SOUND GENERATION. Table 1. Category Phenomenon Examples 1 Volume fluctuating displacing injecting

Transcription

1 SOUND GENERATION 1. Introduction For the engineering acoustics practice and in particular that of noise and vibration control, the physical mechanisms generating sound are of vital importance. It is hence necessary with a thorough understanding of the physics involved for a succesfull handling of the many various noise control situations encountered. Also, these mechanisms are often strongly dependent on the operational conditions for whatever system incorporates them, which makes additional insight required into the associated operational processes. Only with such knowledge is the acoustician or noise control engineer properly equipped to assist in the design process or to assess the most cost efficient means to remedy an existing noise problem. In this context one must distinguish between two concepts, herein referred to as source mechanism and source or better still, source unit. The former denotes the fundamental origin of sound such as an impact whereas the latter is reserved for possibly a conglomerate of mechanisms incorporated in a unit such as a washing machine where e.g. unbalances, cavitation and friction are some of the fundamental mechanisms resulting in noise and vibration output from the machine. A categorization of source units from a physics point of view is therefore rather difficult and will not lead to some unambiguous systemacy. Whilst in practice much noise and vibration are initially caused by mechanical source mechanisms, the following discussion will focus on those generating fluid-borne sound, leaving the structureborne sound to a different context. With respect to fluid-borne sound, the source mechanisms can broadly be organized in three general classes based on their phenomenological origin, see Table 1. This categorization greatly facilitates in noise control work, giving a basis for identification and guidance for predictions of design modifications. Table 1. Category Phenomenon Examples 1 Volume fluctuating displacing injecting Loudspeaker Exhaust pipe Combustion 2 Force or pressure fluctuating Propeller Flow obstructions 3 Shear stress fluctuating Jet engine Pressurized air nozzles Category 1: This category generally contains the most efficient source mechanisms in terms of the ratio of radiated power to supplied mechanical or electrical power. The most common example of sources in this category is the radiation from vibrating bodies where fluid volumes are displaced in an unsteady manner. There is, however, some qualifications necessary for this rule and that is since vibrating bodies simultaneously exert forces on the fluid. Fluctuations in the volume can also come about due to unsteady flow like in pulsating ventilation flows or due to injections of heat as in combustion processes where the temporally varying thermal expansions are 1

2 the underlying fluctuations. Last but not least, one may inject or pump in air in a fluid space which otherwise is at rest. Category 2: In this category fall mechanisms that exert unsteady forces on a fluid without any net volume displacement. Examples are propeller noise, valve noise and the noise from rustling leaves. The physical descriptions for these mechanisms are rather involved since bodies or surfaces at rest cannot do any work on an inviscid fluid. The presence of a rigid body in a moving fluid, however, constrains it locally and a small proportion of the flow energy is converted to acoustic by the forces internal to the fluid. The complexity of the fluid dynamic processes underlying this conversion makes it difficult to give simple qualitative descriptions. The cross-flow over solid bodies gives rise to boundary layers at the body surfaces. This means that the flow will separate from the body at some point downstreams the point of attack. Such a separation, in turn, prevents the flow to close again immediately behind the body and a wake is created. For cylindrical bodies such as pipes and cables exposed to cross-flow the fluid shear layer separating the wake from the fast-mowing outer fluid is unsteady and curls up into discrete vortices being released alternatingly at the two sides of the body. This vortex formation exerts then alternating forces on the cylinder as well as on the fluid. Also, when the cylinder is rigid and does not move, the forces lead to a tonal sound. The phenomenon can easily be observed a windy day in the vicinity of long cables. The tonal character only results by slender bodies whereas for bigger bodies the noise character is broadband. In the latter case moreover, the separated flow is rather turbulent than organised which gives rise to fluctuations in the surface pressure. The category 2 encompasses also forces on a fluid which are stationary in time but which move in space under acceleration. Such a process is involved for instance by fans, propellers and rotors where the noise comes at the harmonics of the so-called blade-passing frequency i.e., the frequency with which the blades pass a certain point in space. Hereby, there are two force components involved, one due to the lift, perpendicular to the plane of rotation and one due to drag in the plane of rotation. In addition of course, there are unsteady time-varying components of the type discussed earlier due to the boundary and shear layers. Category 3: The third category encompasses mechanisms where neither a net volume velocity nor a net force is produced. The most common example of this mechanism is the turbulent flow from a jet as for example the jet engine exhaust. For such a source mechanism, there is no interaction with solid surfaces. It stems from the shear stresses at the interface between fluid elements with differing velocities and which are also instationary in time. The basic problem with such mechanisms is that they cannot be accomodated by the inhomogeneous linearised wave equation. Employing Lighthill s so-called acoustic analogy [1], however, the non-linear features of the mechanism can be incorporated as a source-term in the inhomogeneous wave equation. 2. The inhomogeneous wave equation As for the homogeneous wave equation the derivation of the inhomogeneous version is based on the conservation of mass and momentum as well as the state equation for 2

