- Email: [email protected]
Frequency-dependent, near-pole behavior of acoustic surface waves on a solid sphere
Accepted Manuscript Title: Frequency-Dependent, Near-Pole Behavior of Acoustic SurfaceWaves on a Solid Sphere Author: Bart Van Damme Alessandro Spadoni PII: DOI: Reference:
S0093-6413(14)00064-0 http://dx.doi.org/doi:10.1016/j.mechrescom.2014.05.002 MRC 2865
To appear in: Received date: Revised date: Accepted date:
10-12-2013 11-4-2014 9-5-2014
Please cite this article as: Bart Van Damme, Alessandro Spadoni, Frequency-Dependent, Near-Pole Behavior of Acoustic SurfaceWaves on a Solid Sphere, Mechanics Research Communications (2014), http://dx.doi.org/10.1016/j.mechrescom.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
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Spherical, surface acoustic waves; Antipode phase shift and interference relation; Laser vibrometry; Dispersion relations; Wave focusing.
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unmarked revised manuscript
Frequency-Dependent, Near-Pole Behavior of Acoustic Surface Waves on a Solid Sphere
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a Institute of Mechanical Engineering ´ Ecole Polytechnique F´ ed´ erale de Lausanne CH-1015, Lausanne, Switzerland
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Bart Van Damme1 , Alessandro Spadoni1,∗
Abstract
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A laser vibrometer is used to measure waves generated by a piezoelectric transducer in a solid steel sphere. The spatial and temporal resolution make it possible to verify the theoretical dispersion relation of spherical surface waves. In particular, the area at the source antipode is investigated. The focusing of surface waves results in high-amplitude surface displacements in this area, which is therefore interesting in terms of seismic research and the understanding of the dynamic behavior of granular media. The influence of frequency on the diameter of the focal area is measured using windowed sine bursts as a source. As an example of the complex wave behavior near the poles, the interference of converging and diverging surface waves is demonstrated.
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Keywords: Surface acoustic waves, wave focusing, spheres, dispersion
1. Introduction
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The behavior of elastic waves in spherical bodies gained interest for the study of seismic phenomena [1, 2, 3]. More recently, waves in spherical bodies are being investigated for their contribution to the dynamics of granular solids [4, 5, 6], and for applications such as acoustophoresis [7], gas sensors [8, 9], and optomechanics [10, 11]. Spherical waves on the surface of a sphere in particular can be used for non-destructive testing of bearing balls [12], the development of chemical sensors [8, 9] and optoacoustic-coupling devices capable of generating RF waves [11]. Mechanical wave propagation in granular media is also affected by surface waves; their spectral signature has been detected in the response to high-frequency excitation [4, 5] and it has also been linked to the low-frequency response of a 1D chain of spheres excited by an impact [6]. All these applications are concerned with the characteristics of surface waves, but do not emphasize the convergence of surface waves over spheroidal bodies into narrow, high-amplitude waveforms. This is ∗ Corresponding
author Email address: [email protected] (Alessandro Spadoni) Preprint submitted to Elsevier
relevant for example for nano-printing technology, where ink nano-droplets are pinched off from larger droplets [13]. The theoretical dynamic response of a point source on the surface of an elastic sphere has been derived in [14]. This solution however has a singularity at the source antipode [15]. Among the three predicted wave types generated by a force exerted on the surface moreover, surface or Rayleigh waves are of particular interest since their amplitude is significantly higher than the amplitude of bulk waves (longitudinal and shear) [16]. This is in line with the theoretical energy partition among wave modes in a semi-infinite medium excited by a harmonic point-load, as described in [17], where they found that 67% of the wave energy created by a sinusoidal force on the surface is carried by surface waves. These waves are useful to estimate graded material properties [18] and have some remarkable properties, studied both theoretically and experimentally. A first well-studied feature of spherical surface waves is their dispersive behavior due to the curvature of the surface [1, 19]. This has been experimentally validated in [20]. Second, surface waves favour certain frequencies due to the cyclic symmetry of the wave guide and stress April 11, 2014
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The setup of the experiments consists of a 100 mm-diameter stainless steel sphere (AISI 316), resting on a rotating table with a 0.005◦ rotation angle precision (Fig. 1). The turntable contact zone is much smaller than any wavelength of surface waves and it also coincides with the location with lowest amplitude of surface waves. Piezo actuators are used as an acoustic source. For the low frequency ultrasound range (50-150 kHz), a small piezostack is used (Piezomechanik 2 × 3 × 5 mm3 ). Higher frequencies (up to 350 kHz) can be achieved by a piezoelectric transducer accommodated in a steel casing with a 6 mm diameter (DAKEL midi). The transducers are driven by a Tektronix AFG 3022C arbitrary waveform generator. The source signals are short sinusoidal bursts (1-3 periods, depending on the frequency), modulated by a hamming window to limit the frequency content. The generated waves are measured by a Polytec OFV534 laser vibrometer and recorded with a Tektronix DPO 3012 digital oscilloscope. The source and the laser are aligned so that a great circle through the source and its opposite pole can be scanned by rotating the sphere. The advantage of this arrangement over previously described experiments is the absence of additional mass altering boundary conditions at the antipode of the source. No modifications of the surface, like a thin film [9, 20] or a measuring transducer [24] are required. Moreover, the investigated surface needs no special treatment. However, multiple surface-wave roundtrips are disturbed by piezoelectric transducers used for excitation. These may induce mode conversion at the excitation location, reducing amplitude, but still allow an analysis of the surface-wave frequency content.
