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The exploitation of non-linearity in underwater acoustics
‘j? &und l’%. (1965) 2 (4), 429-434
THE EXPLOITATION
OF NON-LINEARITY
UNDERWATER
IN
ACOUSTICS
D. G. TUCKER Department of Electronic and Electrical Engineering, University of Birmingham, Edgbaston, Birmingham I 5, England (Received 13 April 1965) Non-linearity in electrical and electronic systems has long been exploited for useful ends. Attention is now being given to the question of whether the inherent non-linearity of acoustic propagation in a fluid medium can similarly be exploited. Some of the possibilities are outlined here, with particular reference to low frequency, highly directional beam formation, parametric amplification, and sub-harmonic generation.
INTRODUCTION
the normal textbook derivation (I, 2) of the acoustic wave equation for a fluid medium, an assumption is made that the amplitude of the wave is so low that certain terms may be neglected. This assumption is commonly stated in the form In
i.e. that the second-order term Sa[/ax is negligible. Here S is the condensation and t is the particle displacement in the x-direction. Another form of the assumption is
where p is the density, p. its mean value, and ZJis the particle velocity in the x-direction. Both of these forms are based, in fact, on the simplification that the density changes so little under the acoustic pressure that it may be regarded as having always its static value po. Similarly, it is usually assumed that the bulk modulus K is independent of acoustic pressure and has its static value Kg. Thus the velocity of propagation d/(~~/p~) is regarded as independent of acoustic pressure. Therefore, the wave equation obtained, which is typically written
a2P
at2 =
Ko v2p po
(3)
(where V is the Laplacian operator a2/ax2 + a2/ay2 + a2/ax2), is an equation representing a linear system; by linear is meant that its parameters are independent of the amplitude of excitation. Under this assumption, therefore, the Superposition Theorem applies, and if two or more sinusoidal waves of different frequency exist in the medium, there is no interaction of the kind leading to the production of waves of new frequencies. If waves of high intensity are produced, however, then the assumptions discussed above are not valid, and the ratio ~~~~~ in the wave equation should be replaced by a function of K/P which is therefore itself a function of acoustic pressure and may readily be shown to be expressible as (Ko//‘o)(I+~1~+~2p2+.-.), (4) 429
D. G.TUCKER
430
where al, u2, etc. are constants determined by the parameters of the medium. Thus the relationship between time waveform and spatial waveform of pressure is dependent on a velocity of propagation which is itself, at any point, a function of the instantaneous pressure at that point. This is clearly then a non-linear effect. For example, if a waveform sinusoidal in time is imposed at a particular point in space, then the spatial waveform will not be sinusoidal, and so the waveform at any other point will not be sinusoidal in time. For a plane wave the matter can be discussed in possibly more familiar terms by considering the relationship between acoustic pressure and particle velocity, which are analogous to voltage and current, respectively, in electrical systems. The ratio p/u (where p and u are now regarded as phasors) is related to K and p by P/U = 2/(V),
(5)
and 1/(~p) can evidently be expressed as a power series in p or u in a manner similar to equation (4). Since Z/(K~) is the acoustic resistance, the first impression is that the plane wave system corresponds to an electrical circuit with non-linear resistance. This is misleading, however, as the correct analogy is with an electrical transmission line in which the distributed series L and shunt C are both non-linear, and the characteristic impedance is a non-linear resistance 1/(L/C). This becomes obvious when it is realized that the non-linear acoustic parameters are the elastic parameters of the medium (i.e. the bulk modulus and the changes in density which depend on it) and not the dissipative parameters such as absorption. It is now clear that if two sinusoidal waves of angular frequencies w1 and w2 exist in a particular volume of the medium with finite amplitudes, interaction takes place and waves with frequencies in the series Kor +mw2 are generated; here K takes all integral values from o to to and m takes all integral values (including o) from - 03 to co. Now the coefficients of non-linearity in water are very small; nevertheless it is quite easy in underwater acoustic systems to generate waves of such amplitude that the linear assumption is not justified. In the field of electrical and electronic systems the existence of non-linearity has long been recognized, not only in its r61e of an unwanted complication, but also as something to be exploited to provide desirable and advantageous devices and systems. Examples of these latter are overload protectors, modulators, harmonic and sub-harmonic generators, and parametric amplifiers (3). Recently consideration has been given by a number of workers (4, 5) to the question of whether similar desirable results can be obtained by the exploitation of non-linearity in underwater acoustics. A good deal of work on this topic has been done in the Department of Electronic and Electrical Engineering at the University of Birmingham, and some of this is reported in the other papers in this series (6,7,8). Here we shall attempt some simple physical reasoning and some analogy with well-known electrical systems. It should be emphasized that the non-linear effects discussed here are due to variation of the elastic parameters of the medium with variation of acoustic excitation, and are thus analogous to the effects of non-linear reactance in electric circuits, and not non-linear resistance.
