Distortion of sound field in a resonator

Acta Astronautica Vol.8, No. 5-6, pp. 675--687,1981

0094-5765/81/050675-13502.00[0

Printed in Great Britain

Pergamon Press

Ltd.

Distortion of sound field in a resonator T. Y A M A N A K A

AND H. K A M I M U R A

National Aerospace Laboratory, 1880 Jindaiji-Machi, Chofu, Tokyo, Japan (Received 23 January 1981)

Abstract--An acoustical resonator is useful for manipulating weightless molten material without the material making contact with the walls. A wave form of sound field at the loop of velocity wave, where the material is expected to be placed in position, is distorted by some causes. Characteristics of a speaker driver, design of wave guide, and unmatched resonance are important factors. The area of distorted region and the amplitude spectra of the distorted wave fields have been measured by traversing a microphone probe. An effect of the distortion on the behaviour of liquid drop positioned by the resonator is theoretically discussed.

Introduction SPACE processes cover a varity of fields which have as a common bond the fact that the physical mechanisms involved are adversely affected on Earth by the presence of gravity. They include crystal growth, metallurgy, fluid physics, glass and ceramics technology, and electrophoretic separation. Since the accelerations expected in a spacecraft are not zero, but in the order of 10-4-10-5 g, contactless positioning and manipulation of molten material is required for many processes. Electromagnetic fields can position and shape electrically conducting melts. An acoustical method is useful in the control of any liquid system including those that are electrically nonconducting. An acoustical standing wave excited within a resonator, develops greater pressure at the nodes of the velocity wave than at the loops. Consequently, liquids and particles introduced into the resonator are driven away from the nodes toward the loops, there collecting and remaining until the excitation ceases. It takes a shape of drop or bubble generally, for the molten material levitated in an acoustical resonator. Drops and bubbles have their own natural oscillations of various modes. If a process of solidification; from a liquid phase into a solid phase, is considered in such an environment, the levitated molten material has a possibility of occuring an instability induced by resonance with the acoustical wave fields. Figure 1 shows frequencies of the surface wave of molten glass bubbles and of thin spherical glass shells versus the radii. Young's moduli at high temperatures are assumed a priori as shown in Fig. 2. Natural frequencies of bubbles and thin spherical shells are calculated, respectively, by theories of Lam (1932) and Baker (1961). The shaded frequencies are those of audio sonic waves. Audio sonic and ultra sonic waves are usually used for an acoustical levitation method. In this study, only audio sonic waves are used. Pressure waves of sound field at the loop, where the material is expected to be placed in position, is distorted by some causes. Characteristics of a speaker driver, design of wave guide, and unmatched resonance are important factors. 675

676

T. Yamanaka and H. Kamimura

,o,

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,\,

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Fig. 1. Oscillations of spherical shells and bubbles.

The area of distorted region also varies with these factors. The area of distorted region and amplitude and phase spectra of the distorted wave fields have been measured by traversing a microphone probe through a cylindrical resonator. An effect of the distorted wave field on the behaviour of a liquid drop is analytically discussed. Experimental apparatus Figure 3 shows an experimental apparatus. It consists of a cylindrical

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Distortion o f sound field in a resonator

677

Fig. 3. Experimentalapparatus.

chamber, wave guide, a speaker driver (JBL 2482), microphone probes and traversing driver, and a position reader. The traversing driver consists of a rack and pinion for coarse movement and a fine screw. A condenser microphone cartridge and a preamplifier (B&K 1/8in. Type 4138 and Type 2618) are integrated in the probe. A microphone probe is fixed to a head of the position reader. The head consists of photo diodes, index scales, and photo transisters. The transmitted light rays from the photo diodes pass through the main scale of the linear scale and the index scales of the head, then make a grating of light and dark stripe-pattern. A movement of the head of a rail of the linear scale generates a sinusoidal photo signal. Counting the pulses of the signal by means of the photo transisters, the traversed position of the probe is measured in accuracy of 0.005 mm. The tested resonance chamber was cylindrical. The inner diameter is 70 mm and the length is 100 mm. Two types of wave guide were investigated. One is multi holes type without plug and the other is with plug. The plug may be used for protection of the moving part of the driver from radiation heating of the melt. The resonance chamber and the speaker driver are acoustically coupled through the twelve 8 mm holes which are drilled radially symmetrically around the center of the base plane of the cylinder. In order to investigate the effects of wave travelling orientation on the phase shifts, depending on the radial angles at the loop plane, each hole is drilled obliquely to the circumference. This may induce a helical injection of sound wave into a resonance chamber. The oblique angles have been varied from 0 to 8 degrees. The wave guide with plug has

