Speaker that uses the nonlinearity. sound

Transcription

1 Master's Degree Thesis ISRN: BTH-MT-EX--5/D---SE Speaker that uses the nonlinearity in air to reate sound Johan Fredin Department of Mehanial Engineering Blekinge Institute of Tehnology Karlskrona, Sweden 5 Supervisors: Claes Hedberg, Doent, Ph.D. Meh. Eng. Kristian Haller, M.S. Meh. Eng.

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3 Speaker that uses the non-linearity in air to reate sound Johan Fredin Department of Mehanial Engineering Blekinge Institute of Tehnology Karlskrona, Sweden 5 Thesis submitted for ompletion of Master of Siene in Mehanial Engineering with emphasis on Strutural Mehanis at the Department of Mehanial Engineering, Blekinge Institute of Tehnology, Karlskrona, Sweden. bstrat: The aim of this work was to investigate theoretially and eperimentally the fat that it is possible to reate sound by ug the nonlinearity in air. speaker was built verifying the theory that if at least two high frequenies at high amplitude is propagating through a nonlinear medium, for eample air, the differene frequeny of these two is reated in mid air. Measurements and simulations were performed on what the pressure distribution from a ultrasoni transduer looks like. Keywords: Parametri loudspeaker, Differene frequeny generation, Biharmoni boundary ondition, Nonlinear aoustis

4 knowledgements This work was arried out at the Department of Mehanial Engineering, Blekinge Institute of Tehnology, Karlskrona, Sweden, under the supervision of Claes Hedberg. I wish to epress my gratitude to Dr. Claes Hedberg and Ph.D. student Kristian Haller for their guidane and professional engagement throughout the work. Karlskrona, Marh 5 Johan Fredin

5 Contents NOTTION... 4 INTRODUCTION VIBRTIONS ND SOUND BSICS OF NONLINER COUSTICS DIFFERENCE ND SUM FREQUENCY... 5 RDITION FROM BFFLED PISTON DERIVTION OF RYLEIGH INTEGRL TIME HRMONIC SIGNLS CIRCULR ELEMENTS RECTNGULR ELEMENTS RRY OF ELEMENTS THE SPEKER BUILDING THE SPEKER CRETING THE SIGNL MESURING THE SPEKER DISCUSSION ND CONCLUSION REFERENCES PPENDIX MTLB SCRIPTS STTUSBR.M UPDTESTTUSBR.M PRESSUREXIS_O5.M PRESSUREOFXIS_O5.M PRESSURE_RECT_4O.M

6 Notation a b f g i k Constant Radius or width (m) Height (m) Sound veloity (m/s) Frequeny (Hz) or funtion of outgoing wave Funtion of inoming wave Imaginary number Wave number (rad/m) P Pressure amplitude (kg/ms ) p Q Pressure (Pa) Volume flow (m 3 /s) Q & Soure strength (m 3 /s ) q& v Soure strength per unit volume (/s ) q& a Soure strength per unit area (m/s ) R R r Distane (m) Rayleigh distane (m) Radial diretion S rea (m ) t u Time (s) Veloity (m/s) u& p Piston aeleration (m/s ) y z Diretion Diretion Displaement (m) 4

7 Ratio of speifi heat λ Wavelength (m) Displaement (m) ρ Density (kg/m 3 ) σ Radial diretion τ Time oordinate system (s) φ Veloity potential (m/s) ψ ngle (rad) ngle veloity (rad/s) 5

8 Introdution Often when one thinks about reating sound one thinks about a speaker that uses a piston movement to reate the sound. This way of reating sound is not the only way. The audible sound an be reated in midair by ug ultrasound and the non-linearity in air. If one sends out two frequenies with high frequeny and amplitude into an nonlinear medium, in our ase air, a third frequeny an be heard, namely the differene frequeny. In this thesis this phenomenon will be shown both theoretially and by measurement on a speaker, whih was onstruted within the projet. There are many advantages of reating sound this way instead of the ordinary way. One of the advantages is that the sound that is reated is very direted even of low frequenies. The diretivity of sound is a problem in onventional speaker. The problem is that the low frequenies, is spreading in almost all diretions in the room, and the higher frequenies are more direted. If one instead reates sound by sending out ultrasoni frequenies the nonlinearly reated audible sound get the same diretivity as the ultrasoni frequenies, whih have a very narrow spreading beam. simile an be done with a light bulb and a torh, the onventional speaker radiates the sound like a light bulb is radiates the light, the ultrasoni speaker is on the other hand radiating the sound more like a torh or a spotlight see figure.. Figure.. Light bulb versus torh. The diretivity of a speaker that reates sound ug ultrasound an be a huge advantage in for eample a museum where you only want the people 6

9 in front of a partiular painting to hear the information of that painting, and the rest of the visitors an onentrate on the other parts of the ehibition. nother interesting advantage is that the speaker an be built very thin and that an be of ommerial interest in these days of flat sreen TVs. For a omparison between the thiknesses see figure. Figure.. Comparison of thikness of speaker. In figure. the sound reating units of the speaker is ompared. onventional speaker does not produe sound very well without a speaker bo and the speaker bo need to be quite large. In our ase with the speaker that uses the nonlinearity in air to reate sound there is no need for any speaker bo. 7