A. HIGHLY
DIRECTIVE,
LOW FREQUENCY
TRANSMITTER
Suppose that the two waves at frequencies w1 and o2 are highly collimated along coincident axes, and can thus be regarded as plane waves confined to a narrow tube. Due to the non-linearity discussed above they will interact over a considerable distance determined effectively by the absorption coefficient of the medium at each frequency. If the
EXPLOITATION
OF NON-LINEARITY
IN ACOUSTICS
43’
medium is non-dispersive, as we may assume water to be, both waves will have the same propagation velocity, c, and may be written as p, = A,cos[w,t--(w&)x] and pz = AZ cos [w2 t - (w*/c) x] where x is distance along the axis of propagation. The interaction (a1 - w2) will arise from the product p,p, and so will be proportional
wave of frequency to
cos ((w1 - w2) t - [(WI- w*)/cl x>, where a quadrature phase relationship to the original waves, due to the fact that nonlinear elastic parameters are involved, has been neglected. From this we see that the interaction wave, as observed along the direction of the x-axis, is generated with the correct phase constant, and so propagates in the x-direction; it suffers destructive interference in other directions. We thus see that a highly directional, virtual end-fire array is formed at the difference frequency, which may be very low compared with w1 and w2. The original waves at w1 and w2 may be generated with sufficient directionality by quite small transducers, and thus a highly directional, low frequency source is obtained without the use of the very large transducers which would otherwise be necessary. The attractions of this in sonar work are obvious. The question is, of course, whether the system can operate at readily achievable power levels and with acceptable efficiency. Berktay (6, 7) and Smith (to be published) at Birmingham have shown by experiment as well as theory that it can do so, and have concluded that with quite reasonable power it is possible to obtain an intensity of the low difference frequency at a given point equal to that which would be obtained from the same power applied to transducers of the same size directly at the low frequency, but with the very great advantage of very high directionality-perhaps a beamwidth of the order of I degree in place of an omni-directional radiation. Another most promising application of this system is to electronically scanned, low frequency beams, which, it appears, can be readily achieved if a small array of the high frequency transducers is used. It is worth noting that, as the non-linearity is small, a so-called “ quasi-linear ” approach to the analysis is justified, in which interactions between the difference (or sum) frequency and the original frequencies w1 and w2 are ignored owing to their very small effect. The process is thus regarded as consisting entirely of the interaction of wi with 02, without reaction from the new waves generated; the latter have very small amplitude compared with the original waves. B. TRAVELLING-WAVE
PARAMETRIC
AMPLIFIER
The resemblance between the acoustic system discussed above and the travelling-wave parametric amplifier (TWPA) (3) will b e evident to those who know the latter. The TWPA is potentially attractive in electromagnetic microwave systems since it enables power gain to be obtained with wide bandwidths. It comprises, in essence, a recurrent ladder structure as shown in Figure I in which the capacitances, C, are non-linear in that they have a capacitance value depending on the instantaneous voltage across them. If a large amplitude wave at frequency wp is caused to travel along the structure it causes the value of C to vary periodically. If then a low amplitude signal wave of frequency wq (where wg < LO>)is also caused to travel along the structure, interaction takes place, and (among other things) a difference frequency wave at (wp - w,) is generated. These waves propagate with the correct phase velocities for interaction to be cumulative along the structure.