678

T. Yamanaka and H. Kamimura

straight twelve 8 mm holes which are covered by a cone plug. The effect of cone angles is also investigated by varying the half cone angle from 0 to 60 degrees. Figure 4 shows a resonance chamber and multi holes type wave guides with and without plug. Experimental

results

First, sound pressure levels versus input power to the speaker driver were measured for the wave guides at the well of the sound pressure field of the resonator where the material would be levitated. Because of an one-dimensional resonator, loops of velocity waves are considered to be flat. The sound pressure levels were measured at the two locations, center of the cylinder and near the wall, for the first mode standing wave. Secondly, axial profiles of sound pressure level were measured at the same locations by means of a probe traversing mechanism. At the bottom of the sound pressure field well, levels of higher harmonics of the standing wave become higher, compared with the basic wave. Amplitude spectra and phase spectra of the higher harmonics were measured by means of Fast Fourier Transform Analyzer (H.P. Model 3582 A). Sound intensities required to levitate drops in various gravity fields are estimated in Table 1 (King, 1934). The drops of glass melt can be levitated by the input sound power of 0.16 W against 10-4g environment for a 70 mm diameter cylindrical resonator. If a conversion efficiency of this resonator is assumed to about 0.02t, then about 8 W input to the speaker driver are enough in a spacecraft. Figures 5 and 6 show sound pressure levels at the bottom of the well versus input power to the speaker driver for wave guides without and with plug, respectively. The measurement was carried out at the two locations, i.e. dRc = 0 and r/Rc = 0.8 for the wave guides without plug and r/Re = 0 and r/Rc = 0.57 for L

~70

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. Hehc*l ~,/Jve 6~,de

[ o ~ Pl~ ~ e e 6u~de ~50

Fig. 4. Resonater and wave guides. tDr. T. G. Wang, JPL, U.S.A., has suggested us, at the XXXIst Congress of the IAF '80, Tokyo, that our resonator may have a lower value of Q, and that, if the conversion efficiency of the resonator will be improved, the distortion of the sound field will be a little different from the experimental results. We are now going to improve the resonator, The distortion of the sound field for the improved resonator will be reported later.

679

Distortion o f sound field in a resonator

Table 1. Sound intensity required to levitate drops Required sound intensity (W/m 2) Ig 10-3 g 10 -'1 g 10-5 g

Density (kg/m 3) Water Glass melt

1000 2200 3600

12o I10

12 x liP 25 × liP 40 x liP

12 x 10 25 x 10 40 x 10

12 25 40

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Fig. 5. Minimum sound pressure vs input power (without plug).

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---> Input Power to Driver ( ~ a t t s ) Fig. 6. Minimum sound pressure vs input power (plug type).

680

T. Yamanaka and H. Kamimura

plug type, where r is a radial lenth and Rc is the inner radius of the cylinder. We can find out little differences of the minimum sound pressures which are due to radial locations, helical angles and cone angles in Figs. 5 and 6. Sound pressure profiles along the chamber axes are shown in Figs. 7-9. Figure 7 shows that pressure profiles along a symmetrical axes of the cylindrical chamber are not affected by helical angles. If we remark the data expressed by triangles, i.e. r[Rc = 0.8, in Fig. 7, it can be seen that there are no radial distortion of sound pressure profiles for a = 0 ° wave guide, axially straight multi holes type. Figure 8 shows a distortion of sound pressure profile due to helical angles. Compared the data expressed by circles; i.e. a = 0 °, r/Rc = 0 , along a symmetrical axis of cylindrical chamber, with those expressed by triangles and squares; a = 4 ° and 8 °, respectively, along an offset axis of the cylindrical chamber at r/Rc = 0.8, it can be found out that the helical angles induce a radial distortion of sound pressure profile along an offset axis. The minimum pressure on an offset axis tends to be a little higher than that on a symmetrical axis. Sound pressure profiles along the chamber axes for plug type wave guides are shown in Fig. 9. We can not find out any difference and any distortion of sound profile due to cone angles. But the maximum sound pressure of plug type are a little lower than those without plug. The loss of pressure levels due to the plug is about 3 or 4 dB. Distortions of sound field in f r e q u e n c y domain were measured by means of two channels F F T analyzer. Figures 10 and 11 show amplitude spectra and phase angle spectra at the bottom of the well. There, 0 is the phase angle, B is the amplitude of sound pressure wave, suffixes A and B show a location faced to the plane transmitting sound waves in the chamber at r/R~ = 0.8 and a location along the symmetrical cylinder axis, respectively, suffixes 0 and 1 show the basic and the second harmonic f r e q u e n c y of the sound pressure wave, respectively, z and L are the axial length originated from the half length of the cylindrical chamber