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boundary-conditions which lead to a discrete spectrum. [21, 10, 19, 22]. Finally, surface waves travelling across the source antipode undergo a π/2 phase shift [23], and their experiments show a signal that is much more complex in the poles than on any other point of the surface. Most of the experimental work carried out so far has focused on determining the wave (phase or group) velocity and the amplitude variation away from poles, usually by using a short excitation source generated by a pulsed laser [15, 20, 24]. This approach however is well suited for high-frequency spectra given the pulse duration is in the ns range, while the aim of our investigations is the behavior of surface waves in the kHz range. In previous studies furthermore, the area near the source antipode has mostly been ignored because of the complexity of the local wavefield. Seismological studies have shown however that a small area around the antipode provides significant information about the various wave types and reflections in a sphere due to geometric focusing [2, 25]. This paper presents an experimental technique to investigate all of the aforementioned properties of spherical surface waves. The proposed experiments provide flexibility regarding amplitude and frequency of the wave source and both spatial and temporal-measurement accuracy. The use of windowed, sinusoidal bursts allows for the investigation of frequency dependence of these phenomena. The measured dispersion, frequency content and frequency-dependent focusing are compared to theoretical models predicting the dynamic response for the precise excitation signals we used. The focusing of surface waves at the antipode is of particular interest to investigate the complex wave behavior in that area.
3. Results and Discussion A typical response signal resulting from a short sinusoidal burst is shown in Fig. 2 at the antipode (a) and away from the antipode (b). The highamplitude pulses are surface waves arriving with regular intervals defined by their wave speed and the circumference of the sphere. For a 100 mmdiameter stainless-steel sphere and a pulse with central frequency 206 kHz, this interval is measured to be 107 µs. The measured surface waves can be visualized as a ring with the pole-to-pole line as an axis, with a radius and a particle amplitude that vary as the wave travels over the sphere. Due to conservation of energy, the amplitude is lowest at
2. Material and methods
Transducer
Laser vibrometer
Turntable Figure 1: Setup of the experiments. A sphere is positioned on a precision turntable to allow the laser vibrometer to scan a line along a great circle centered around the rotation axis.
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Phase6velocity6Ekm/sx
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Figure 2: Time histories of signals measured in the antipode of an acoustic source generating a short sinusoidal burst (a) and at 25◦ away from the antipode (b). The source signal is a 3-period 206 kHz hanning-windowed sinusoidal burst. The periodic arrivals of Rayleigh wave pulses are clearly visible.
