19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 NONLINEAR DYNAMICS IN PARAMETRIC SOUND GENERATION

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1 9 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 7 NONLINEAR DYNAMICS IN PARAMETRIC SOUND GENERATION PACS: 43.5.Ts, y V.J. Sánchez Morcillo, V. Espinosa, I. Pérez-Arjona and J. Redondo Departamento de Física Aplicada, Escuela Politécnica Superior de Gandia, Universidad Politécnica de Valencia, Crta. Natzaret-Oliva s/n, 4673 Grau de Gandia, Spain. victorsm@fis.upv.es ABSTRACT We consider the dynamic behaviour of ultrasound, parametrically generated in an acoustic interferometer. Depending on the geometry, determining the aspect ratio of the resonator, we can distinguish two cases. In a small aspect-ratio configuration, the system can be theoretically described by a large but finite set of time dependent, coupled amplitude equations for the interacting frequency components. Although, in general, the weak dispersion in the system implies the presence of many higher harmonics of the excitation frequency, in the neigbourhood of parametric instability only the second harmonic has an appreciable amplitude, as experimental results reveal. Thus we propose a theoretical analysis in terms of a three-mode model, including the fundamental (pump), subharmonic and second harmonic. The analysis shows the existence of secondary instabilities leading to complex temporal dynamics, including self-pulsing and chaotic behaviour. In the large aspect ratio case (decreasing, e.g. the length of the resonator), diffraction effects can play an important role, leading to spatial instabilities of the on-axis plane waves. As a result, different pattern formation scenarios have been predicted including spatially periodic and localized structures in the transverse cross section of the resonator. We have designed an experimental setup in order to explore the obove-mentioned effects. The results are in qualitative agreement with the predictions of the model. A. INTRODUCTION The parametric generation of ultrasound in acoustic resonators, since it discovery four decades ago, has attracted great attention, being relevant both in their fundamental and applied aspects. The phenomenon, consisting in the excitation of modes with frequencies below the excitation frequency, has been observed in a variety situations, and different theories have been proposed to describe the observations. One theoretical approach is based in the Mathieu equation, while others in the modal expansion for each of the interacting components. In the latter case, the theory is handeable if some dispersion mechanism is considered, which reduces the effectivity of the (otherwise unavoidable) energy flow into the higher harmonics. A good agreement with modal theories have been obtained using resonators with lateral boundaries 3 or with finite-thickness walls 4,5. On the other side, the acoustical resonator can be considered as a complex physical system driven far from equilibrium 6, and from this point of view one can expect different spatio-temporal dynamical regimes, e.g. chaotic behaviour and self-organization. These dynamical regimes of parametric sound generation have been theoretically considered in some recent works 7,8. Here we present our advances on this topic, and the experimental results supporting the theory. B. THEORY The system under study consists in an acoustic interferometer of length L, composed by two parallel plane walls with high reflectivities, containing a fluid medium inside. The resonator is pumped by an external acoustic field of amplitude p in and frequency ω. Assuming the existence of some dispersion mechanism, which supress or reduces the harmonic cascade, the process of parametric sound generation can be described with enough accuracy in terms of the interaction of three field modes, the driving or fundamental (ω ) and subharmonic frequencies (ω,ω ), satisfying ω = ω + ω, assuming that any of them is close to a resonator eigenfrequency. We will consider here the particular degenerate case, where ω = ω.