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Distortion of sound field in a resonator

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and the inner length of the chamber, respectively, and .f0 is a frequency of most efficient first mode standing wave. Here a frequency of most efficient first mode standing wave is experimentally determined by finding out a minimum sound pressure level at the bottom of the well under a constant input power to the speaker driver. As long as a frequency of most efficient first mode standing wave is applied to a resonance chamber without plug, amplitude spectra of the higher harmonics are lower than that of the basic sound wave; smaller than - lOdB, as shown in Fig. 10. But the plug induces high levels of higher harmonics as shown in Fig. 11, even if a resonant matching is obtained in a chamber. Helical angles of a wave guide without plug and cone angles of a wave guide with plug have little influence on the levels of the second higher harmonic sound pressure waves, the phase angles of the basic

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T. Yamanaka and H. Kamimura

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sound pressure waves and the distorted area, but some of the phase angles of the second higher harmonic sound pressure wave. If an unmatched frequency is applied, amplitudes of the higher harmonics grow in the higher unmatched frequency region and a distorted sound field area also increases. An effect of unmatched frequency is shown in Figs. 10 and 11. An effect of distorted sound field on drop dynamics A sound pressure profile in an acoustical resonator system is schematically shown in Fig. 12 for the first mode standing wave of the basic sound wave. Here, L is the axial length of the levitation chamber, l0 is a characteristic length with a speaker driver unit, l = 3]4. A0+ lo, where A0 is the wave length of the basic

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Distortion of sound field in a resonator

F

683

L

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Fig. 12. Sound pressure profile in a cylindrical resonator.

sound wave, ¢ and z are axial coordinates. The sound pressure in the resonance chamber is given by

Pi(t, ~)

~, P. cos k.(~: - 1) eJ~O.t+o.). .=o/" cos (k. 1)

where P is an amplitude of sound pressure wave at ¢ = 0, K is a phase constant; 2w/A, co is an angular frequency, 0 is an initial phase angle, and the suffix n is an index number of higher harmonic wave. Rewriting the equation gives,

pi(t,

Po sin (2~rz/Ao) eJ(Oo~+Oo)+Pj cos (2rrz/A i) z) = sin (2zrlo/Ao) cos (2rrlO/Al)

eS~O,,~+o,)

+ P_2s i n (2zrz/A2) eJt~2t+o2) + . sin (2zrlo/A2) ""'

(1)

where ;to = (n + I)A, and n is an integer as n = 1,2 . . . . An ideal acoustical levitation resonance chamber would be a spherical one with a radial standing wave. A triaxial acoustical levitation resonane chamber is one of the practical applications (Wang, 1974; Wang, 1977; Jacobi, 1979; Lagomarsini, 1979). In a triaxial acoustical levitation resonance chamber, the pressure profile of sound wave field acts on drop dynamics as a complicated tensor. If the radial resonance pressure wave condition is assumed to be expressed by eqn (1) and (I) is approximated for a triaxial resonance chamber, the drop dynamics may solve the basic instability problem caused by the distorted sound field. Based on such a simple assumption, the following analysis is derived by letting z = r radial length of a spherical coordinate system, in eqn (1). A liquid drop is supposed to be leviated at the node of triaxial sound pressure waves r = 0. The liquid drop is immersed in another fluid, gas, which occupies the resonance chamber. The sound waves do not propagate into the liquid drop because the density of the liquid, pf, and the sound velocity of the liquid,, q , are