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Time6Emsx
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Theory Experiment Rayleigh6wave6speed
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the equator and highest at the poles, as the ring focuses theoretically in a single point [19]. Although viscous damping and dispersion of the pulses make the amplitude decay, up to 50 roundtrips can be measured. The amplitude is much lower for signals away from the antipode, as is shown in Fig. 2 (b). At 25◦ away from the pole, the measured velocity is almost reduced by a factor 10. Each high amplitude peak now consists of two minor peaks. The first is the signal still converging towards the pole, whereas the second already traveled over the pole and is diverging [20]. The authors first verified the dispersion relation and the surface wave frequencies of the considered sphere by measuring the time history of surface signals along a line with a 0.04◦ step over a total angle of 120◦ . The sampled time history of the surface velocity at each location is assembled into a matrix s (xk , tl ), where xk in the location of the kth station and tl is the lth time step. A two-dimensional, discrete Fourier transform Mr,m =
0.1
0
(c) 3.6 Group6velocity6Ekm/sx
0
300
T206
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0.01
200 Frequency6EkHzx T50
S6 Eω,t)
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(b) Velocity (m/s)
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(a)
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(a)
0
100
200 Frequency6EkHzx
300
Figure 3: Measured phase velocity dispersion (a) and group velocity dispersion (c) of surface acoustic waves on the surface of a 100 mm-diameter stainless steel sphere. The white (blue) line shows the theoretical values. The green dotted line is the theoretical Rayleigh wave speed on a plane surface. Two measurements have been overlapped to cover a larger frequency range, hence the difference in color in part (a). The convolution signal used to determine the group velocity is shown for 50 kHz and 206 kHz (b).
Eq. (1) is implemented numerically in Matlab via the function fft2. The magnitude of complex coefficients Mr,m provides the amplitude for each wave number-frequency combination of the measured signals. The highest amplitude from these data is used to evaluate the phase-velocity (cph = f λ) dispersion relation which is shown in Fig. 3(a). The experiment was carried out twice, with excitations at 60 kHz and 200-kHz-central frequency in order to span a wide frequency range. The agreement between the theoretical prediction [1] (white line) and the measured wave speeds is remarkable, especially for the higher frequencies. Discrete frequencies with high energies are well-guided, Rayleighwave frequencies as predicted from theory [1]. The corresponding wavelengths are determined by symmetry of the waveguide leading to discreteness. An
s(xk , tl )e−i2πk/K/λm e−i2πfr l/L ,
k=0 l=0
(1) where i = −1, r, m are integers fk and λm are the frequency and wavelength respectively; K = 3000 and L = 16384. The bandwidth of signals obtained with the laser vibrometer extends to 3 MHz. √
3
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focal area is clearly visible, and the opening angle of the focus diminishes with frequency. For the 61 kHz pulse, the full focus angle at half height of the focus is approximately 20◦ , whereas it is only 7◦ for the excitation at 206 kHz. The time of arrival of the maximum amplitude is comparable for both frequencies at approximately 50-60 µs, half of the time it takes a surface wave to travel around the entire circumference. The low-frequency wave arrives slightly earlier due to dispersion. The focusing effect is particularly visible in the 206 kHz plot Fig.4(c): low-amplitude waves converge from both sides of the antipode and diverge afterwards, resulting in a X-shaped pattern. Before and after the high-amplitude focus in time, a low-amplitude ripple is visible consisting of the various bulk waves. Just like surface waves, bulk waves reach their highest amplitudes at the poles due to reflections. An interesting observation is the presence of spatial side lobes in the focus area for both frequencies. These can be explained by interference of surface waves. Since surface waves consist of a train of several periods of a sine, the early part of the signal is already diverging when the last part of the signal is still traveling towards the antipode. The diverging and converging parts of the waves interfere, thereby resulting in a typical interference pattern with clear amplitude minima. These minima occur when the converging and diverging waves are out of phase, i.e. when the difference in the traveled path is a multiple of the half wavelength. A similar phenomenon occurs when waves in a medium are scattered by a rigid spherical obstruction [26, 27]. Although the interference of different wave types is of large interest for seismic research [2, 28], the interactions of waves of the same type have not been studied extensively. The interference behavior can be modeled by geometric considerations. The converging and diverging parts of the wave encounter destructive interference if their travelled path differs by a multiple of half the wavelength. At an angle θ away from the source antipode, the converging wave has travelled over a distance
