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# Crossing of acoustic envelope solitary waves in homogeneous elastic rods: Experiments

Michel de Billy

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Michel de Billy. Crossing of acoustic envelope solitary waves in homogeneous elastic rods: Experiments. Ultrasonics, 2016, 72, pp.42-46. 10.1016/j.ultras.2016.07.009. hal-01459720

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# Crossing of acoustic envelope solitary waves in homogeneous elastic rods: Experiments

M. de Billy

Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert - Site de St Cyr, 2 place de la Gare de Ceinture, 78210 St Cyr l'Ecole, France

In this letter we focused on the problem in which two shear envelope solitary waves traveling in 1-D homogeneous system (elastic PMMA rod of constant cross-section) and incoming in opposite directions, meet one another and head-on collide. The properties of incident acoustic waves after collision are investigated. Two cases are experimentally examined: incoming waves with close amplitudes and quite large difference in magnitude. No phase shift and no variation of polarization are experimentally detected after the collision.

## 1. Introduction

Since many years the research on solitary waves was one of the most field developed in various area such as nonlinear dynamics, optics, biophysics and atomic physics. The property commonly admitted for presenting this wave is the following: it is a wave spatially localized within a finite region and which arises from a balance between nonlinearity and dispersive effects. This wave propagates with a permanent form and a constant velocity. Two types of waves were mainly analyzed as solution of the Korteweg-de Vries (KdV) or the nonlinear Schrödinger (NLS) equations. The NLS equation gives rise to envelope solitary waves exhibiting both a carrier and a modulation [1–9]. Many theoretical works and few experimental studies have been devoted to the collision of these types of non-linear waves. Such head-on collisions were studied in water [10–17], in nonlinear dusty plasma [18–22], in rods [23–27] and in superimposed elastic thin films [28]. The objective of this letter is to experimentally investigate the collision of two transversally polarized envelope solitary waves propagating in opposite directions in polymethylmetacrylate (PMMA) rods and compare the experimental observations with those available for other laboratory conditions.

## 2. Experiments and results

The experimental set-up (sketched in Fig. 1) is fundamentally the same as the one described in Ref. [8]. It is well known that

the generation of solitary waves is based on the balance between the non-linearity and dispersion. In the present laboratory work, the rod is ended with small glass bead of 8 mm in diameter and the whole sample (rod + beads) is clamped with two shear contact transmitters driven by the same acoustic impulse and submitted to a static compression. At the contact points between the spherical beads and the front face of the emitters a structural nonlinearity is caused and is described by the nonlinear Hertz' law. So it is natural to consider (and it is experimentally verified) that the transmitted waves are affected by the non-linearity. In addition the propagation in bars of small diameter (with comparison to the wavelength) is dispersive. As the displacement of these wave trains is at the surface, it is possible, as they propagate, to record it with a third shear transmitter (receiver) in contact with the surface of the bar and acting over a linear distance equal to the diameter of the transmitter. The spatial localization of this PZT transducer may vary along the rod and its direction of polarization has to be maintained parallel to the common direction of the polarization of the two emitters. This simple experimental set-up was used to investigate the colliding problem of two envelope solitary waves propagating in the same bar and traveling in opposite directions (bidirectional collision). The waveguides used for experiments are cylindrical PMMA bars whose length (l) is less than 66 cm. The diameter of the rods (d) is 10 mm. The choice of glass beads is justified by the fact that with this material short acoustic signals are transmitted.

In Fig. 2 are reproduced experimental data of temporal wave signal profiles, as a function of the spatial location (d) of the receiver. The distances (d) are measured by the middle of the rod taken as the origin and the vertical axis describes the displacement of the

<footnote>E-mail address: mdebilly@univ-paris-diderot.fr</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic diagram of the experimental set-up.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Time evolution of the recorded signal as a function of the position (d) of the receiver; case of two incident acoustic signals of close amplitude. d varies from -25 cm to 25 cm by step of 5 cm from the top to the bottom of the figure. The values of the length and diameter of the PMMA rod are respectively 65.3 cm and 10 mm.</figcaption></figure>

surface of the rod in arbitrary unit. Systematic records on "off-center" positions have been registered from $d = -25$ cm (top of the figure) to 25 cm (bottom of the figure) by step of 5 cm. The length value of the PMMA bar is 64.5 cm.