10 3 Vibrations and sound medium that is in omplete rest is silent i.e. there is no sound. When for eample people talk, vibrations our to voal ords. s a result of these vibrations sounds are reated. This means that the vibrations are the soure of sound. Other soures of sound are loudspeakers and all kinds of musi instrument. But all vibrations are not audible, they have to be in the right frequeny and amplitude. The audible sound for a human lies in the range of about to, Hz with pressure range about * -5 to about Pa. Usually sound pressure is given in deibels, db. Deibel is not a unit in the sense that a meter or a gram, a deibel is a relationship between two values of power. In this ase the relationship is often between the lowest hear able sound pressure namely about -5 Pa and the measured sound pressure []. The formula for deibels is: db Power PowerB (3.) Power of sound varies as the square of pressure and if a value is squared its logarithm is doubled, so the deibels formula beomes. Sound db log Sound pressure pressure B Sound log Sound pressure pressure B (3.) s mentioned previously sound pressure B is often the lowest sound pressure that is possible to hear, namely * -5 Pa. sound pressure of Pa is 33 db (re µpa). log 33 5 db (re µpa) (3.3) The pain threshold is about db (re µpa). Sound with frequenies lower than Hz is often alled infrasound and sound with higher frequenies than khz is alled ultrasound. Ultrasound is going to be used a lot in this thesis. 8

11 Sound travels in an elasti medium in a form of wave motion. It is often alled sound wave. Waves an be of various types the most ommon types are longitudinal waves and transverse wave. Sound waves propagating in air are longitudinal waves. The wave length depends on the frequeny and the sound veloity. λ f (3.4) Ultrasoni sound is often reated with piezoerami elements that transform an eletri signal to sound. The sound beam the elements reate an be divided in a ouple of zones []. The zone nearest the element is alled the near field or the Fresnel zone, in this zone the amplitude hanges substantially. The amplitude hanges is due to that the same wave is sent from eah part of the element, the phase differene between all these waves affet the amplitude. On greater distane from the element these phase differenes beomes less important. Therefore the wave gets more stable and spread more like a spherial wave in the so alled far field or Fraunhofer zone. The length of the Fresnel zone depends both on the frequeny and the size of the element, this will be shown in hapter 4. 9

12 4 Basis of nonlinear aoustis The wave equations, whih are usually used in solutions to aoustial problems, are valid only when the signal propagation is relatively small. In the derivation of these formulas the maimum displaement of the air partiles is assumed to be small ompared to the wavelength. This makes the density appear to be a linear funtion of the pressure. When these assumptions are no longer valid the wave will hange shape as it propagates in the medium. Eah part of the wave travels with a veloity that is the sum of the signal veloity and the partile veloity [3]. In other words the peaks travel with a higher veloity than the rest of the wave. This makes the wave deforming as it propagates. Figure 4.. Wave as it propagates through the medium In this thesis the fat that the non-linearity in air reates new frequenies if at least two frequenies are sent in to air is very important. In the following eample this fat will be shown.

13 4. Differene and Sum Frequeny We start with three equations of hydrodynamis for one-dimensional plane motion [4]. The first one is a generalization of the seond Newton law with respet to a ontinuous medium. p ρ (4.) t Where the variable (,t) the initial position. is the displaement of the medium partiles from The seond one is the law of onservation of mass written down in differential form. ρ ρ nd the third one is the equation of state. (4.) p p ( ρ) p ρ ρ (4.3) (4.) an be rewritten. ρ ρ Put (4.4) into (4.3) (4.4) p p ( ρ ) ρ p (4.5)

14 ρ ρ ρ ρ p p p p (4.6) (.6) into (.) ρ p t (4.7) ρ p t rewriting ρ p t (4.8) This equation is the non-linear Earnshaw equation. The denominator an be rewritten with the approimate relationship in form of the general type nonlinearity epansion in terms of power nonlinearities. ( ) ( ) ( )( ) (4.9) Substituting this epression in Earnshaws equation gives. ( ) ( )( ) p t ρ

15 3 This an be written: ( ) ( )( ) t (4.) where ρ p is the equilibrium sound veloity. ( ) ( )( ) t (4.) This an be rewritten to a nie nonlinear wave equation ontaining quadrati nonlinear and ubi nonlinear terms. ( ) ( )( ) t (4.) To investigate if any new frequenies are reated when a ouple of frequenies are propagating through air we are doing a simple alulation. The alulations are made with the biharmoni boundary ondition at : t t (4.3) This is a signal whih has two frequenies before it has propagated in air. where is the amplitude of the first frequeny the amplitude for the seond one. The frequenies and are in radians. We start to investigate the quadrati non-linearity. ( ) t (4.4) The left hand side is linear so it is the right hand side that is important in this ase.

16 4 First of all we hange the oordinate system to a oordinate system that travels with the wave with the sound veloity, t τ. The signal is written with the new oordinate system. τ τ τ τ Develop eah part individually. os os τ τ τ τ (4.5) τ τ ll parts together beomes

17 5 ( ) ( ) ( ) 3 3 os os os os os os t τ τ τ τ τ τ τ τ τ τ τ τ If we rewrite this one more we an see that the first harmoni of the frequenies has been reated.

18 6 ( ) ( ) τ τ τ τ τ τ 3 3 os os t (4.6) The rest of the solution is treated separately.