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energy source and an energy sink at antipodes are employed to find physical solutions of the equation of motion [19]. Solutions to the characteristic equation resulting from symmetry conditions indicate the lowest possible frequency that can travel over a sphere: the cut-off frequency in the case of a 10 cm steel sphere is 27 kHz. The wave speeds evaluated from experiments are not very accurate for low frequencies. Since the maximum distance over which the time signals can be measured is limited to half a circumference of the sphere, the wave number cannot be determined accurately with the method of Eq. (1). To resolve this experimental challenge, a different approach was used to calculate the group velocity of each frequency component. In a first step, the measured time signal s(t) is convolved with a windowed sine burst fω (t) with the frequency of interest ω: Z S(ω, t) = fω (τ − t)s(τ )dτ . (2)
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The number of periods in fω (t) is taken equal to the number of periods in the source signal. This results in a signal S(ω, t) still showing the repeated arrival of the pulses at the location of the measurement, but the convolution acts as a narrow-band filter (Fig. 3(b)). In a second step, the time lag Tω between the maximum of two consecutive pulses is used to calculate the group velocity of the pulse containing the selected frequency only, since the traveled distance is known to be one circumference. A comparison of theoretical and experimental group velocities is shown in Fig. 3(c). Errorbars are determined by the standard deviation on the mean time interval between two maxima of the signal. The authors’ second point of interest is the dynamic behavior of the antipode. In order to visualize the influence of the frequency on the wavefocusing area in the antipode, a line scan was performed measuring the normal surface velocity over a 60◦ angle centered around the pole. Surface waves around the antipode can be easily distinguished. Due to multiple reflections, bulk waves, which travel faster than surface waves, arrive at the antipode as a long train of low amplitude disturbances. Surface-wave pulses on the other hand, are guided and they reach the antipode almost undisturbed. Consequent arrivals of the latter can therefore be easily distinguished. The maximum velocity in each point is shown in Fig. 4(a,c) for a 61 kHz and a 206 kHz sinusoidal burst, before and after the first arrival of the surface wave. In both cases, the
d1 = R(π − θ),
(3)
and the wave that already passed the antipode has gone a distance R(π + θ). In [23], it is shown that a passage over the antipode additionally results in a phase shift of the wave, inducing a reduction of the travelled wave path by a quarter of the wavelength 4
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61 kHz
(b) interference model measurement
0.10 0.05
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interference model measurement
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Figure 4: Surface velocity around the source antipode, before and after the first surface wave arrival (a,c). Results for two frequencies reveal that the focusing area of higher frequencies is smaller than for low frequencies. Interference pattern near the antipode of a point source on a sphere for two frequencies (b,d). Measurements and analytic results are shown. The minima are reached at the angles predicted by geometric considerations, shown as red dashed lines.
the material, geometry and colatitudinal angle. To achieve reasonable fidelity of the wave response to a windowed sine burst, the first 50 modes are taken into account including their overtones up to 1 MHz. The calculations show that even a point source results in spatial side lobes around the source antipode, confirming that the size of the source does not affect the interference. By tuning the calculated displacements to the measurements, the stress applied by the piezo transducers is estimated to be approximately 2 kPa. The measured and calculated interference patterns are shown in Fig. 4(b,d).
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λ. The total travel distance of the diverging wave is thus d2 = R(π + θ) − λ/4. (4)
3λ 8R
(5)
pt
θmin =
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The difference of both wave paths |d1 − d2 | is half a wavelength for the first interference minimum, which results in an angle
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away from the source antipode. In the case of the 206 kHz wave, the first two minima are expected at 6.3◦ and 12.5◦ which agrees well with the experiment. In the case of the 61 kHz wave the minimum is expected at 20◦ . The measurements however show a smaller angle of 17◦ due to a very strong frequency component at 80-90 kHz as shown in Fig. 3, which agrees well with this first minimum. The diameter of the source transducer might introduce a wave-path difference as well, and so supplies an alternative explanation to the presence of the interference pattern. In order to check if the side lobes are indeed due to the interaction of the converging and diverging part of the wave pulse, an analytical calculation of the radial surface motion is carried out. The wave propagation in a sphere is a superposition of the spectral components of the source signal and of the possible vibration modes of the sphere under consideration [1, 16, 22].The amplitude of each spectral component depends on