D. G. TUCKER
432
Now it can be readily shown, as on p. 155 of reference 3, that the non-linear capacitances controlled in this way behave effectively as elements having a negative resistance component in their impedance. This means that the signal wave is amplified as it progresses along the structure. It is important to observe that in this case the interaction between the wave at (op - w,) and the pump wave not only cannot be regarded as negligible, but is in fact vital to the generation of the negative resistance. There is no question of a “ quasi-linear ” approach, as mentioned above, being justifiable here because the new wave is of amplitude entirely comparable with that of the applied signal wave at wq. The physical mechanism of the process can be envisaged as follows. As the variation of a capacitance which has a voltage established across it requires work to be done (in contradistinction to the variation of a resistance, which requires no work), power is drawn from the high amplitude wave at c+, and this must necessarily be transferred to the signal and interaction waves. Thus the high amplitude wave is called the “ pump ” wave
Figure
I.
Basic structure of travelling wave parametric amplifier.
(hence the suitability of subscript p). The system would be equally effective with pumped inductance, or indeed any purely-reactive elements, however complex, but obviously not with pumped resistance. The negative resistance effect, and the consequent gain at the signal frequency, is greatest when wq N 42, and diminishes to zero as o,Jwp -+ o. It is apparent that the system described under (A) above will be broadly analogous to the TWPA if, firstly, one of the waves is of very large amplitude and thus becomes the pump wave, and, secondly, the elastic or energy storing components of the parameters of the medium (corresponding to the reactive elements in the TWPA) are large enough in comparison with the dissipative components. The second condition suggests that the frequencies used should not be too high. The first condition can be met by providing a high power local source. The acoustic system is not very closely analogous to the TWPA, however, because, whereas the latter has a cut-off frequency (due to its lumped character) above which waves are not effectively propagated, the acoustic system is continuous and has no cut-off. Thus it is possible in the TWPA to ensure that the only waves which propagate are those at wp, wp and (We- we) ; the sum frequency (wp + w,J may be arranged to be above the cut-off if wq is not too small compared with wp. It is not difficult to show that in a parametric amplifier the presence of the sum frequency reduces the negative resistance effect and thus reduces the gain offered to the signal frequency We The effect of the sum frequency signal in reducing this gain can be seen from the expressions on p. 40 of reference 3 to be small when wq is around one-half of wp, but very large as w,Jw,, becomes small. The gain at wI becomes zero (in decibels) as w,Jwp approaches zero. It can now be seen that if wq is of the order of wJ2, then gain is obtained at the signal frequency, and, so long as the pump wave retains a significantly large amplitude, the signal amplitude will increase along the collimated pump beam. As stated earlier, the “quasilinear” approach must not be used in calculating the performance of this system. There may be some applications for this effect in a practical acoustic system, but none are envisaged at present and it has therefore not been very fully studied. But there appears to be a most important application for a system in which wq is small and the output is
EXPLOITATION
OF NON-LINEARITY
IN ACOUSTICS
433
taken (by means of a probe transducer) at the sum frequency (op + w,). This would enable a directional receiver to be made for receiving low frequency signals where a normal transducer of adequate directivity would be prohibitively large. In this case any gain at the signal frequency wq is utilized only insofar as it causes a consequential increase of the amplitude of the sum-frequency component produced by successive elements of the beam; and as to be really useful the system would have o,Jw* tending to zero, this gain can reasonably be ignored in the analysis since it is likely to be very small compared with the attenuation of the waves due to absorption losses. Under these conditions the reaction of the sum- and difference-frequency waves on the pump and signal waves may also reasonably be ignored, and the “ quasi-linear ” approach justified, as a first approximation in the calculation of performance. The system examined by Berktay (7) in his paper below is thus based on the concept of small amounts of the signal and pump power being diverted at each elemental crosssection of the beam into the sum frequency, the eventual sum frequency output being the summation of these elemental contributions along the effective length of the pumped region. He has been able to show that significant power gain can be obtained-with practical pump powers-in this way as well as the advantages of directivity. The system appears attractive therefore as a highly directive amplifying transducer for low frequencies. The analogy with the TWPA is now seen to be a little less close than would at first appear. Naturally there are some practical difficulties in the system, and one of these is the separation of the wanted output at (wp + w,) from the residual pump wave at wp which will inevitably be at a very much higher power level. Since wp < wp, these frequencies are very close. Another difficulty is that the pump transducer cannot be transparent to the signal wave and will therefore hinder its entry into the pumped beam. Berktay (7) suggests a method of overcoming both these difficulties by having a small angle between the directions of the received wave and the pump wave. The output at (wp + w,) will then be taken at an angle to the pump wave, and the separation of the waves will be more readily effected.