19 7 ( ) ( ) ( ) ( ) ( ) ( ) os os os os os os os os os os τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ

20 8 ll parts together gives. ( ) ( ) ( ) ( ) ( ) 3 3 τ τ t (4.7) Here we an see that the first harmoni of the two initial frequenies, the sum of the frequenies and the differene frequeny is reated when the signal is propagating through the medium.

21 4.. Differene Frequeny The most important one of the reated frequenies is, in this thesis, the differene frequeny. This is the frequeny that will be audible if the original frequenies lies less than khz from eah other. The differene frequeny is reated due to that the sound waves are propagating through the nonlinear air, and it therefore takes a while for the differene frequeny to be reated. The distane it takes for the differene frequeny to be reated depends among other things on the amplitudes of the original frequenies; higher amplitudes mean a shorter distane. The amplitude of the differene frequeny is only a fration of the amplitude of the original frequeny, due to the fat that the reated sound is only a biprodut and the most energy is still in the original frequenies. Calulations an be done with Burgers equation to look further in to these fats. The results of suh alulations are very hard to ompare with measurements beause we are not able to measure the amplitudes with the auray needed. For eample we do not know the amplifiation of the mier used in the measurements and we also do not know the sensitivity and frequeny response of the mirophone. The alulations with Burgers equation is therefore left out in this thesis. 9

22 5 Radiation from a baffled piston The differene frequeny is radiating as the beam of the ultrasoni sound. This fat makes it interesting to know how the ultrasoni beam look likes, in order to understand how the audible sound radiates. There are many variables that affet how the beam of ultrasound looks like. Size of the element or the array of elements Frequeny Shape of the element or the array of elements ll these variables will be investigated further below baffled piston, a piston surrounded of a rigid plane, radiates sound a little different from a piston whih is not baffled. baffled piston radiates only in the hemisphere in front of the piston not the whole spherial room. 5. Derivation of Rayleigh Integral baffled piston an be seen as a large number of point soures []. So first of all we will investigate how a point soure radiates sound. The derivation does not need to onsider nonlinearity, it is linear. It an pretty easily be shown that the wave equation for spherial waves φ rr φr φ tt (5.) r where φ is the veloity potential defined by has the general solution ( r t) g( r t) rφ f u φ (5.) Where the first term is an outgoing spherial wave and the seond term is an inoming spherial wave. In our ase the inoming wave does not eist due to we have no refletion and things like that, the solution is onveniently epressed as

23 ( t r ) f φ (5.3) r We now look at a small pulsating sphere. The volume flow of fluid is Q ( t) from the soure. ( r ( ) 4π a u ) ( a, t) Q t (5.4) where a is the radius of the sphere and u (r) is the radial omponent of the veloity. In the ase with the baffled piston the radiation is restrited to the hemisphere in front of the piston not the whole spherial spae. Therefore the fator 4 π in equation (5.4) is replaed by π, this gives the equation. ( r ( ) π a u ) ( a, t) Q t (5.5) The radial omponent of the veloity an be epressed with equation (5.3) u ( r ) ( a, t) φ a a f ( t a ) f ( t a ) f ( t a ) a This epression is put into equation (5.5) a a (5.6) ( t) Q π a f ( t a ) f ( t a ) a a Q ( t) π f ( t a ) f ( t a ) a (5.7) To onsider the sphere as a point soure the radius a has to be very small therefore we take the limit as the radius a. ( ) πf ( t) lim Q t a (5.8) The pressure is

24 p ρ φ ρ Equation (5.8) gives f ( t) t ( ) and the derivative is f ( t r ) r (5.9) Q t (5.) π ( ) Q& t f ( t) (5.) p Equation (5.9) gives the pressure p ρ Q& t πr ( r ) (5.) This is the pressure for one point soure. The quantity Q & is often alled the soure strength. If the element we are interested in is not small enough to be epressed as a point soure we may divide it into small elements whih individually at as point soures. The radiation reeived at the field point (,y,z) is the sum of the radiations from point soures. p ( y, z, t) (, y, z ; t R ) q& ρ π R, V d dy dz v (5.3) R is the distane from the point on the soure (, y, z ) to the field point (,y,z) see figure 5..

25 (,y,z ) Radiation Body r r R (,y,z) z y Figure 5.. Radiation from body of arbitrary shape. where q& v is the soure strength per unit volume. The time R is the time it takes for the wave to travel from the soure to the field point. In most ases the soures are flat and represent an area instead of a volume, this gives the pressure p ( y, z, t) q& (, y ; t R ) ds a, ρ πr S The piston is in the,y plane so all z is zero, see figure 5.. (5.4) 3

26 (,y,z) R ds (,y,) z Piston y Figure 5.. Radiation of flat piston of arbitrary shape. In this ase q& a is the soure strength per unit area or the volume aeleration per unit area, it is simply the piston aelerationu& p. p (, y, z, t) ρ S u& p (, y ; t R ) ds πr (5.5) This equation is often alled the Rayleigh integral. 4

27 5 5. Time Harmoni Signals The vibrations in a piston like element are often time harmoni [], that is t i p e u u (5.6) The aeleration term in equation (5.5) beomes ( ) ( ) ( ) ( ) ;, R t i R t i p e u i e u t R t y u & (5.7) ommonly used epression is the wave number that is defined as k (5.8) The angle veloity an then be epressed as k (5.9) The aeleration term an be epressed as ( ) ikr t i ikr t i p e u k i e u i R t y u ;, & (5.) The pressure then beomes ( ) ds R e u k i t z y p S ikr t i π ρ,,, (5.) We now move all the onstant terms outside the integral. ( ) ds R e e u k i t z y p S ikr t i π ρ,,, (5.) The pressure amplitude u ρ is denoted P ( ) ds R e e P k i t z y p S ikr t i π,,, (5.3)

