On the relationship between sound intensity and wave impedance

Transcription

1 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century PROCEEDINGS of the nd International Congress on Acoustics Sound Intensity and Inverse Methods in Acoustics: Paer ICA On the relationshi between sound intensity and wave imedance Domenico Stanzial (a), Carlos E. Graffigna (a,b) (a) Italian National Research Council, Institute of Acoustics and Sensors O. M. Corbino, c/o Physics Deartment, University of Ferrara, Room C, v. Saragat 1, I-441 Ferrara, Italy, domenico.stanzial@cnr.it (b) Universidad Nacional de Chilecito (Argentina) and University of Ferrara, International Doctorate Program, Room G115, v. Saragat 1, I-441 Ferrara, Italy, carlos.graffigna@idasc.cnr.it Abstract Following a recent aer by one of the author [ On the hysical meaning of the ower factor in acoustics, J. Acoust. Soc. Am. 131(1), 69 8 (1)] where the concet of comlex intensity has been fully develoed from the hysical oint of view and its sectral roerties have been highlighted, the resent communication focuses on the relationshi between sound intensity and wave imedance. It will be shown how the sectrum of the comlex sound intensity magnitude is directly connected with the sectrum of the wave imedance in some model fields (lane quasi-stationary waves and sherical waves) so refiguring a new methodology for measuring the sound intensity. Keywords: Wave Imedance, Sound Intensity, Acoustic Simulations.

2 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century On the relationshi between sound intensity and wave imedance 1 Introduction The connection between the comlex sound intensity and the energy admittance as defined by one of the author in a revious aer (see Eq.. 73 in [1]) is here revisited in order to establish a direct lin between sound intensity and wave imedance. The term "wave imedance" will be used in the following to indicate the acoustic imedance ratio Z / c as defined by Morse (see. 37 in Ref. [6]), but maing the orientation n, of the elementary area ds determining the so called volume flow vs v n S ds, to coincide with the tangent direction t of the ower streamline at any oint x of the sound field. This way, S can be interreted as the cross-sectional area of any ower tube shaed by equally saced energy streamlines [4],[5], and one can roerly define the wave-imedance field in the ( x, ) domain as the ratio of the secific acoustic imedance to the characteristic resistance of the medium. Based on the above definition of the acoustic wave imedance, its connection with the radiating intensity (commonly nown as active intensity ) and the oscillating intensity (see Eqs. 13, in []) it will be stated for the monochromatic case in the next section. The validity of this relationshi will be grahically demonstrated in the following, for the two canonical case-studies of quasi-stationary lane waves and monoole fields. Finally, authors ositively conclude that the here reformulated equation between the acoustic resistance and the c-normalized ratio of active intensity to the time-stationary average of inetic energy density, can be validated in any general sound field thus refiguring an alternative methodology for sound intensity measurements. The lin between sound intensity and wave imedance: the WiSi-connection Let s start writing down the formula for the c-normalized seed of the energy article,, as it can be easily inferred from the general definition given in Eq.5,. 93 of Ref. [3]: I I : cw c( W W) where I is the active intensity magnitude, c the sound seed, W the time-stationary energy density given by the sum of inetic and otential energy densities W and W resectively. As demonstrated in [1], Eq.(.1), once multilied by the characteristic resistance c, can be also interreted as the energy conductance of any general sound field with comlex intensity (.1)

3 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century S( x) : I( x) iq( x ) (.) as given in Definition 1,. 73 of the same aer. Now the question is: how this new vision of sound intensity theory can be lined to the concet of wave imedance as usually modelled and calculated in classical literature? The answer to this question can be obtained simly by grahically analysing the behaviour of energy conductance with resect to W and W searately. This analysis will be accomlished in the following Sections 3 and 4 so to demonstrating, in the reorted cases, the existing lin between the Wave imedance and the Sound intensity (WiSi-connection). To this aim the following indicators have to be further defined I I Q : : : (.3) cw cw cw and comared, together with (.1), to the wave imedance Z( x, ). As stated in the introduction of the resent aer, anyway, a secial direction of the medium vibrations has to be considered here in order to define the volume flow of the air article: the direction t : I( x) / I( x ). This condition is naturally fulfilled by the fields analyzed in Sections 3 and 4, thus the comletely usual definition of Z( x, ) given by the ratio of the comlex amlitude of the ressure solution of the wave equation to the comlex amlitude of the velocity solution has been adoted there. This way, it turns out that the following fundamental relationshi always holds in the studied cases: I Q Z(, ) : R(, ) ix (, ) c ic x x x i c. (.4) cw cw The above equation can be generalized by stating that in every monochromatic field the acoustic resistance R( x, ) is equal to the inetic energy conductance and the acoustic reactance is equal to the inetic energy suscetance. Let s now demonstrate grahically this statement for two monochromatic model fields of ractical interest: the divergent (no reflections) sherical wave field generated by a monoole source in Section 3, and a quasi-stationary lane wave field (lane wave with reflections) in Section 4. 3 Divergent Sherical Wave Case The model chosen for the divergent sherical wave field is the ulsating shere with a radius a. For this model all the relevant quantities used for the grahical demonstration of Eq. (.4) are reorted below. 3