The two groups of wave trains with close amplitudes income from opposite directions (see top of Fig. 2) and interact at the head-on collision point ($d = 0$) where a single group of wave trains

is formed during the collision. Then the process of reconstruction of the original wave trains is observed: two groups of wave trains appear and, as the distance d increases, move away (see bottom of Fig. 2).

What happens when one of the train waves has a lower amplitude than the other? Experimental records, reproduced in Fig. 3, illustrate this case. They are obtained with the sample above described; on the surface and from the right end of the bar glue was continuously put over 6 cm in length. One clearly verified that the incident acoustic wave train of low amplitude (coming from the right) crosses the other wave train (with higher amplitude and coming from the left) and propagates towards the left direction with low magnitude after collision.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Time evolution of the recorded signal as a function of the position (d) of the receiver; case of two incident acoustic signals of different amplitude. d varies from -20 cm to 20 cm by step of 5 cm from the top to the bottom of the figure. The values of the length and diameter of the PMMA rod are respectively 63.8 cm and 10 mm.</figcaption></figure>
### 3. Discussion

Before making comments on the results above presented we have to ask an important question: how the nonlinearity can be relevant for collisions of the wave trains in the rod when they have moved from the beads into the rods?

To answer this question we suggest to experimentally observe the joint amplitude of the interacting wave trains whose maxima coincide in time domain (see Fig. 4). The two first plots represent the signals detected by the receiver located at the middle of the bar one of the two emitter being off. The receiver is very slightly moved (one or two mm) to avoid phase shift between the two extremum of the two signals. Then as the experimental conditions (length of propagation, attenuation, dispersion, frequency response) are the same, the shapes are very similar and the measured amplitude are $5.7 \pm 0.1$ and $6.0 \pm 0.1$ mV respectively. The sum of these magnitudes may be compared to the amplitude of the third signal ($9.8 \pm 0.1$ mV) observed when the two emitters are on. We notice that this amplitude is smaller than the linear sum of the two incoming signals. This is in contradiction with what would happen if the two waves overlapped linearly and *the principle of linear superposition is not fulfilled* which indicates that the head-on collisions of the pulse trains are affected by nonlinearity. However we have to underline again that we have to be careful because effects as dispersion, attenuation and limitation in accuracy measurement may perhaps contribute to a phase shift. This is the reason why we have chosen the position of the receiver for which the maximum of the two signals are collapsing in the time domain.

Another argument for justifying the fact that nonlinearity exists in the collision of the acoustic wave trains may be related to the frequency analysis of the detected signal. In the top of Fig. 5 is represented the wave train resulting from the interaction of two colliding incident waves propagating in a PMMA rod of 65.3 cm in length. The associated frequency power is reported in the bottom of Fig. 5. This spectrum clearly points out peaks at 30 and

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Wave trains detected by the receiver: top: the emitter at the left end of the rod is out; middle: the emitter at the right end of the rod is out; bottom: the both emitters are working.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Frequency analysis: case of a PMMA rod (10 mm in diameter and 65.3 cm in length). The top of the figure represents the wave train in the middle of the rod; the bottom shows the associated frequency power spectrum. The peaks are observed for values of the frequency: 14, 30, 44 and 58 kHz.</figcaption></figure>

58 kHz. A more careful analysis allows us to identify maximum around 14 and 44 kHz. An analysis of this series of values confirm the existence of two fundamental modes generated by the glass beads ($f_1 = 30$ and $f_2 = 44$ kHz) acting as a low frequency filter. In the presence of non linearity, the power spectra should include combinations of the fundamental frequencies $f_1$ and $f_2$ (frequencies transmitted by the beads) such as $f_2 \pm f_1$, and the higher harmonics $2f_1, 2f_2, ...$ [29]. Then the first value (14 kHz) may be identified as the combination $f_2 - f_1$ and the last one (58 kHz) may represent the second harmonic of the first fundamental frequency $2f_1$. Usually the frequency combinations $2f_2$ and $f_1 + f_2$ are not observed in the spectra. This missing is probably due to the low response of the transmitter in the frequency region which corresponds to the values higher than 80 kHz. We also experimentally noticed that the shape of the power spectrum slightly depends on the position of the receiver on the rod. In our opinion, these frequency responses clearly demonstrate that non linearity plays a role in the collision process even for large propagating distances.