4. Conclusions In this letter, a non-contact experimental technique for the investigation of surface-wave coalescence over a solid sphere is presented. Theoretically-predicted dispersion of surface waves and the characteristics of their focusing at the pole opposite to the excitation location are validated experimentally with a laser vibrometer. Measurements of bandwidth-limited signals indicate a significant frequency-dependence for the focusing of surface waves. A model to predict interference patterns based on geometric notions and the antipode phase-shift of surface waves on a sphere is proposed. The knowledge of surface waves in spheriodal bodies is important for the dynamics of granular solids, 5
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[18] S. Chirit¸a ˘, Thermoelastic surface waves on an exponentially graded half-space, Mechanics Research Communications 49 (0) (2013) 27 – 35. [19] I. A. Viktorov, Rayleigh and Lamb waves: physical theory and applications, Vol. 147, Plenum press New York, 1967. [20] P. Otsuka, O. Matsuda, M. Tomoda, O. Wright, Interferometric imaging of surface acoustic waves on a glass sphere, Journal of Applied Physics 108 (12) (2010) 123508–123508. [21] L. Rayleigh, Cxii. the problem of the whispering gallery, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 20 (120) (1910) 1001– 1004. [22] E. R. Lapwood, T. Usami, Free oscillations of the Earth, Cambridge University Press, Cambridge, 1981. [23] J. Brune, J. Nafe, L. Alsop, The polar phase shift of surface waves on a sphere, Bulletin of the Seismological Society of America 51 (2) (1961) 247–257. [24] Y. Tsukahara, N. Nakaso, H. Cho, K. Yamanaka, Observation of diffraction-free propagation of surface acoustic waves around a homogeneous isotropic solid sphere, Applied Physics Letters 77 (2000) 2926. [25] E. P. Chael, D. L. Anderson, Global q estimates from antipodal rayleigh waves, Journal of Geophysical Research: Solid Earth (1978–2012) 87 (B4) (1982) 2840– 2850. [26] L. Rayleigh, A. Lodge, On the acoustic shadow of a sphere. with an appendix, giving the values of legendre’s functions from p {0} to p {20} at intervals of 5 degrees, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203 (359-371) (1904) 87–110. [27] J. Foxwell, The diffraction of sound round a rigid sphere from a point source on the sphere, Journal of Sound and Vibration 11 (1) (1970) 103–113. [28] H. Kawase, The cause of the damage belt in kobe:the basin-edge effect, constructive interference of the direct s-wave with the basin-induced diffracted/rayleigh waves, Seismological Research Letters 67 (5) (1996) 25– 34.
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for applications such as acoustophoresis, gas sensors, optomechanics, and nano-printing. Understanding the convergence of surface waves into narrow, high amplitude waveforms sheds new light on energy transmission between contacting particles, but can also be used to enhance performance of devices exploiting surface waves.
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[1] Y. Sat¯ o, Basic study on the oscillation of a homogeneous elastic sphere, Central Meteorological Observatory of Japan, 1962. [2] J. Rial, V. Cormier, Seismic waves at the epicenter’s antipode, Journal of Geophysical Research: Solid Earth (1978–2012) 85 (B5) (1980) 2661–2668. [3] B. A. Bolt, J. Dorman, Phase and group velocities of rayleigh waves in a spherical, gravitating earth, Journal of Geophysical Research 66 (9) (1961) 2965–2981. [4] M. De Billy, Experimental study of sound propagation in a chain of spherical beads, The Journal of the Acoustical Society of America 108 (4) (2000) 1486–1495. [5] J. Anfosso, V. Gibiat, Elastic wave propagation in a three-dimensional periodic granular medium, Europhysics Letters 67 (3) (2004) 376. [6] B. Van Damme, A. Spadoni, Intra-particle dynamics and the impact response of granular chains, European Physics Letters (2013) 48003. [7] I. Leibacher, W. Dietze, P. Hahn, J. Wang, S. Schmitt, J. Dual, Acoustophoresis of hollow and core-shell particles in two-dimensional resonance modes, Microfluidics and Nanofluidics (2013) 1–12. [8] T. Nakamoto, K. Aoki, T. Ogi, S. Akao, N. Nakaso, Odor sensing system using ball saw devices, Sensors and Actuators B 130 (2008) 386–390. [9] K. Yamanaka, N. Nakaso, D. Sim, T. Fukiura, Principle and application of ball surface acoustic wave (saw) sensor, Acoustical science and technology 30 (1) (2009) 2–6. [10] J. Zehnpfennig, G. Bahl, M. Tomes, T. Carmon, Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres, Optics Express 19 (15) (2011) 14240–14248. [11] A. B. Matsko, S. A. A., V. S. Ilchenko, D. Seidel, L. Maleki, Surface acoustic wave frequency comb, in: Proceedings of SPIE, Vol. 8236, 2012, pp. 82361M–1. [12] K. Yamanaka, H. Cho, Y. Tsukahara, Precise velocity measurement of surface acoustic waves on a bearing ball, Applied Physics Letters 76 (19) (2000) 2797–2799. [13] P. Galliker, J. Schneider, H. Eghlidi, S. Kress, V. Sandoghdar, D. Poulikakos, Direct printing of nanostructures by electrostatic autofocussing of ink nanodroplets, Nature communications 3 (2012) 890. [14] H. Lamb, On the vibrations of an elastic sphere, Proceedings of the London Mathematical Society 1 (1) (1881) 189–212. [15] D. Clorennec, D. Royer, Investigation of surface acoustic wave propagation on a sphere using laser ultrasonics, Applied physics letters 85 (2004) 2435. [16] S. Ishikawa, H. Cho, Y. Tsukahara, N. Nakaso, K. Yamanaka, Analysis of spurious bulk waves in ball surface wave device, Ultrasonics 41 (1) (2003) 1–8. [17] G. Miller, H. Pursey, On the partition of energy between elastic waves in a semi-infinite solid, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 233 (1192) (1955) 55–69.
