariations along the ie axis. 3 Conclusions A comlete energetic analysis of both internal and external sound field of an organ flue ie has been accomlished. The exerimental data was obtained by intensimetric monitoring of the sound field and collecting measures of standard and non-standard energetic uantities defined according to the new concets of actie (radiating) and oscillating intensities. These new concets hae been theoretically introduced for general sound fields and hysically interreted in the case study of the organ flue ie. The erformed analysis has essentially confirmed well known acoustic behaiors of the organ flue ies so allowing a deeer understanding of the hysical model of the organ ie acoustic system from the energetic oint of iew. 645

6 Forum Acusticum 005 Budaest New exerimental concets as the non-return oint for sound radiation allows to monitor the real satial extension of any hysical acoustic source. In the case of musical instruments and secifically for organs this indicator may be of secial interest. In fact, being this indicator strongly affected by the oscillating intensity it accounts both for the acoustical couling between any single source and the enironment, and the mutual acoustical couling between two or many instruments. Finally adanced concets as the ressure of radiation fully introduced in the frame of linear acoustic theory with the hel of the 4-D Minkowskian formalism has been exerimentally alidated by monitoring the organ flue ie with intensimetric techniues. No doubt about it: the flue organ ie is a real aradigm of acoustics. radiation ressure, Acta Acustica/Acustica, Vol. 89. No (Ar 003) References [1] W. C. Sabine, Collected Paers on Acoustics, Peninsula Publishing, Los Altos, California, USA, ISBN (199) [] F.J. Fahy, Sound Intensity, Second Edition, E&FN SPON, Chaman & Hall, London, ISBN (1995) [3] A. Runnemalm, O. Johansson, Sound intensity measurements of an oen organ ie, Proc. Inst. of Acoustics 1997, ISMA97 Conference, Edinburgh, Vol. 19, Part 5, (1997) [4], D. Bonsi, N. Prodi, Measurements of newly defined energetic indicators of the steady sound field inside an organ ie, Proc. Inst. of Acoustics 1997, ISMA97 Conference, Edinburgh, Vol. 19, Part 5, (1997) [5] G. Schiffrer,, Energetic roerties of acoustic fields, J. Acoust. Soc. Am., Vol. 96. No (Dec 1994) [6], N. Prodi, G. Schiffrer, Reactie acoustic intensity for general fields and energy olarization, J. Acoust. Soc. Am., Vol. 99. No (Ar 1996) [7], N. Prodi, Measurements of newly defined intensimetric uantities and their hysical interretation, J. Acoust. Soc. Am., Vol. 10. No (Oct 1997) [8], D. Bonsi, N. Prodi, Measurement of new energetic arameters for the objectie characterisation of an oera house, J. of Sound and Vib. 3 (1), (000) [9], D. Bonsi, G. Schiffrer, Fourdimensional treatment of linear acoustic fields and 646