684

T. Yamanaka and H. Kamimura

much greater than those of the occupied fluid, p0 and Co, respectively, i.e. the acoustical impedance of the liquid is much greater than that of the occupied gas. The pressure field of the sound wave is considered to act on the drop dynamics as an evnironment. The origin of a spherical coordinate system is taken at the center of the spherical interface R. The liquid and the gas occupying the perturbed region exterior to this sphere are assumed to immiscible, incompressible and nonviscous. It is presumed that the ratio gLR2Vp/tr is sufficiently small that gravitational forces are negligible (gL is the local acceleration of gravity, Ap = IP~- P01, tr is the interfacial tension). The equations of motion and continuity in the liquid sphere and in the perturbed region exterior to this sphere, then become 9v/cgt + V p / p = 0,

(2)

V. v = 0

(3)

Consider a distortion of the interface from R to rs, rs = R + a Y~

(4)

where Y~ is a spherical harmonic of degree v, and a is a function of time such that la(t)l ~ R. The stability analysis given here will be limited to the first order in a. To this order, the fluid particle at the interface in the radial direction is given by u = d a / d t • Y~

(5)

Here, the movement of the center of the drop and the growth or collapse such as cavitation bubbles are not considered for simplicity, i.e. d R / d t = 0. From eqns (2) and (3), one obtains V2p = 0.

(6)

The equation (6) has solutions of spherical harmonics. If one considers a perturbed pressure solution due to the deformation of the drop, remembering the solutions derived by Lam (1932), Plesset (1954), Miller (1968) and others, and eqn (5), the second term of eqn (2) becomes Vp/plr_r ' = /3,2 aY~

(7)

where /St is the perturbed pressure contributing to a restoring force to the deformation of the liquid sphere, and /3,2 = (v - 1)v(v + 1)(v + 2)o" {Vpo+ (v + 1)pl}R 3 ' where/3* is the frequency of oscillation for two inviscid fluids.

(8)

Distortion of sound field in a resonator

685

The liquid sphere and the gas occupying the perturbed region exterior to this sphere are immersed in the gas occupying the resonance chamber. The sound pressure field of standing waves is assumed to be independent from the deformation of the drop because of the externally applied intensive sound field. But the drop dynamics is inversely susceptible to the applied sound pressure field, which is assumed to play a similar role as eqn (7) in the drop dynamics. The differential equation governing the perturbation of the sphere is approximately given by c~u/ c~t + V p,/ pl r=r, = -- V ~f/ Polr=,,,

(9)

where/~t is a sum of the terms contributing to the perturbed drop dynamics in eqn (1), and one obtains

Pltol 27r VP//PO[r=rs =

a Yv

e.~(,,,t+o,)

PoCo A~ cos (2zr/o/Ai)

+ the higher harmonics,

(10)

where 27rr,/A0 < 1 and 2~rr,/A~ < 1 are assumed. The remaining terms of Vp/p in eqns (1) and (2) govern the dynamics of the drop translational motions together with Bernoulli integral constants and the unperturbed interfacial tension term (Plesset, 1954). Neglecting the higher harmonics of n _->2 in eqn (10), one obtains the following differential equation from eqns (5) and (7)-(10), d2a/dt2 + {/372

2~rtOjPl (tolt + O0}a O. p0c0A~ cos (k~lo)cos =

(11)

Letting th = (tOlt + 01)/2,

(12)

h 2 = 16~rP1/pocotOlAj cos (kll0),

(13)

b = (2/3*/tOl)2 + h2/2,

(14)

and

equation (11) becomes the Mathieu equation as follows, d2a/d~b2+ (b - h 2 cos 2 ~b)a = 0.

(15)

Figure 13 shows stability solutions of the Mathieu equation for various values of b and h (Morse, 1953; Abramowitz, 1965). If tOo= tOj/2 =/3*, i.e. the frequency of the basic sound wave is equal to the frequency of the surface wave

686

T. Yamanaka and H. Kamimura

i

y

S t ~ t e 5olut~ons

3

0

÷

>6

Fig. 13. Solutions of the Mathieu equation. of the levitated drop or bubble, one obtains from eqn (14) as, b = 1 + h2/2.