2 Under this condition, the pressure p inside the resonator can be described as a superposition of standing waves ( k z) c.. iω j t p( x, y, z, t) = p j ( x, y, t) e cos j + c () where z is the coordinate along the longitudinal direction, (x,y) define the plane transverse to the resonator axis, and p j is the complex pressure amplitudes with frequency ω j, and k j is the wave number of a cavity mode. In the limit of sufficiently small losses (highly reflecting mirrors) one can assume slowly varying envelopes of the mode amplitudes, and consider an effective loss parameter for each mode. Then the pressure amplitude evolution for each mode is given by the equations 7 p = γ t p = γ t ( + i ) c p iβp + ia p + pin, L * ( + i ) p iβp p + ia p. together with their complex-conjugates. In Eqs. (), cavity, i ( i i ) i () γ i are the decay rates of each mode in the c = ω ω / γ is the detuning between the frequency of the field ω j and the closest frequency of the cavity c ω j, normalized to the decay rate. The diffraction parameter is defined as ai = c i / ωi. Finally, the nonlinear coupling coefficient is β = εω / 4ρ c, being ε the nonlinearity parameter. Note that the transverse laplacian operator, accounting for diffraction, plays no role in the D case describing plane waves propagating along the axis. However, other more complex distributions can be supported in general, as shown in Ref. 7. Two stationary homogeneous solutions are sustained by Eqs. (): the trivial (subharmonicoff), where cp P = and Lγ P =, (3) + and the nontrivial solution, where the subharmonic field is switched on, given by γ = + β / cβp P, (4) β L P and = γ γ ( + ) ± γ γ ( + ) existing above a given threshold pump value, P th, obtained by equating the second of Eqs.(4) to zero. At this pump value the injected energy overcomes the losses, and as a consequence the trivial solution Eq. (3) becomes unstable and bifurcates into the nontrivial one. The character of the bifurcation depends on the detuning values: it is supercritical (monostable) for <, and subcritical (hysteretic), for >. Both situations, illustrated in Fig. (left), can be accomplished in a real system since the resonator eigenmodes are not equally spaced when the walls have a finite thickness 5. It has been also shown in Ref. 7 that (only) in the former case, the stationary solution Eq. (4) can loose the stability and enter into a dynamical regime. At a given (second) threshold value, a Hopf bifurcation can develope, resulting in a low-frequency auto-modulation of the intracavity fields. At even higher inputs, a period-doubling route to chaos is predicted. An spectrogram with a typical complex temporal scenario for increasing pump is shown in Fig. (right). 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID

3 FIGURE. Left: Bifurcation diagram of trivial solution, showing mono- and bistable operation of the subharmonic field. Continuous (dashed) lines represent stable (unstable) branches. (b) Spectrogram of the subharmonic field, showing the cascade on instabilities. The dense regions corresponds to chaotic domains. When the Fresnel number of the resonator F = R / λl (determining its aspect-ratio) is large, then diffraction terms can play an important role, as demonstrated in Ref. 7. The threshold for subharmonic generation is lower for plane waves propagating at some angle with respect to the axis. A nonlinear competition results in a pattern selection, which is periodic along the transverse section of the resonator. An example of numerical simulation of Eqs. () showing this proccess is given in Fig.. After a transient state (a and b) a final stationary transverse stripped pattern (c) results. FIGURE. Numerical simulation of Eqs. (), showing the transverse subharmonic distribution at three instants of time, for detuning values =.6, =. In the next section we describe the experiment setup where the results of the previous section are being investigated. C. EXPERIMENT In order to check the validity of the above theoretical results, we have designed an experimental setup, consisting in an acoustic resonator, formed by two plane parallel and highly reflecting walls with radius.5 cm and variable separation. One of the walls is an ultrasonic transducer resonant at MHz acting as a source (pump). A signal generator and an amplifier (ENI4L) drives the transducer. The resonator is mounted inside a metacrilate tank filled with water, previously degassed to avoid cavitation effects. A broadband needle hydrophone measures the pressure inside the cavity, and a further processing of the signal allows to obtain its spectral content. 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID 3