28 5.3 Cirular elements The elements that are used in the onstrution of the loudspeaker in this thesis are irular. It is therefore interesting to investigate how the radiation looks like from a irular piston. In this ase the surfae integral an be hanged to a double integral over the radius of the piston and the angle ψ see fig 5.3. a ds R (,,z) ψ s r z ds y Figure 5.3. Radiation from irular piston. Due to symmetry we an loate the field point in the,z plane. The symmetry also makes it possible to loate two idential elements on the soure, one when ψ goes in the positive diretion and one when it goes in the negative diretion. These two elements have the same distribution to the pressure in eah field point so instead of alulating the distribution from eah of the two elements the pressure is alulated for one of the elements and then doubled to get the total pressure. The two elements have the ombined area ds σ dσ dψ (5.4) The pressure radiation from a irular piston beomes 6

29 p π a it ikr i k P e e, π (5.5) R ( y, z, t) σ dσ d ψ The distane R is still the distane from the soure point to the field point. In this ase it an be alulated from the two vetors r and r. R r r (5.6) Often the Rayleigh distane is mentioned, it is the distane whih roughly marks the end of the near field. Rayleigh distane is denoted [] S R λ (5.7) Where S is the area of the element. For a irular piston this beomes. R S π a k a λ λ (5.8) Vetor r is the vetor from the origin to the field point. r i j kz (5.9) Note that i is not the imaginary unit in this ase. Vetor r is the vetor from the origin to the soure point. r i σ osψ j σ ψ k (5.3) The distane R then beomes. ( osψ ) σ z R r r σ ψ (5.3) The pressure is alulated with use of MTLB in a ouple of ases. 7

30 5.3. Sound field on ais s told in hapter 3 the field in front of the element an be divided in the near field and the far field. The near field is the part with heavily varying amplitudes, in this ase with irular elements the amplitude has nulls along the distane see figure 5.4. The number of nulls depends on how long the wavelength is ompared to the radius of the element. If the wavelength is half the radius there are two nulls and so on, see figure 5.4 and figure 5.5. The speial ase on ais an be solved analytially. [] The infinitesimal element an be seen as a ring around the entre of the piston. The area of the element is ds π σ dσ (5.3) The distane from the soure point to the field point is simply R r σ (5.33) The Rayleigh integral an now be rewritten p ikr e, dσ R it ( y, z, t) i k P e σ a (5.34) The integral an now be hanged from one over σ to one over R with the fat thatσ d σ RdR []. The integral is now p r r [ ] ma it ikr i( t kr ) i( t krma ) (, y, z, t) i k P e e dr P e e where r (5.35) ma r a (5.36) This integral is solved for every point on the ais in a MTLB sript that an be seen in appendi

31 Figure 5.4. Pressure distribution on ais. Figure 5.5. Pressure distribution on ais. 9

32 5.3. Sound field off ais One thing that an not be seen on the ais is how direted the sound beam is, therefore the pressure distribution off ais is alulated. Off ais the pressure is alulated from equation (5.5) in a little more omple MTLB sript that an be seen in appendi 9.4. The result an be seen in figure 5.6. The result is of ourse the same on the ais. very interesting and useful thing to notie is that when the wavelength dereases, the sound beam gets more direted. When the wavelength is relatively large ompared to the piston radius the sound is radiating spherially, but when the wavelength get smaller the sound is radiating more like a beam. Figure 5.6. Pressure distribution off ais. 3

33 5.4 Retangular elements ds dy d r (,,z) R r R (,,z) z Piston y Figure 5.7. Radiation from retangular piston. We start with the Rayleigh integral []. p (, y, z, t) ρ S u& p (, y ; t R ) ds πr (5.37) We use the same epression for the aeleration term as before and in this ase the two elements have the ombined area ds d dy (5.38) We get the pressure equation p π a it ikr i k P e e, (5.39) R ( y, z, t) d dy π The distane from the soure point to the field point is in this ase desribed as ( ) ( y y ) z R (5.4) 3

34 Rayleigh distane for a retangular piston is S a b R (5.4) λ λ where a and b is the height and width of the element Sound field on ais To alulate the sound field on ais for a retangular piston the Rayleigh integral is used. There is no simplifiation done, as in the ase of irular piston. To make it easier to ompare with the result from the irular piston the outer dimension is the same as for the irular piston, and the piston is square. The result from the alulations an be seen in figure 5.8 and figure 5.9. Figure 5.8. Pressure distribution on ais. 3

35 Figure 5.9. Pressure distribution on ais. One signifiant differene from the irular piston is that the nulls, that where present in the ase with irular elements, is now only dips in the amplitude. The near field is therefore muh smoother for a retangular piston than for a irular piston. 33

36 5.4. Sound field off ais To alulate the pressure distribution off ais for the retangular element the same MTLB sript was used as for retangular on ais. The results from the alulations are similar to the ones from irular element off ais. The greatest differene an be seen on the near field, see figure 5.. The same onlusions about that a smaller wavelength makes a narrower sound beam an be made here. Figure 5.. Pressure distribution off ais for retangular element. 34