C. SUB-HARMONIC HIGH
GENERATION USING ACOUSTIC INTENSITY
LOCALIZED
It is well known in electrical systems that if a non-linear reactive element is subjected to a sufficiently large excitation at, say, an angular frequency w, then sub-harmonics at frequencies w/n (where n is an integer) may be generated (3). It is easiest to consider the generation of w/z, and this may be explained in physical terms as follows. Suppose a signal at frequency o/z is present initially. Then the larger signal at frequency w acts as a “pump ” on the non-linear inductance or capacitance as in any parametric amplifier, and the interaction of W/Z with w generates amplification of the signal at W/Z, the differencefrequency being coincident with the signal frequency. This is equivalent to saying (as can easily be shown mathematically) that a negative-resistance component is introduced into the effective impedance of the pumped element (at frequency o/2), and power is drawn from the source at frequency w and transferred to the signal at o/2. When the pumping action is strong enough, the condition may be reached when the negative resistance cancels out all the positive resistance in the circuit, and then clearly the signal source at w/2 may be removed and the circuit continues to generate (or oscillate at) the sub-harmonic w/2. To ensure the correct order of sub-harmonic, some tuning of the circuit is usually required when the order (n) of the sub-harmonic is other than 2. The assumption that the sub-harmonic must exist initially to interact with the signal at w seems implicit in the explanation of all sub-harmonic generation by any type of system; but in practice it is 29
434
D. G. TUCKER
often found that there are ample disturbances (if only by noise) to start the sub-harmonic oscillation. It is clear that with the same condition as in Section B, namely that the elastic or energy storing parameters of the medium are dominant, an acoustic parametric system analogous to the sub-harmonic generator described above can be realized. It would seem that a localized high intensity region would be most effective for this purpose, and the experiments carried out at Birmingham by Dunn, Kuljis and Welsby (8) confirm that subharmonics of half the excitation frequency can indeed be generated by this mechanism in water. It appears, unfortunately, that the power level which has to be used corresponds to that required for cavitation, and thus the system is inevitably noisy. It is beginning to be clear that cavitation is actually essential in water to provide a sufficient degree of nonlinearity, but that it may be possible to operate the system in other fluid media without cavitation. Bergmann (9) quoting the work of Lange and Esche, reports the production of subharmonics in water and tentatively explains their formation in terms of pulsating resonant bubbles. It is possible that some such mechanism is really necessary if sub-harmonics other than the half frequency are to be produced; but there seems to be only meagre evidence of such high-order sub-harmonics. For generating the half frequency it seems necessary only to have sufficient non-linearity, and the role of cavitation is more likely to be to increase the non-linear effects by introducing a relatively large change in the effective (or volume average) elastic parameters. ACKNOWLEDGMENTS
These ideas are largely the result of stimulating H. 0. Berktay and B. K. Gazey.
discussions with Drs V. G. Welsby,
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9.
L. E. KINSLERand A. R. FREY 1962 Fundamentals ofAcoustics. New York: John Wiley. P. M. MORSE 1948 Vibration and Sound. New York: McGraw-Hill Book Company, Inc. D. G. TUCKER1964 Circuits with Periodically Varying Parameters. London: Macdonald. P. J. WESTER~ELT1963J. acoust. Sot. Am. 35, 535. Parametric acoustic array. J. L. S. BELLIN and R. T. BEYER 1962J. acoust. Sot. Am. 34, 1051. Experimental investigation of an end-fire array. H. 0. BERKTAY 1964 J. Sound Vib. 2, 435. Possible exploitation of non-linear acoustics in underwater transmitting applications. H. 0. BERKTAY 1964 J. Sound Vib. 2, 462. Parametric amplification by the use of acoustic non-linearities and some possible applications. D. J. DUNN, M. KULJIS and V. G. WELSBY 1964.7. Sound Vib. 2,471. Non-linear effects in a focused underwater standing wave acoustic system. L. BERGMANN1954 Der Ultraschall. Ziirich: Hirzel Verlag.