4 The subharmonic generation has been observed both in the degenerate ( ω = ω = ω / ) and nondegenerate ( ω = ω + ω ) cases. Increasing the transducer input and measuring the peak amplitude of fundamental and subharmonic modes the bifurcation diagram can be obtained for a set of parameters. Some examples of experimentally obtained bifurcation diagrams are shown in Fig. 3. In some cases there is agreement, especially notable close to the threshold, with the above theory (e.g. the pump value saturates beyond the parametric instability). There the conditions where closer to the theoretical premises (left picture). In other cases the behaviour of the fundamental was different. We note that in these cases, an extended theory including the effect of the second harmonic (to be presented elsewhere) results in agreement with the experimental observations (right picture). Mode amplitude (mv) Driving amplitude (V) FIGURE 3. Bifurcation diagrams obtained under different detuning values. (a) nondegenerate case (only the highest subharmonic is plotted) (b) degenerate case. We also explored the existence of the dynamical regimes predicted by the theory. The Hopf (automodulation) bifurcation described in the previous section corresponds, in a spectral representation, to a pair of low frequencies placed symmetrically with respect to the unstable subharmonic mode. In Fig. 4 we show the single, isolated subharmonic field, at f=.8 MHz, at a given driving voltage (a), and the sideband frequencies emerging after a slight increase of the driving amplitude (b). Such frequencies appear with a finite amplitude, suggesting that the Hopf bifurcation has a subcritical character. (a) (b) Amplitude (mv)..5 Amplitude (mv) frequency (MHz) FIGURE 4. Spectrum of the resonator, in the neigbourhood of the subharmonic, at a voltage input of.8 V (a) and. V (b) The observation of spatial structures different than plane waves (transverse patterns) require a decrease in the Fresnel number obtained, e.g. by reducing the resonator length. The field distribution can be measured by scanning the transverse plane with the hydrophone (then we have quantitative information) or by means of optical techniques, which allow to visualize the density variations inside the whole resonator volume. We have implemented in the previous 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID frequency (MHz)

5 setup a Schlieren system, and investigated the changes in the field distribution as the parameters are varied. In this way we can get images of the complete pressure distribution inside the resonator. We were able to observe qualitative changes in the transverse profile, as Fig. 5 show. In all the pictures the input pressure was kept constant, and the frequency was varied. In Fig. 5(a) the resonator was below the subharmonic threshold, and a plane wave propagating along the axis is observed (the squared shadow of the hydrophone is shown at the top). Figs. 5(b) and (c) show the formation of structure as the frequency of the source (and correspondingly the detuning) was slightly changed. In these cases, the hydrophone marked the appearance of subharmonics. We expect to find some correlations of these results with the theory described above. Work in this direction is in progress. (a) (b) (c) FIGURE 5. Schlieren images of the resonator at a fixed input and increasing source frequency (a) Plane stationary waves below parametric threshold. (b) and (c) Spatial patterns above threshold. C. CONCLUSIONS The dynamical behaviour of parametric sound generation in resonators has been investigated, both theoretically and experimentally. Experiments show that the spectrum near threshold is complex (many subharmonic pairs are present) but at higher inpus only one or few of the subharmonics (often the degenerate one-half) survives. In this case a model in terms of coupling of few modes is justified. But at such high inputs the effect of the higher harmonics becomes important, and one needs to include at least the coupling with the second harmonic (4ω) and the sum frequency modes. The inclussion of these modes in the theory results in a better agreement with the experiments in certain cases. Different dynamical regimes and structure formation have been observed in the experiment. The comparison of these results with the prediction of the theory is in progress. Acknowledgments This work was supported by the project FIS5-793-C3- of the MEC of the Spanish Government. References. Korpel A. and Adler R., "Parametric phenomena observed on ultrasonic waves in water", Appl. Phys. Lett. 7, 6-8 (965).. Adler L. and Breazeale M.A., "Generation of fractional harmonics in a resonant untrasonic wave system ", J. Acoust. Soc. Am. 48, (97). 3. Ostrovsky L.A. and Soustova I.A., "Theory of parametric sound generators", Sov. Phys. Acoust., (976). 4. Zarembo L.K. and Serdoboloskaya O.Y., "Parametric amplification and generation of sound waves",sov. Phys. Acoust., (975). 5. Yen N., Experimental investigation of subharmonic generation in an acoustic interferometer J. Acoust. Soc. Am. 57, (975). 6. M.C. Cross and P.C. Hohenberg, "Pattern formation outside of equilibrium", Rev. Mod. Phys. 65, 85- (993). 7. Pérez-Arjona I. and Sánchez-Morcillo V.J., Pattern formation in parametric sound generation, Phys. Rev. E 76, 66 (5). 8. Sánchez-Morcillo V.J., Espinosa V., Redondo J and Alba J., Nonlinear dynamics and chaos in parametric sound generation, Acta Acustica united with Acustica 9, -6 (6). 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA7MADRID 5