37 5.5 rray of elements n array is when many elements are put in a pattern. The way this pattern is designed determines how the sound field will look like. n advantage of ug an array is that a higher output level is possible than with a gle element. Beause of the larger dimensions the diretivity of an array is higher than for a gle element. disadvantage is that the near field, due to the larger dimensions, is more heavily hanging amplitude. One way to get a smoother near field is to plae the elements in a retangular pattern, this has the same effet on the near field as for a gle element that is retangular instead of irular. To alulate the pressure distribution from an array the pressure distribution from many gle elements are simply summed in eah point that are oiniding. alulation of a five element line array has been done to ompare the near and far field of an array with the same outer dimension as a large element. The frequeny is 4 khz and the elements in the array have the radius five millimetres, the distane between the elements is ten millimetres. The gle element that the omparison is made with is five times fifty millimetres. The MTLB sripts for this alulation an be seen in appendi 9.6 to 9.8. In figure 5. and figure 5. the results from the alulations are presented. The most obvious differene is that the near field is not the same for the two alulations. The plots of the pressure distributions on ais may be misleading, beause in the array the whole area is not produing pressure so the pressure looks pretty muh the same on ais but if we look at the pressure between two elements it is far from the same. This means that an array annot be simplified as one element when it is interesting what happens in the near field. In some ases the diretivity in the far field an be alulated with the simplifiation with one gle element, but only when only the far field is of interest. The simplifiation is only valid when the distane between the elements is small. If the distane is so small that the elements in the array over the whole area, the simplifiation is then of ourse valid in both the near field and the far field. 35

38 Figure 5.. Pressure distribution from 5 element line array. Figure 5.. Pressure distribution from large piston. 36

39 To ompare the results on ais further the plots from the two alulations are put in the same figure, see figure 5.3. Here it an be seen that the pressures follow eah other very well. The amplitude differene is due to that the gle large element has a larger area than the array. Figure 5.3. Pressure distribution on ais in the two ases. 37

40 6 The Speaker 6. Building the speaker There are a ouple of parameters to think about when a speaker of this kind should be built. Size s large as possible to get a narrow beam, but not to big beause of a larger element gives a longer near field Shape retangular array makes the near field less jumpy Distane between elements If the distane is great the phase differene is large. Manageable number of elements in the array It takes some time to solder all elements Two fields It is not possible in this ase to play both frequenies in all of the elements. If this is done the nonlinearity arise in the element instead of in the air in front of the speaker. If possible; four fields Four fields an make it possible to make eperiments on Disappearing Sound. The speaker was built before all alulations were made, so some assumptions made before the alulations may be inorret. The speaker is built with 96 piezo erami elements whih are onneted in parallel in two groups. The reason to divide the elements into two groups is as said in the list above that it is a great risk that the differene frequeny is reated in the elements due to nonlinearity in these instead of in the air. It is possible by a simple operation to divide the elements into four groups. This an be used to make eperiments on Disappearing sound. Disappearing sound is when two pairs of high frequeny at high amplitude are reating the same 38

41 differene frequeny but one frequeny in one of the pairs is in another phase. This makes the differene frequeny audible only in a limited area. The finished speaker an be seen in figure 6.. Figure 6.. Front and bak of the speaker. 39

42 6. Creating the Signal It is of great interest that the speaker an be used as a demonstration unit, to show that it is really possible to reate sound by ug the nonlinearity in air. To make the speaker more interesting as a demonstration unit it has to be able to hange frequeny, beause it is getting very annoying to listen to a gle frequeny for a long time. This is done by programming the signal generator with the program LBVIEW in a omputer. One of the two frequenies is the same at all times and the other frequeny is hanged up and down randomly in the hromati sale. It is not really interesting to make a more detailed eplanation of how the signal generator was programmed, but a sreenshot from the program is presented in figure 6.. Figure 6. Sreenshot from LBVIEW. 4

43 6.3 Measuring the speaker Some measurements are made of the speaker. The biggest reason for this is to ensure that it really is the differene frequeny that appears. One interesting thing to measure is how the frequenies hange on different distanes from the speaker. ouple of different measurement set ups where tried, the hardest task was to move the mirophone in a straight line in front of the speaker. The final solution was to use a robot that holds a mirophone and makes a perfet sweep over the measurement domain. The signal from the mirophone first ran through a mier that amplified the signal, and then the signal was filtered and sent to a lok in amplifier that gave the amplitude and phase of the seleted frequeny. LBVIEW was used to ollet the data. shemati piture of the measurement set up an be seen in figure 6.3. Figure 6.3 Shemati piture of measurement set up. 4

44 Three measurement results are presented in figure 6.4 to figure 6.6. The figures show the amplitude at the ais for the speaker both for the differene frequeny ( khz) and for the two initial frequenies (4 and 4 khz). Figure khz. Figure khz. 4

45 Figure 6.6. khz. s an be seen in figure 6.4 and 6.5 the ultrasoni frequenies are damped out quite quikly. It is in the high peak in the ultrasoni field that the hearable khz tone is reated. The audile tone in figure 6.6 seems to die out quite fast as well, but this an be a little misleading. If one listens in front of the speaker the differene frequeny an be heard at great distanes, beause ears are etremely sensitive detetors. The ears sensitivity for sound amplitude is logarithmi. For the same reason the sound level is often measured in deibels []. It is a good idea to look at the result of the differene frequeny in logarithmi sale on the amplitude ais. Figure 6.7 shows the same as figure 6.6 but the amplitude is presented in logarithmi sale. 43