THE EXPLOITATION
OF NON-LINEARITY
UNDERWATER
IN
ACOUSTICS
D. G. TUCKER Department of Electronic and Electrical Engineering, University of Birmingham, Edgbaston, Birmingham I 5, England (Received 13 April 1965) Non-linearity in electrical and electronic systems has long been exploited for useful ends. Attention is now being given to the question of whether the inherent non-linearity of acoustic propagation in a fluid medium can similarly be exploited. Some of the possibilities are outlined here, with particular reference to low frequency, highly directional beam formation, parametric amplification, and sub-harmonic generation.
INTRODUCTION
the normal textbook derivation (I, 2) of the acoustic wave equation for a fluid medium, an assumption is made that the amplitude of the wave is so low that certain terms may be neglected. This assumption is commonly stated in the form In
i.e. that the second-order term Sa[/ax is negligible. Here S is the condensation and t is the particle displacement in the x-direction. Another form of the assumption is
where p is the density, p. its mean value, and ZJis the particle velocity in the x-direction. Both of these forms are based, in fact, on the simplification that the density changes so little under the acoustic pressure that it may be regarded as having always its static value po. Similarly, it is usually assumed that the bulk modulus K is independent of acoustic pressure and has its static value Kg. Thus the velocity of propagation d/(~~/p~) is regarded as independent of acoustic pressure. Therefore, the wave equation obtained, which is typically written
a2P
at2 =
Ko v2p po
(3)
(where V is the Laplacian operator a2/ax2 + a2/ay2 + a2/ax2), is an equation representing a linear system; by linear is meant that its parameters are independent of the amplitude of excitation. Under this assumption, therefore, the Superposition Theorem applies, and if two or more sinusoidal waves of different frequency exist in the medium, there is no interaction of the kind leading to the production of waves of new frequencies. If waves of high intensity are produced, however, then the assumptions discussed above are not valid, and the ratio ~~~~~ in the wave equation should be replaced by a function of K/P which is therefore itself a function of acoustic pressure and may readily be shown to be expressible as (Ko//‘o)(I+~1~+~2p2+.-.), (4) 429
D. G.TUCKER
430
where al, u2, etc. are constants determined by the parameters of the medium. Thus the relationship between time waveform and spatial waveform of pressure is dependent on a velocity of propagation which is itself, at any point, a function of the instantaneous pressure at that point. This is clearly then a non-linear effect. For example, if a waveform sinusoidal in time is imposed at a particular point in space, then the spatial waveform will not be sinusoidal, and so the waveform at any other point will not be sinusoidal in time. For a plane wave the matter can be discussed in possibly more familiar terms by considering the relationship between acoustic pressure and particle velocity, which are analogous to voltage and current, respectively, in electrical systems. The ratio p/u (where p and u are now regarded as phasors) is related to K and p by P/U = 2/(V),
(5)
and 1/(~p) can evidently be expressed as a power series in p or u in a manner similar to equation (4). Since Z/(K~) is the acoustic resistance, the first impression is that the plane wave system corresponds to an electrical circuit with non-linear resistance. This is misleading, however, as the correct analogy is with an electrical transmission line in which the distributed series L and shunt C are both non-linear, and the characteristic impedance is a non-linear resistance 1/(L/C). This becomes obvious when it is realized that the non-linear acoustic parameters are the elastic parameters of the medium (i.e. the bulk modulus and the changes in density which depend on it) and not the dissipative parameters such as absorption. It is now clear that if two sinusoidal waves of angular frequencies w1 and w2 exist in a particular volume of the medium with finite amplitudes, interaction takes place and waves with frequencies in the series Kor +mw2 are generated; here K takes all integral values from o to to and m takes all integral values (including o) from - 03 to co. Now the coefficients of non-linearity in water are very small; nevertheless it is quite easy in underwater acoustic systems to generate waves of such amplitude that the linear assumption is not justified. In the field of electrical and electronic systems the existence of non-linearity has long been recognized, not only in its r61e of an unwanted complication, but also as something to be exploited to provide desirable and advantageous devices and systems. Examples of these latter are overload protectors, modulators, harmonic and sub-harmonic generators, and parametric amplifiers (3). Recently consideration has been given by a number of workers (4, 5) to the question of whether similar desirable results can be obtained by the exploitation of non-linearity in underwater acoustics. A good deal of work on this topic has been done in the Department of Electronic and Electrical Engineering at the University of Birmingham, and some of this is reported in the other papers in this series (6,7,8). Here we shall attempt some simple physical reasoning and some analogy with well-known electrical systems. It should be emphasized that the non-linear effects discussed here are due to variation of the elastic parameters of the medium with variation of acoustic excitation, and are thus analogous to the effects of non-linear reactance in electric circuits, and not non-linear resistance.