(16)

The solutions for eqn (16) are shown in Fig. 13. It shows an instability of perturbation growth for a finite value of h. The experimental results of Figs. 10 and 11 show this condition. C o n c l u d i n g remarks

Sound fields of an acoustical cylindrical resonance chamber have been experimentally investigated by means of microphone probe and F F T analyzer for different kinds of wave guides. An effect of the distorted wave field on the behavior of liquid drop positioned by the resonator is theoretically discussed. In the limited range of our experiment and from a simple analysis, the followings can be concluded. In the measurement of sound pressure levels at the bottom of the well versus input power to the speaker driver for wave guides with and without plug, differences can not be found out which are due to radial locations, helical angles, and cone angles. But in the measurement of sound pressure profiles along chamber axes,the minimum pressure on an offset axis tends to be a little higher than that on a symmetrical axis. Sound pressure profiles along a symmetrical axis of cylindrical chamber are not affected by helical angles. As long as an axially straight multi holes type wave guide is used, no radial and axial distortion of sound pressure profile can not be found out for wave guides with and without plug. But helical angles induce a radial distortion into sound pressure profiles. The maximum sound pressures of plug type are a little lower than those without plug. The loss of pressure levels due to the plug is about 3 or 4 dB. As long as a f r e q u e n c y of most efficient first mode standing wave is applied to a resonance chamber without plug, amplitude spectra of the higher harmonics

Distortion of sound field in a resonator

687

are lower than that of the basic sound wave; smaller than -10dB, but a plug i n d u c e s high l e v e l s o f h i g h e r h a r m o n i c s . If an u n m a t c h e d f r e q u e n c y is a p p l i e d , a m p l i t u d e s o f t h e h i g h e r h a r m o n i c s g r o w in the h i g h e r u n m a t c h e d f r e q u e n c y r e g i o n , this c o r r e s p o n d s to a t e m p e r a t u r e i n c r e a s e of the gas in the c h a m b e r . Assuming a radial resonance sound wave condition by an one-dimensional e q u a t i o n , a n a p p r o x i m a t e i n s t a b i l i t y e q u a t i o n o f d r o p d y n a m i c s l e v i t a t e d in a n a c o u s t i c a l r e s o n a n c e c h a m b e r is d e r i v e d . A g r o w t h c o n d i t i o n o f s u r f a c e w a v e t y p e p e r t u r b a t i o n o n the d r o p is d i s c u s s e d b y s t a b i l i t y s o l u t i o n s o f the M a t h i e u e q u a t i o n . T h e s e c o n d h i g h e r h a r m o n i c s o u n d w a v e p l a y s a n i m p o r t a n t role f o r the i n s t a b i l i t y in the a n a l y t i c a l i n v e s t i g a t i o n . Acknowledgements--A part of this work was carried out under a co-operative research project

contracted by National Aerospace Laboratory and National Space Development Agency. Portions of the data were obtained for this paper from this source. The authors wish to express their thanks to Dr. T. G. Wang, JPL, U.S.A., for his kind discussions. They thank Mr. H. Nagasu, Director of Space Research Group, NAL, for his guidance, suggestions, and criticisms during the preparation of this paper.

References Abramowitz M. and Stegun I. A. (Eds.) (1965) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 4th Edn. National Bureau of Standards, Washington D. C. Baker W. E. (1961) J. Acoust. Soc. Am. 33, 1749-1758. Jacobi N., Tagg R. P., Kendal J. M., Elleman D. D. and Wang T. G. (1979) AIAA Paper 79-0225, AIAA 17th Aerospace Sciences Meeting, New Orleans, Louisiana. King L. V. (1934) Proc. R. Soc. (London) A147, 212-240. Lagomarsini G. and Wang T. G. 0979) AIAA Paper 79--0369, AIAA 17th Aerospace Science Meeting, New Orleans, Louisiana. Lain H. 0932) Hydrodynamics, 6th ED. Cambridge University Press, Cambridge, Mass. Miller C. A. and Scriven L. E. (1968) J. Fluid Mech. 32, 417-435. Morse P. M. and Feshbach H. (1953) Methods o[ Theoretical Physics, Part l, McGraw-Hill, New York. Plesset M. S. (1954) J. Appl. Phys. 25, 96-98. Nagai S. et al. [Eds.] 0960) Kogyo Zairyo Binran. Toyokeizai Shinpo, Inc., Tokyo. Nagai S. et al. [Eds.] (1965 Mukikagaku Handbook. Gihodo, Inc., Tokyo. Wang T. G., Saffren M. M. and Elleman D. D. (1974) AIAA Paper 74-155, AIAA 12th Aerospace Sciences Meeting, Washington, D. C. Wang T. G., Saffren M. M. and Elleman D. D. (1977) Prog. Astronaut. Aeronaut. 52, 151-172.