46 Figure 6.7 khz, logarithmi amplitude sale. 44

47 7 Disussion and onlusion The most interesting thing with the speaker is not any alulations or measurement results, it is to atually listen to the speaker. It is very lear that the sound is very direted. You do not hear the sound if you are not in the beam of sound and if there are no refletions. The speaker has already been used at a demonstration at a loal shool playing simple one tone melodies, and the students all seemed very amazed by the fat that the sound they heard did not ome from the speaker but from the wall that refleted the sound. I take that reation as a proof that I have built an interesting demonstration unit. It is possible to move the point on the wall that reflets the sound by simply rediret the speaker in another diretion. This an be used to fool the listener that the sound omes from an objet that an not produe sound itself. To be used ommerially the speaker has to be able to play any sound, not only simple usoidal tones. This an be done with some signal proesg and some additional theory. Due to lak of ertain equipment, suh as digital to analogue onverters that an manage very high frequenies, and also due to some lak of time, no attempts on reating more omple sound has been done. 45

48 8 Referenes. Kinsler L.E., Frey.R.,Coppens.B. & Sanders J.V., (), Fundamentals Of oustis, fourth edition, John Wiley & Sons, New York.. Blakstok D.T., (), Fundamentals Of Physial oustis, John Wiley & sons,new York 3. Croft J.J. & Norris J.O., (), Theory, History and The dvanement of Parametri Loudspeakers, merian Tehnology, San Diego 4. Gurbatov S.N. & Rudenko O.V., oustis In Problems, to be published. 5. Enflo B.O. & Hedberg C.M., (), Theory Of Non-linear oustis In Fluids, Kluwer ademi Publishers, Dordreht. 6. Rudenko O.V. & Soluyan S.I., (977), Theoretial Fundations Of Nonlinear oustis, Consultants Bureau, New York 46

49 9 ppendi MTLB sripts There are a ouple of MTLB funtions that are used in almost every sript in this thesis. These two funtions shows the status of the progress in the sripts, the reason is that the sripts an take a while to run. 9. Statusbar.m % STTUSBR - statusbar.m makes a figure that shows the status of o proess % open a figure for progress bar % E: [fighndl statushndl] statusbar('calulating LS Per Tone...'); % E # [fighndl statushndl] statusbar('calulating LS Per Tone...',); if it is the seond window in the same file % to update the status bar write for eample this in your loop that you want to know the status of% % E: for j:t % updatestatusbar(statushndl,j,t); % end funtion [fighndl, statushndl] statusbar(namestring,n) if nargin n; end % open figure fighndl figure('name',namestring,... 'IntegerHandle','off',... 'NumberTitle','off',... 'MenuBar','none',... 'position',[33 5-(n-)*7 4 4]); % setup aes statushndlaes(... 'Units','normalized',... 'Position',[ ],... 'Bo','on',... 'UserData',,... 'Visible','on',... 47

50 'XTik',[],'YTik',[],... 'XLim',[ ],'YLim',[ ]); 9. Updatestatusbar.m % Used in statusbar to update funtion updatestatusbar(statushndl,status,mastatus) % make the aes with the handle statushndl ative aes(statushndl); % fill the bar one additional step path[ status status ]/(mastatus); ypath[ ]; path(path,ypath,'r','edgecolor','none','erasemode','none'); view(); % save the urrent status of the bar set(statushndl,'userdata',status); drawnow; 48

51 9.3 Pressureais_o5.m %% Sript that plots pressure distribution on ais for two ases %% one ase with radius of wavelength and one with radius of 5 wavelength lear all; l; lose all; global dsigma k ti u.5; f4; 343; lambda/f; rho.; alambda*; %Radius of element % On the ais on different distane w*pi*f; kw/; R(k*a^)/; %Rayleigh distane. rsteps;% steps along ais r[e-7:r/rsteps:r]; rsqrt(r.^a^); t; Prho**u; %% a lambda * %%% p(,:)p*(ep(i*(w*t-k*r))-ep(i*(w*t-k*r))); rgrr/r; Pplotabs(p)/P; figure; plot(rgr,pplot,'k'); ais([.3]); label('distane/r'); ylabel('pressure/p'); tet(.5,,'radius * Wavelength'); %% a lambda * 5 %%% 49

52 alambda*5; %Radius of element % On the ais on different distane w*pi*f; kw/; R(k*a^)/; %Rayleigh distane. rsteps;% steps along ais r[e-7:r/rsteps:r]; rsqrt(r.^a^); p(,:)p*(ep(i*(w*t-k*r))-ep(i*(w*t-k*r))); rgrr/r; Pplotabs(p)/P; figure; plot(rgr,pplot,'k'); ais([.3]); label('distane/r'); ylabel('pressure/p'); tet(.5,,'radius 5 * Wavelength'); 5