A. HIGHLY
DIRECTIVE,
LOW FREQUENCY
TRANSMITTER
Suppose that the two waves at frequencies w1 and o2 are highly collimated along coincident axes, and can thus be regarded as plane waves confined to a narrow tube. Due to the non-linearity discussed above they will interact over a considerable distance determined effectively by the absorption coefficient of the medium at each frequency. If the
EXPLOITATION
OF NON-LINEARITY
IN ACOUSTICS
43’
medium is non-dispersive, as we may assume water to be, both waves will have the same propagation velocity, c, and may be written as p, = A,cos[w,t--(w&)x] and pz = AZ cos [w2 t - (w*/c) x] where x is distance along the axis of propagation. The interaction (a1 - w2) will arise from the product p,p, and so will be proportional
wave of frequency to
cos ((w1 - w2) t - [(WI- w*)/cl x>, where a quadrature phase relationship to the original waves, due to the fact that nonlinear elastic parameters are involved, has been neglected. From this we see that the interaction wave, as observed along the direction of the x-axis, is generated with the correct phase constant, and so propagates in the x-direction; it suffers destructive interference in other directions. We thus see that a highly directional, virtual end-fire array is formed at the difference frequency, which may be very low compared with w1 and w2. The original waves at w1 and w2 may be generated with sufficient directionality by quite small transducers, and thus a highly directional, low frequency source is obtained without the use of the very large transducers which would otherwise be necessary. The attractions of this in sonar work are obvious. The question is, of course, whether the system can operate at readily achievable power levels and with acceptable efficiency. Berktay (6, 7) and Smith (to be published) at Birmingham have shown by experiment as well as theory that it can do so, and have concluded that with quite reasonable power it is possible to obtain an intensity of the low difference frequency at a given point equal to that which would be obtained from the same power applied to transducers of the same size directly at the low frequency, but with the very great advantage of very high directionality-perhaps a beamwidth of the order of I degree in place of an omni-directional radiation. Another most promising application of this system is to electronically scanned, low frequency beams, which, it appears, can be readily achieved if a small array of the high frequency transducers is used. It is worth noting that, as the non-linearity is small, a so-called “ quasi-linear ” approach to the analysis is justified, in which interactions between the difference (or sum) frequency and the original frequencies w1 and w2 are ignored owing to their very small effect. The process is thus regarded as consisting entirely of the interaction of wi with 02, without reaction from the new waves generated; the latter have very small amplitude compared with the original waves. B. TRAVELLING-WAVE
PARAMETRIC
AMPLIFIER
The resemblance between the acoustic system discussed above and the travelling-wave parametric amplifier (TWPA) (3) will b e evident to those who know the latter. The TWPA is potentially attractive in electromagnetic microwave systems since it enables power gain to be obtained with wide bandwidths. It comprises, in essence, a recurrent ladder structure as shown in Figure I in which the capacitances, C, are non-linear in that they have a capacitance value depending on the instantaneous voltage across them. If a large amplitude wave at frequency wp is caused to travel along the structure it causes the value of C to vary periodically. If then a low amplitude signal wave of frequency wq (where wg < LO>)is also caused to travel along the structure, interaction takes place, and (among other things) a difference frequency wave at (wp - w,) is generated. These waves propagate with the correct phase velocities for interaction to be cumulative along the structure.