3 an ideal gas - the gas law. Allowing for in- or out-flux of mass, the conservation of mass yields,!#" + # 0 $ % u =!m!t!t where m is the mass per unit volume which is supplied, u the particle velocity vector and the density,! =! 0 +!", is made up of the density of the unperturbed gas plus its perturbation. From the gas law, the time differrentiated density perturbation can be substituted by the pressure differential to give, 1!p c 2!t + " # $ u =!m 0!t Similarly the linearized equation for conservation of momentum is altered to accomodate also any external force or force distribution and becomes,!p + " 0 #u #t = f 3 wherein the external force per unit volume is denoted f. By means of a differentiation of 2 with respect to time and a differentiation of 3 with respect to space, the set of expressions, 1! 2 p c 2!t + " 0# $ 2 %!u!t * =!2 m,!t 2 1 2! 2 % p + " 0! # $u $t * =! # f, results, yielding the inhomogeneous wave equation as,! 2 p " 1 c 2 # 2 p #t 2 = " #2 m #t 2 +! $ f. If, in addition, the possibility that the fluid is moving violently is to be accounted for such as in a jet exhaust, a convective part must be incorporated in the momentum equation and, following Lighthills analogy [1-3], a third source term is introduced such that the general form of the inhomogeneous wave equation for non-viscous fluids becomes,! 2 p " 1 c 2 # 2 p #t 2 = " #2 m #t 2 +! $ f + Q 4 in which the so-called Lighthill-tensor is given by 3 3! 2 "u i u j Q = ##. 5 i=1 j =1!x i!x j 3

4 This term is acoustically equivalent in the sense that it replaces source mechanisms due to the flow with a source in a fluid at rest. It is often called the Reynolds stresses and becomes significant only for large Reynolds numbers. 3. Elementary sources 3.1. Monopole zero order radiator Whilst most sources or source units are extended in practice it is in the analysis more convenient to work with distributions of elementary sources. The simplest source corresponding the mass fluctuations in equation 4 which in a free field generates a spherically symmetric sound field is known as a point source or monopole. Upon employing the Dirac delta function defined by the integral, # $!x " x 0 dx = 1, "# the mass acceleration per unit volume generated by the monopole at a point x 0 in space can be written as,! 2 m!t 2 = M "x # x 0 6 The point source may, however, be approached in a more physical way by assuming that the mathematical point is substituted by a very small breathing sphere, thus instead of introducing mass to the fluid at rest, realising a volume displacement. By assuming that the small sphere of radius r 0 has a harmonic radial velocity of v 0 where, in order to obey acoustic linearity, the displacement amplitude must be assumed much smaller than the radius, the relationship between particle velocity and pressure,!"p = # dv dt can be employed to give,!p!r r =r0 = " j#$v 0 Note that owing to the spherical symmetry, the vectorial notation can be omitted. With the complex notation, the solution to the wave equation can be written as, pr = A r e- jkr when only outward travelling waves are present. Hence the amplitude A can be found to be given by, A = j!"v 0r jkr 0 e jkr 0 4