From these important considerations we may conclude that the collisions of the incident wave trains support non linearity. In fact once the non linear waves are formed they propagate through the rod whose spatially small structure yields dispersion and change the non linear waves to envelope solitons. Then as the shape of the time signals exhibit both a carrier (high frequency) and a modulation (low frequency) it should be consistent to claim that the experimentally transmitted short pulse trains may be identified as envelope solitary waves (ESW) or NL solitons.

From an analysis of the experimental records we notice that after collision the pulse shape survives (cf. Figs. 2 and 3). Also we verified that the velocity of propagation does not depend on the pulse height [30] and is constant. We noticed that no post collision phase shift is observed (or too small to measure) as it is observed in the plots of Fig. 6 which represent the variations of the time distance of the echoes as a function of the distance d ($l = 64.5$ cm). Mention that for compressional solutions of the NLS equation (envelope solitary waves), no phase shift is theoretically observed in plasma dusty [22]. Besides it was noted a variation of the amplitude of the signals as the distance d varies; this is consistent with the presence of dissipative process particularly sensitive in PMMA material [31,32]. We also focused on the polarization of the solitary waves and it was experimentally verified that the direction of polarization of the incident waves is not affected by the collision.

Another question may be formulated: how the contact non linearity at the beads can affect pulse collisions in the center of the
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Arrival time of the two colliding envelope solitary waves as a function of the displacement (d).</figcaption></figure>

rod? In spite of the large amount of theoretical works on the subject of understanding the dynamics of several systems, the problem of dynamics of the alignment of beads and rod was not studied yet in our knowledge and looks to be difficult for theoretically modeling: the specimen is neither a pure linear arrangement of identical spheres nor a rod alone. The space between the transmitter and the rod can be described by three contacts: plane/bead (of 8 mm in dia.), bead/tube (of 5 mm in height) and tube/rod. Then the dynamic of the generation and propagation of the physical non linearity due to Hertz contact (first contact) should be considered with care.

At an intuitive level the dynamics of the specimen could be solved by considering a single bead and then – after move – by modeling the rod (and the tube) as a chain of beads immersed in a medium whose the physical properties are the same that in the rod. In other words and intuitively the dynamics of the small tube and the rod could be solved by modeling the both as a chain of beads whose the diameters of the beads and rod should be equal. The first Hertzian contact (transmitter/bead) behaves as a nonlinear system initially pre-compressed to insure the propagation of the short signal which is transmitted trough the tube and the rod under non-linear surface acoustic wave. In the present experimental work, short non linear pulses suffer dispersion due to the geometry of the rod. When non-linearity is balanced by dispersion these waves become envelope soliton and the dynamics of the whole problem can be governed by a usual nonlinear Schrödinger equation at the surface of the rod. This situation was solved for longitudinal excitation in a rod. In the present theoretical model it should be necessary to consider a shear excitation. The difference of the material between the beads and the rod could be neglected in a first approximation.

In the laboratory work above described (head-on collision) two short nonlinear pulses propagate in the rod in opposite directions. Then we are in presence of two almost monochromatic envelope solitary waves and the whole problem should be modalize as the sum of the two analytic equations of the two opposite solitary waves, solutions of the two NLS equations, propagating with opposite velocities and overlapping at the middle of the rod.

## 4. Conclusion

In the present letter we considered an experimental device devoted to the analysis of the head-on collision of two incoming transversally polarized wave trains simultaneously excited and propagating in opposite directions in PMMA rods. We have verified that during collision envelope solitary waves exist. Our study clearly shows that the two incident envelope solitary waves

emerge after collision, maintain their shape, propagate at a constant velocity and does not modify their direction of polarization. We have noticed that the principle of linear superposition is not fulfilled and that frequency combinations of two initial frequencies are observed as predicted in the case of non linear interaction. It was also confirmed that no phase shift is observed after collision which a characteristic of the solution of the NLS equation. We suggest, in a very first approximation, to solve the dynamics problem by considering the whole specimen as a chain of beads and by modelling the collision as the sum of the two non linear dispersive solitary waves each of them being described by a usual NLSE propagating with opposite direction.

## Acknowledgments

The author thanks the reviewers for their help and the many suggestions and comments for improving the manuscript.

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