D. G. TUCKER
432
Now it can be readily shown, as on p. 155 of reference 3, that the non-linear capacitances controlled in this way behave effectively as elements having a negative resistance component in their impedance. This means that the signal wave is amplified as it progresses along the structure. It is important to observe that in this case the interaction between the wave at (op - w,) and the pump wave not only cannot be regarded as negligible, but is in fact vital to the generation of the negative resistance. There is no question of a “ quasi-linear ” approach, as mentioned above, being justifiable here because the new wave is of amplitude entirely comparable with that of the applied signal wave at wq. The physical mechanism of the process can be envisaged as follows. As the variation of a capacitance which has a voltage established across it requires work to be done (in contradistinction to the variation of a resistance, which requires no work), power is drawn from the high amplitude wave at c+, and this must necessarily be transferred to the signal and interaction waves. Thus the high amplitude wave is called the “ pump ” wave
Figure
I.
Basic structure of travelling wave parametric amplifier.
(hence the suitability of subscript p). The system would be equally effective with pumped inductance, or indeed any purely-reactive elements, however complex, but obviously not with pumped resistance. The negative resistance effect, and the consequent gain at the signal frequency, is greatest when wq N 42, and diminishes to zero as o,Jwp -+ o. It is apparent that the system described under (A) above will be broadly analogous to the TWPA if, firstly, one of the waves is of very large amplitude and thus becomes the pump wave, and, secondly, the elastic or energy storing components of the parameters of the medium (corresponding to the reactive elements in the TWPA) are large enough in comparison with the dissipative components. The second condition suggests that the frequencies used should not be too high. The first condition can be met by providing a high power local source. The acoustic system is not very closely analogous to the TWPA, however, because, whereas the latter has a cut-off frequency (due to its lumped character) above which waves are not effectively propagated, the acoustic system is continuous and has no cut-off. Thus it is possible in the TWPA to ensure that the only waves which propagate are those at wp, wp and (We- we) ; the sum frequency (wp + w,J may be arranged to be above the cut-off if wq is not too small compared with wp. It is not difficult to show that in a parametric amplifier the presence of the sum frequency reduces the negative resistance effect and thus reduces the gain offered to the signal frequency We The effect of the sum frequency signal in reducing this gain can be seen from the expressions on p. 40 of reference 3 to be small when wq is around one-half of wp, but very large as w,Jw,, becomes small. The gain at wI becomes zero (in decibels) as w,Jwp approaches zero. It can now be seen that if wq is of the order of wJ2, then gain is obtained at the signal frequency, and, so long as the pump wave retains a significantly large amplitude, the signal amplitude will increase along the collimated pump beam. As stated earlier, the “quasilinear” approach must not be used in calculating the performance of this system. There may be some applications for this effect in a practical acoustic system, but none are envisaged at present and it has therefore not been very fully studied. But there appears to be a most important application for a system in which wq is small and the output is
EXPLOITATION
OF NON-LINEARITY
IN ACOUSTICS
433
taken (by means of a probe transducer) at the sum frequency (op + w,). This would enable a directional receiver to be made for receiving low frequency signals where a normal transducer of adequate directivity would be prohibitively large. In this case any gain at the signal frequency wq is utilized only insofar as it causes a consequential increase of the amplitude of the sum-frequency component produced by successive elements of the beam; and as to be really useful the system would have o,Jw* tending to zero, this gain can reasonably be ignored in the analysis since it is likely to be very small compared with the attenuation of the waves due to absorption losses. Under these conditions the reaction of the sum- and difference-frequency waves on the pump and signal waves may also reasonably be ignored, and the “ quasi-linear ” approach justified, as a first approximation in the calculation of performance. The system examined by Berktay (7) in his paper below is thus based on the concept of small amounts of the signal and pump power being diverted at each elemental crosssection of the beam into the sum frequency, the eventual sum frequency output being the summation of these elemental contributions along the effective length of the pumped region. He has been able to show that significant power gain can be obtained-with practical pump powers-in this way as well as the advantages of directivity. The system appears attractive therefore as a highly directive amplifying transducer for low frequencies. The analogy with the TWPA is now seen to be a little less close than would at first appear. Naturally there are some practical difficulties in the system, and one of these is the separation of the wanted output at (wp + w,) from the residual pump wave at wp which will inevitably be at a very much higher power level. Since wp < wp, these frequencies are very close. Another difficulty is that the pump transducer cannot be transparent to the signal wave and will therefore hinder its entry into the pumped beam. Berktay (7) suggests a method of overcoming both these difficulties by having a small angle between the directions of the received wave and the pump wave. The output at (wp + w,) will then be taken at an angle to the pump wave, and the separation of the waves will be more readily effected.