5 such that the pressure is obtained as, pr = jkr 0!c " r 0 % 1+ jkr 0 # $ r v 0 e jkr r 0 7 By introducing the concept of volume velocity defined by, q = v 0 4!r 0 2 e j"t, the pressure can be rewritten as, pr = q jk"c e - jkr- r 0 4!r 1+ jkr 0 8 From equation 7 it is seen that the impedance that the monopole sees looking into the fluid space is, Z = pr 0 jkr =!c 0 9 v 0 1+ jkr 0 This is the acoustic impedance which involves the characteristic impedance!c of the fluid and has a behaviour for small Helmholtz numbers, kr 0, which is essentially mass controlled and with an increasing resistance. In Figure 1 the normalised impedance is displayed to real and imaginary part. 5

6 Re[Z/!c] Helmholtz number, kr Im[Z/!c] Figure Helmholtz number, kr 0 Real and imaginary part of acoustic impedance presented to a monopole source. As can be observed the real part active grows with Helmholtz number up to a plateau of unity whereas the imaginary reactive part increases for small Helmholtz numbers but decreases for large. The power radiated to the fluid space is given by W = 1 2 Re[ pr 0 v! 2 1 " r 0 ]ds = 4#r 0 2 Re[ pr 0 v! r 0 ] S which can be conveniently expressed in terms of the volume velocity as, W = q 2!ck 2 8" 1+kr This means that as Helmholtz number tends to zero, the power is proportional to the frequency squared. 6

7 Now, with the limiting value process, the pressure from a theoretical monopole can be developed as, jkr e$ lim pr = j"#q r 0!0 4%r kr 0!0 11 From this asymptotic form, the free-field Green s function is readily established by employing a unit strength monopole yielding, gx x 0 = e- jkr 4!R where x is the observation point and R = x - x 0 the distance between this point and the source position. From 12 moreover it is clear that reciprocity holds for the freefield Green s function i.e., gx x 0 = gx 0 x so that it does not matter if the locations are interchanged. As mentioned earlier, the elementary source is a helpful tool in analysis since distributed sources can be synthetized by superposition. The fact that the monopole is here defined for harmonic motion constitutes no restriction since for all physically realizable signals, a Fourier transform or series can be developed and thence, the calculations can be made in the frequency domain and if necessary subsequently inversely transformed back to the time domain to yield an eventual transient time history Dipole first order radiator When a small disk oscillates in a fluid, there is negligible volume displacement and essentially only pressure fluctuations. Such a source that then exerts forces on the fluid belongs to the second category. The simplest arrangement to analyse such a source is to combine two monopoles but with opposite phases. This means that whilst one is expanding the other is retracting such that a portion of a control volume around the pair is compressed and the other is rarefied without moving the enclosed fluid. Therefore, upon assuming that two equally strong monopoles are located close to each other, one of strength q at a position +d 2 and the other of q at - d 2, the Green s function approach may be employed for the pair giving, [ ] pr = j!"q gr +d / 2- gr +d / 2 wherein the vectorial notation is introduced also for the point source positions. In order to construct a compact source, which operates at the source location r 0, it is instructive to consider first the two-dimensional problem. 12 7

8 Figure 2. Two closely spaced monopoles. For this situation, the pressure at the receiver position can be explicitly written as, pr = j!"q e# jkr 1 # e# jkr2 % * 4$r 1 4$r 2 which in the case that d is much smaller than r, can be simplified to, lim pr = j#$q e% jkr r!" 4r. 0 jkd cos 2 e 0 0 1% d cos + * 2r, - / 0 1 jkd cos 2 3 e% % 1 + d cos * 2r, - 23 If in addition, kd is substantially smaller that unity or equivalently, the distance between the monopoles are substantially smaller that a sixth of the wavelength, then the exponential functions can be expanded to give, jkr e$ d lim pr = j"#q kd!0 4%r r + jkd * + cos, where high order terms have been omitted. Often, the distance between the monopoles is combined with the volume velocity of the point sources such that the pressure from the acoustically compact dipole is written as, jkr e" lim pr = D kd!0 4#r $ 1 % r + jk cos* 13 in which case D denotes the dipole strength or sometimes dipole moment. With this slight detour over the two-dimensional problem it is fairly straightforward to generalize the derivation to three dimensions. Upon applying a Taylor series expansion to the Green s function, gr r 0 +!r = gr r 0 +!r "#gr r