53 9.4 Pressureofais_o5.m %% Sript that plots pressure distribution off ais for two ases %% one ase with width and height of wavelength and one with width and height of 5 lear all; l; lose all; global dsigma k ti u.5; f4; 343; lambda/f; rho.; a*lambda; %%Radius of element for ase with radius * wavelength %a*lambda; %%Radius of element for ase with radius * wavelength %a5*lambda; %%Radius of element for ase with radius 5 * wavelength % On the ais on different distane w*pi*f; kw/; R(k*a^)/; %Rayleigh distane. rsteps5;% steps along ais z[e-7:.6735/rsteps:.6735]; %rsqrt(r.^a^); t; %time ant vary in this program. Its only the amplitude thats interesting so it doesn t matter Prho**u; astep5; dsigmaa/astep; sigma:dsigma:a; istep5; dipi/istep; i:di:pi; tot3*a; step; dtot/step; :d:tot; 5

54 [fig status]statusbar('calulating Pressure for eah z'); for n:length(z); updatestatusbar(status,n,length(z)); [fig status]statusbar('calulating Pressure over for eah z',); for m:length(); %ti updatestatusbar(status,m,length()); Rfeval('R3',z(n),(m),i,sigma); psum(r,); p(n,m)i*k*p/pi*ep(j*w*t)*sum(p*di,); % to end lose('calulating Pressure over for eah z') end lose('calulating Pressure for eah z') % figure; % plot(z/r,abs(p)/p); ais([ ]); % to %save rekt3g; P(:,step:*step)p; P(:,:step)flipdim(p,); X(:,step:*step); X(:,:step)-flipdim(,); mesh(z(:,:rsteps)/.6735,x/a,abs(p(:rsteps,:))'/p); view([ 9]) ais equal ais([ - ]) %tet(.6,.4,'radius * wavelength') %% For the ase with radius * wavelength %tet(.6,.4,'radius * wavelength') %% For the ase with radius * wavelength %tet(.5,.,'radius 5 * wavelength') %% For the ase with radius 5 * wavelength 5

55 save bild %save bild %save bild3 Funtions for Pressureofais_o5.m R3.m funtion RR3(Z,X,i,sigma); global k dsigma %[fighndl statushndl] statusbar('calulating the R matri over r'); Rzeros(length(sigma),length(i));%Rzeros(length(r),length(teta),length(s igma),length(i)); %for k:length(r); % updatestatusbar(statushndl,k,length(r)); %[fighndl statushndl] statusbar('calulating the R matri over teta',); %for n:length(teta); %updatestatusbar(statushndl,n,length(teta)); for m:length(i); for :length(sigma); R(m,)sqrt((Xsigma()*os(i(m)))^sigma()^*((i(m)))^Z^); R(m,)sigma()*ep(-i*k*R(m,))/R(m,)*dsigma; end end %end %lose ('Calulating the R matri over teta'); %end %lose('calulating the R matri over r'); 53

56 9.5 Pressure_ret_4o.m %% Sript that plots pressure distribution for two ases %% one ase with width and height of 4 wavelength and one with width and height of lear all; l; lose all; global da k u.5; f4; 343; lambda/f; rho.; %alambda*; %Height of element in ase a lambda* %alambda*4; %Height of element in ase a lambda*4 %alambda*; %Height of element in ase a lambda* ba; %Width of element w*pi*f; kw/; R(a*b)/lambda; %Rayleigh distane. zstep5; % Steps along z-ais %ztotr; % Size of room in z diretion in the ase on ais ztot.8575; % Size of room in z diretion in the ase off ais where R is.8575 in the last ase z[e-7:ztot/zstep:ztot]; step; % Steps along -ais tot*a; % Size of room in diretion [-tot:tot/step:tot]; % In the ase off ais %; % In the ase On ais t; %time ant vary in this program. Its only the amplitude thats interesting so it doesn t matter 54

57 Prho**u; astep7; daa/astep -a/:da:a/; bstep7; dbb/bstep; B-b/:db:b/; [fig status]statusbar('calulating Pressure for eah z'); for n:length(z); updatestatusbar(status,n,length(z)); [fig status]statusbar('calulating Pressure over for eah z',); for m:length(); updatestatusbar(status,m,length()); Rfeval('R3',z(n),(m),B,); psum(r,); p(n,m)i*k*p/pi*ep(i*w*t)*sum(p*db,); end lose('calulating Pressure over for eah z') end lose('calulating Pressure for eah z') figure; % plot(z/r,abs(p)/p,'k'); ais([ ]); % in the ase on ais % label('distane/r'); % in the ase on ais % ylabel('pressure/p'); % in the ase on ais mesh(z(:,:zstep)/.8575,/a,abs(p(:zstep,:))'/p); view([ 9]) %tet(.5,.7,'width and height * Wavelength'); %in the ase a lambda* %tet(.5,.7,'width and height 4 * Wavelength'); %in the ase a lambda*4 55

58 %tet(.5,.7,'width and height * Wavelength'); %in the ase a lambda* %save bild %save bild %save bild3 Funtions for Pressureais_ret_4o.m R3.m funtion RR3(Z,X,,B); global k da for s:length(b); for :length(); R(s,)sqrt((X-())^(B(s))^Z^); R(s,)ep(-i*k*R(s,))/R(s,)*da; end end 9.6 Comparison.m To get any results from this sript it is neessary to run the two net oming sripts first, Large_element.m and Small_element.m %% Sript that reads and plots results to ompare rray with large element. %% The first operation is to make the sumation to an array lear all; l; lose all; load rektwithfarfield; P(:,3:6)p; P(:,:3)flipdim(p,); X(:,3:6); X(:,:3)-flipdim(,); distane.%.5 MTdistane/d; 56