4 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century 3.1 Model The free field velocity otential rt, for a shere centred at r, whose radius a varies from the equilibrium osition a lie a sinusoid of circular frequency can be written as: rt A ˆe r a b e 1 ia i t r ia, Aˆ (3.1) where the comlex amlitude  deends from the amlitude of vibration b a, and / c is the roagation constant in radians er meter. Clearly the field has a sherical symmetry and our analysis will be focused here only for r a so to disregard the singularity at r. The observables quantities i.e. the ressure and velocity erturbation fields are then derived as: i a tr e ( rt, ) ia b ˆ r, t r, t ˆ r, t t 1ia r ia tr e ir 1 1 ( rt, ) a b vˆ r, t vr, t vˆ( r, t) r ia r and of course the direction of the air article velocity (3.3) is always coincident with the radial direction e,,. [7]. r 3. Imedance Calculation Following the usual method, the imedance is calculated from the ratio of the comlex ressure to comlex velocity amlitudes so to obtain: X r ˆ ir r r Z( r, ) : R Z( r, ) Z(, ) vˆ ir 1 1 r 1 r (3.4) 1 X ( r, ) 1 1 tan tan. Rr (, ) r Here R is the acoustic resistance, X is the reactance and is the hase difference between ressure and velocity for any circular frequency. 3.3 Calculation of Energetic Quantities and Indicators The energetic quantities are calculated directly from Eqs (3.) and (3.3) using the stationary 1 T time-average functional : lim dt T T. This way the following results can be obtained: T (3.) (3.3) 4

5 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century v 1 b a : I( r) r a r b a a b 1 c b a c r : 4 4 r c 1 a 4 1 b a c c r r : 4 4 c 1 a r v : W : W 4 4 a r 4 a c r W W W W W L (3.5) (3.6) J : c WW Q : J I (3.7) where Ir () is the active intensity, W, W and W are resectively the inetic, otential and total energy densities, L is the Lagrangian density, J is the aarent intensity and Q is the oscillating intensity (see Ref. [1]). (a) (b) Figure 1: Grahical behaviour of imedance quantities and energy quantities: resistant art (a) and reactance art (b). From Eqs.(3.5), (3.6) and (3.7), all sound intensity indicators of interest are then derived: 5

6 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century I r c : cw c r I r I c : : cw c (1 r ) cw Q r c : cw c r Q r Q c : : cw c (1 r ) cw r J WW cr 1 r : cw W c r J WW r J WW c 1 r : : cw W c 1 r cw W r r 1 r 1 : cos. (3.8) (3.9) (3.1) (3.11) (a) Figure : Grahical behaviour of imedance quantities and energy quantities: imedance magnitude (a) and hase difference (b). (b) 6