9 and retaining only the first order term, which is valid under the criteria of compactness discussed previously, the pressure is obtained as, lim pr = " D # $gr r 0 14 kd!0 r 0 Thereby it should be noted that the dipole vector is D = j!"q#r which can be arbitrarily orientated and,!gr r 0 T = " { sin# cos$ sin# sin$ cos$ } e" jk r"r0 1 4% r " r 0 r " r 0 + jk+ * 15 In comparison with that from the monopole, the farfield pressure from a dipole is featured by two distances and two angles. The particle velocity in the dipole case cannot be purely radial but also tangential components exist. The latter, however, can readily be shown to be 90º out of phase with respect to the pressure, v! = " 1 1 %p j#$ r %! = D e " jkr 1 j#$ 4r 2 r + jk * +, sin! and does not give any active power to the farfield. Hence, it is sufficient to consider the radial components only as far as the farfield goes whereby v r =! 1 $p j"# $r = k 2 D e! jkr 4%r 1+ 2 kr 1 kr! j * + * + cos, The radial acoustic impedance seen by the dipole is thus given by Z = pr,! v r r,! 4 + kr 4 # kr 4 + jkr 2 + kr 2 = "c% $ 16 which accordingly turns out to be independent of the angle. In Figure 3 is shown the impedance as function of Helmholtz number. It is observed that the decrease for small 9

10 Re[Z/!c] Helmholtz number, kr Im[Z/!c] Figure Helmholtz number, kr 0 Real and imaginary part of acoustic impedance presented to a dipole source. Helmholtz numbers is rather steep for the real part and that this, alike that for the monopole, levels at unity in the upper range. In contrast, the imaginary part is almost the same as that for the monopole and exhibits the same features. From the radial velocity component and the pressure the farfield intensity can be determined as, I r = 1 2 Re [ pr,!v* r,!] = " 2 32#c 3 $ 2 r 2 D 2 cos 2! from which the radiated power can be obtained by integrating over a hypothetical spherical surface, 2! W d = 2!r 2 # I r sin" d" = $ 2 D 2 24!%c

11 In order to get some impression of the efficiency of the dipole, the power radiated can be compared with that of a monopole, W d = kd2 W m 3 in which the dipole lever reappears. As could be expected from the impedance comparison, it is clear that the dipole is significantly less efficient than the monopole in a wide and practically interesting frequency range since kd is assumed always much less than unity. It is important to note, however, that this analysis is only valid for observation points sufficiently remote from the dipole that the condition r >> d is fulfilled. 3.3 Quadrupoles second order radiator In addition to the two elementary radiators considered above, also more complicated sources can be present. To handle fluctuating shear stresses where neither net volume flow nor net pressure fluctuation occur, yet a third elementary source configuration is required. Since two anti-phased, closely spaced monopoles have been shown to cancel any volume displacement but rendering a net pressure fluctuation, a combination of two closely spaced dipoles in anti-phase operation could realize the desired configuration. Such a configuration would thus simulate mechanisms falling in the third source category, the quadrupoles. In order to get a physical impression of a quadrupole, one may consider a small rectangular block centred on a shaft as depicted in Figure 4. When it is turning back and forth around the shaft, the adjacent fluid layers experience a relative motion, giving rise to fluctuating shear stresses with no net pressure fluctuation in addition to the fact that no net volume is displaced. Figure 4. Quadrupole source realized by a rectangular parallelepiped rotating around a shaft. For the analysis, two different models of acoustic quadrupoles have to be distinguished. The first is the so-called longitudinal quadrupole and the second is the lateral, see Figure 5. The realization in Figure 4 is of the lateral type. 11