59 X-d*(64*MT)/:d:d*(64*MT)/-d; S(:,:,)P; S(:,MT:6MT,)P; S(:,*MT:6*MT,3)P; S(:,3*MT:63*MT,4)P; S(:,4*MT:64*MT,5)P; Ssum(S,3); figure(); subplot(,,); mesh(z(:,:5),x/.5,abs(s(:5,:)'/p)); view([ 9]); ais([.3 - ]); title('pressure in plane'); ylabel('radial distane/radius of small element') figure(); subplot(,,); plot(z(:5),abs(s(:5,(64*mt)/)/p),'k'); ais([.3 ]); title('pressure on ais');label('distane from soure [meter]'); ylabel('pressure/p') figure(3) plot(z(:5),abs(s(:5,(64*mt)/)/p),'k'); ais([.3 ]); label('distane from soure [meter]'); ylabel('pressure/p') hold on; lear all; load rektangularg3 figure() subplot(,,) plot(z(:),abs(p(:,3)/p),'k'); ais([.3 ]); title('pressure on ais');label('distane from soure [meter]'); ylabel('pressure/p') figure(); subplot(,,); mesh(z(:),/.5,abs(p(:,:)/p)'); view([ 9]); ais([.3 - ]); title('pressure in plane'); ylabel('radial distane/radius of small element') %z(:)/r, figure(3) plot(z(:),abs(p(:,3)/p),'k-.'); ais([.3 ]); legend('rray','large element') 57

60 9.7 Large_element.m %% Sript that alulates pressure distribution from a retangular element, %% with the dimensions a and b lear all; l; lose all; global da k ti u.5; f4; 343; lambda/f; rho.; a.5; %Height of element b.55; %Width of element w*pi*f; kw/; R(k*a^)/; %Rayleigh distane. zstep; % Steps along z-ais ztot.336; % Size of room in z diretion z[e-7:ztot/zstep:ztot]; step3; % Steps along -ais tot.5; % Size of room in diretion [-tot:tot/step:tot]; t; %time ant vary in this program. Its only the amplitude thats interesting so it doesn t matter Prho**u; astep7; daa/astep -a/:da:a/; bstep; dbb/bstep; B-b/:db:b/; 58

61 [fig status]statusbar('calulating Pressure for eah z'); for n:length(z); %ti updatestatusbar(status,n,length(z)); [fig status]statusbar('calulating Pressure over for eah z',); for m:length(); %ti updatestatusbar(status,m,length()); Rfeval('R3',z(n),(m),B,); psum(r,); p(n,m)i*k*p/pi*ep(i*w*t)*sum(p*db,); %to end %to lose('calulating Pressure over for eah z') end lose('calulating Pressure for eah z') to save rektangularg3; Funtions for Large_element.m R3.m funtion RR3(Z,X,,B); global k da for s:length(b); for :length(); R(s,)sqrt((X-())^(B(s))^Z^); R(s,)ep(-i*k*R(s,))/R(s,)*da; end end 59

62 9.8 Small_element.m %% Sript that alulates pressure distribution from a irular element, %% with the radius a lear all; l; lose all; global dsigma k ti u.5; f4; 343; lambda/f; rho.; a.5;%.5; %Radius of element % On the ais on different distane w*pi*f; kw/; R(k*a^)/; %Rayleigh distane. rsteps5;% steps along ais z[e-7:35*r/rsteps:35*r]; %rsqrt(r.^a^); t; %time ant vary in this program. Its only the amplitude thats interesting so it doesn t matter Prho**u; astep; dsigmaa/astep sigma:dsigma:a; istep5; dipi/istep; i:di:pi; tot*a; step3; dtot/step; :d:tot; [fig status]statusbar('calulating Pressure for eah z'); 6

63 for n:length(z); updatestatusbar(status,n,length(z)); %ti [fig status]statusbar('calulating Pressure over for eah z',); for m:length(); %ti updatestatusbar(status,m,length()); Rfeval('R3',z(n),(m),i,sigma); psum(r,); p(n,m)i*k*p/pi*ep(j*w*t)*sum(p*di,); % to end lose('calulating Pressure over for eah z') %to end lose('calulating Pressure for eah z') %figure; %plot(z/r,abs(p)/p); ais([ ]); to save rektwithfarfield; Funtions for Small_element.m R3.m funtion RR3(Z,X,i,sigma); global k dsigma %[fighndl statushndl] statusbar('calulating the R matri over r'); Rzeros(length(sigma),length(i));%Rzeros(length(r),length(teta),length(s igma),length(i)); %for k:length(r); % updatestatusbar(statushndl,k,length(r)); %[fighndl statushndl] statusbar('calulating the R matri over teta',); 6

64 %for n:length(teta); %updatestatusbar(statushndl,n,length(teta)); for m:length(i); for :length(sigma); R(m,)sqrt((Xsigma()*os(i(m)))^sigma()^*((i(m)))^Z^); R(m,)sigma()*ep(-i*k*R(m,))/R(m,)*dsigma; end end %end %lose ('Calulating the R matri over teta'); %end %lose('calulating the R matri over r'); 6

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66 Department of Mehanial Engineering, Master s Degree Programme Blekinge Institute of Tehnology, Campus Gräsvik SE Karlskrona, SWEDEN Telephone: Fa: ansel.berghuvud@bth.se