C. SUB-HARMONIC HIGH
GENERATION USING ACOUSTIC INTENSITY
LOCALIZED
It is well known in electrical systems that if a non-linear reactive element is subjected to a sufficiently large excitation at, say, an angular frequency w, then sub-harmonics at frequencies w/n (where n is an integer) may be generated (3). It is easiest to consider the generation of w/z, and this may be explained in physical terms as follows. Suppose a signal at frequency o/z is present initially. Then the larger signal at frequency w acts as a “pump ” on the non-linear inductance or capacitance as in any parametric amplifier, and the interaction of W/Z with w generates amplification of the signal at W/Z, the differencefrequency being coincident with the signal frequency. This is equivalent to saying (as can easily be shown mathematically) that a negative-resistance component is introduced into the effective impedance of the pumped element (at frequency o/2), and power is drawn from the source at frequency w and transferred to the signal at o/2. When the pumping action is strong enough, the condition may be reached when the negative resistance cancels out all the positive resistance in the circuit, and then clearly the signal source at w/2 may be removed and the circuit continues to generate (or oscillate at) the sub-harmonic w/2. To ensure the correct order of sub-harmonic, some tuning of the circuit is usually required when the order (n) of the sub-harmonic is other than 2. The assumption that the sub-harmonic must exist initially to interact with the signal at w seems implicit in the explanation of all sub-harmonic generation by any type of system; but in practice it is 29
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often found that there are ample disturbances (if only by noise) to start the sub-harmonic oscillation. It is clear that with the same condition as in Section B, namely that the elastic or energy storing parameters of the medium are dominant, an acoustic parametric system analogous to the sub-harmonic generator described above can be realized. It would seem that a localized high intensity region would be most effective for this purpose, and the experiments carried out at Birmingham by Dunn, Kuljis and Welsby (8) confirm that subharmonics of half the excitation frequency can indeed be generated by this mechanism in water. It appears, unfortunately, that the power level which has to be used corresponds to that required for cavitation, and thus the system is inevitably noisy. It is beginning to be clear that cavitation is actually essential in water to provide a sufficient degree of nonlinearity, but that it may be possible to operate the system in other fluid media without cavitation. Bergmann (9) quoting the work of Lange and Esche, reports the production of subharmonics in water and tentatively explains their formation in terms of pulsating resonant bubbles. It is possible that some such mechanism is really necessary if sub-harmonics other than the half frequency are to be produced; but there seems to be only meagre evidence of such high-order sub-harmonics. For generating the half frequency it seems necessary only to have sufficient non-linearity, and the role of cavitation is more likely to be to increase the non-linear effects by introducing a relatively large change in the effective (or volume average) elastic parameters. ACKNOWLEDGMENTS
These ideas are largely the result of stimulating H. 0. Berktay and B. K. Gazey.
discussions with Drs V. G. Welsby,
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9.
L. E. KINSLERand A. R. FREY 1962 Fundamentals ofAcoustics. New York: John Wiley. P. M. MORSE 1948 Vibration and Sound. New York: McGraw-Hill Book Company, Inc. D. G. TUCKER1964 Circuits with Periodically Varying Parameters. London: Macdonald. P. J. WESTER~ELT1963J. acoust. Sot. Am. 35, 535. Parametric acoustic array. J. L. S. BELLIN and R. T. BEYER 1962J. acoust. Sot. Am. 34, 1051. Experimental investigation of an end-fire array. H. 0. BERKTAY 1964 J. Sound Vib. 2, 435. Possible exploitation of non-linear acoustics in underwater transmitting applications. H. 0. BERKTAY 1964 J. Sound Vib. 2, 462. Parametric amplification by the use of acoustic non-linearities and some possible applications. D. J. DUNN, M. KULJIS and V. G. WELSBY 1964.7. Sound Vib. 2,471. Non-linear effects in a focused underwater standing wave acoustic system. L. BERGMANN1954 Der Ultraschall. Ziirich: Hirzel Verlag.