12 Figure 5. Longitudinal a and lateral b quadrupole models. Following the procedure above leading to the dipole, the quadrupole is obtained from superposing two dipoles yielding, pr = D! " gr r 0 + d 2 # gr r 0 # d 2 r0 In the limit as d << r and kd <<1, the expression above can be approximated by, lim pr = D "#d "#gr r 0 d!0 r0 kd!0 Upon evaluating this expression for the longitudinal type orientated along the z axis it is found that, e " jkr * $ pr,! = Q zz, 1 " 3cos 2! 4#r % + " jk r " 1 r 2 + k 2 3 " k 2 - / where Q zz = Dd and the angle! is between the quadrupole axis and the vector for the receiver position. Similarly for the lateral type where the dipole vectors are orientated in the x direction whilst the distance vector lies in the y direction, the pressure is given by, e # jkr % pr,!," = Q xy 4$r sin2! cos" sin" 3 r # k jk 2 r * 19 Obviously, one may have three of each type in a general case such that the complete description must be given a tensor form, " $! = $ # $ Q xx Q xy Q xz Q yx Q yy Q yz Q zx Q yz Q zz % 12

13 but such situations fall beyond the present scope. It remains to consider the far-field intensity. Since at large distances where any wave tends towards a plane wave, the intensity equals, I r = 1 2 pr 2!c the intensity can readily be found to be given by, and I r, long = k cos!4 Q xx 2"cr 2 I r, lat = k sin!4 cos" sin" 2 Q xy 2#cr 2 respectively. 4. Distributed sources 2 2 With the elementary sources discussed above, it is possible to handle also acoustic sources, which are large in comparison with the wavelength. Thereby, rather source units are under study than sources. This could for example be a washing machine as a source of airborne sound where the machine casing vibrates and radiates to the ambient air. The first simplification introduced for the present analysis is that the source unit is placed in free space i.e., there are no outer boundaries or other objects interfering. Intuitively, the casing vibration can be represented by a distribution of discrete monopoles where each one represents the motion of a small surface element. Based on the superposition principle, the distribution can be made continuous such that the monopoles only represent the motion of an infinitesimally small area ds. With a normal velocity v n x, y, the volume velocity becomes, qx, y = v n x,yds, and in the light of the analysis above, the pressure at a receiver point can be written as pr = j!"gr r s v n r s ds 20 Had it been so that this surface element was the only one, then the pressure would have been correctly calculated by means of this expression. The fact that there are neighbouring surface elements present which, for one thing, make the washing machine casing acoustically non-transparent a point source is always acoustically transparent since it does not occupy any space, and may scatter, diffract and absorb the radiated sound, makes the fluid exposed also to forces from the surface elements. In addition to the monopoles therefore, force sources must be included. The inclusion of both volume velocity and surface force correctly describes the boundary conditions at the interface between the casing and the fluid. Such force sources are, as seen 13

14 above, equivalent to dipoles such that the resulting contribution to the receiver pressure from the surface force becomes, pr = p s x, yds!gr r s!n With the contributions of equations 20 and 21, the total sound pressure of the vibrating surface is given by, 21 # px R = px S!gx x R S " gx R x S!p % $!n!n S ds 22 S since,!"p = # dv dt, as discussed initially. In equation 22 the free field Green s function as well as its normal derivative vary both with source and receiver position. The first term corresponds to the dipole action and gives the influence of the obstruction to the sound that the casing realises. The second term presents the monopole action, which thus is proportional to the normal acceleration of the surface. For problems such as the emission from the washing machine where the casing constitutes a closed surface, the solution given by equation 22 is only valid for the free unbounded outer space. For the cavity inside the machine, the free field Green s function is of course not valid. Therefore, it is customary to distinguish between the inner and the outer problem. If however the source unit is open, such that the one side of the vibrating panel can communicate with the opposite, there will be monopole sources on either side which effectively are collocated and cancel implying that only the dipole term, remains representing the pressure difference between the two sides. 5. References [1] M.J. Lighthill, Proceedings of the Royal Society A211, On sound generated aerodynamically. [2] P.M. Morse and K.U. Ingard, Theoretical acoustics. Mc-Graw Hill, New York. [3] D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl and F.G. Leppington, Modern methods in analytical acoustics. Springer Verlag, London. 14