7 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century With all of these indicators in our hands, the lots of Figures 1 and are obtained, showing a erfect coincidence only for the quantities and aearing in Eq. (.4) whose validity is thus grahically roven. 4 Quasi-Stationary Plane Wave Case Differently from the case of Section 3, the quasi-stationary lane wave field considered here, results from the suerosition of a ure rogressive and a ure regressive wave which are modelled as incident and reflected waves resectively. This means of course a more simle 1-D geometry but the introduction of a real reflection coefficient R with its hase angle, maes this case more cumbersome to be wored out. 4.1 Model Following the algorithmic rocess already introduced in Section 3, the fields are then calculated:, i t x i t x x t Ac e Re (4.1) ( xt, ) ix i( x ) it ˆ x, t i Ac e R e e t ˆ x t x, t, ( xt, ) ix i( x) it vˆ x, t iac e R e e x v x, t (, ) vˆ x t where now the satial variable is x defines also the direction of the air vibration, and A stands for the real amlitude of the incident wave. 4. Imedance Calculation The imedance, calculated as usual, gives exlicit results that cannot be reorted here due to their excessive length and thus only their inert form is indicated below: ix i( x ) ˆ ˆ e R e Z( x, ) : c R ( ) Z ˆ( x, ) X Zˆ ( x, ) ix i x vˆ e R e X( x, ) R( x, ) 1 tan. 4.3 Calculation of Energetic Quantities and Indicators The energetic quantities and their derived indicators are calculated as follows. Exlicit results are reorted only when of reasonable length, only indicating their inert form when necessary. (4.) (4.3) (4.4) 7

8 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century 1 I( x) A c1 R (4.5) W x 1 4 A R x R W x 1 4 A R x R W x W x W x 1 A R L x W x W x A R x ( ) 1 cos ( ) 1 cos ( ) ( ) ( ) 1 ( ) ( ) ( ) cos (4.6) J : c W W Q : signum L x J I I 1 R : cw 1 R I 1 R : cw 1 Rcos( x ) R I 1 R : cw 1 Rcos( x ) R (4.7) (4.8) Q Q Q : : : (4.9) cw cw cw J J J : : : (4.1) cw cw cw 1 R : 4 1 R 4 R cos ( x ) R 1 : signum L x cos. (4.11) Clearly all above inert forms have been actually comuted to numerical values and the obtained lots are reorted in Figures 3 and 4. As already found in Section 3, even in this case, a erfect match exists when the comarison is done between the grahs of the acoustic resistance R and the equivalent c units of the inetic energy conductance c or between the reactance X and the equivalent c units of the inetic energy suscetance c. This match is clearly visualized in Figures 3 and 4, so demonstrating once more the validity of Eq. (.4). 8

9 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century (a) Figure 3: Grahical behaviour of imedance quantities and energy quantities: resistant art (a) and reactance art (b). (b) (a) Figure 4: Grahical behaviour of imedance quantities and energy quantities: imedance magnitude (a) and hase difference (b). (b) 9

10 nd International Congress on Acoustics, ICA 16 Buenos Aires 5 to 9 Setember, 16 Acoustics for the 1 st Century 5 Conclusions The fundamental relationshi between Wave imedance and Sound intensity also named here WiSi-connection, has been resented in this aer for two canonical model fields: the field generated by a ulsating shere and the quasi-stationary lane wave field. The analysis has been restricted to the monochromatic case, but the authors are ositive that the relationshi can be generalized by simly choosing the direction of active intensity as the referred direction along which to calculate the volume velocity of the wave imedance. The validity of the relationshi in the two case-studies reorted here has been roved by the grahical matching of the acoustic resistance with the inetic energy conductance and of the reactance with the inetic energy suscetance, all quantities measured in MKS-Rayl. Due to the vectorial nature of sound intensity, the authors are confident that the Wisi-connection could serve as an alternative measurement method of 3-D active sound intensity, starting from the exerimental measurement of wave imedance along three orthogonal directions with a common origin at every oint of general fields. References [1] Stanzial, D. Sacchi G. Schiffrer G. On the hysical meaning of the ower factor in acoustics. J. Acoust. Soc. Am., 131(1), 1, [] Stanzial, D. Prodi, N. Schiffrer G. Reactive acoustic intensity for general fields and energy olarization. J. Acoust. Soc. Am., 99 (4), 1996, [3] Stanzial, D. Schiffrer G. On the connection between energy velocity, reverberation time and angular momentum. Journal of Sound and Vibration, 39, 1, [4] Waterhouse, R.V. Yates, T.W. Feit, D. Liu, Y.N. Energy streamlines of a sound source. J. Acoust. Soc. Am., 78(), 1985, [5] Waterhouse, R.V. Feit, D. Equal-energy streamlines. J. Acoust. Soc. Am., 8(), 1986, [6] Morse, P., McGraw-Hill, NY (USA), nd edition, [7] Towne, D. Wave Phenomena, Addison-Wesley, Massachusetts (USA),