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S0093-6413(14)00064-0 http://dx.doi.org/doi:10.1016/j.mechrescom.2014.05.002 MRC 2865
To appear in: Received date: Revised date: Accepted date:
10-12-2013 11-4-2014 9-5-2014
Please cite this article as: Bart Van Damme, Alessandro Spadoni, Frequency-Dependent, Near-Pole Behavior of Acoustic SurfaceWaves on a Solid Sphere, Mechanics Research Communications (2014), http://dx.doi.org/10.1016/j.mechrescom.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
te
d
M
an
us
cr
ip t
Spherical, surface acoustic waves; Antipode phase shift and interference relation; Laser vibrometry; Dispersion relations; Wave focusing.
Ac ce p
Page 1 of 7
unmarked revised manuscript
Frequency-Dependent, Near-Pole Behavior of Acoustic Surface Waves on a Solid Sphere
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a Institute of Mechanical Engineering ´ Ecole Polytechnique F´ ed´ erale de Lausanne CH-1015, Lausanne, Switzerland
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Bart Van Damme1 , Alessandro Spadoni1,∗
Abstract
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A laser vibrometer is used to measure waves generated by a piezoelectric transducer in a solid steel sphere. The spatial and temporal resolution make it possible to verify the theoretical dispersion relation of spherical surface waves. In particular, the area at the source antipode is investigated. The focusing of surface waves results in high-amplitude surface displacements in this area, which is therefore interesting in terms of seismic research and the understanding of the dynamic behavior of granular media. The influence of frequency on the diameter of the focal area is measured using windowed sine bursts as a source. As an example of the complex wave behavior near the poles, the interference of converging and diverging surface waves is demonstrated.
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Keywords: Surface acoustic waves, wave focusing, spheres, dispersion
1. Introduction
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The behavior of elastic waves in spherical bodies gained interest for the study of seismic phenomena [1, 2, 3]. More recently, waves in spherical bodies are being investigated for their contribution to the dynamics of granular solids [4, 5, 6], and for applications such as acoustophoresis [7], gas sensors [8, 9], and optomechanics [10, 11]. Spherical waves on the surface of a sphere in particular can be used for non-destructive testing of bearing balls [12], the development of chemical sensors [8, 9] and optoacoustic-coupling devices capable of generating RF waves [11]. Mechanical wave propagation in granular media is also affected by surface waves; their spectral signature has been detected in the response to high-frequency excitation [4, 5] and it has also been linked to the low-frequency response of a 1D chain of spheres excited by an impact [6]. All these applications are concerned with the characteristics of surface waves, but do not emphasize the convergence of surface waves over spheroidal bodies into narrow, high-amplitude waveforms. This is ∗ Corresponding
author Email address: [email protected] (Alessandro Spadoni) Preprint submitted to Elsevier
relevant for example for nano-printing technology, where ink nano-droplets are pinched off from larger droplets [13]. The theoretical dynamic response of a point source on the surface of an elastic sphere has been derived in [14]. This solution however has a singularity at the source antipode [15]. Among the three predicted wave types generated by a force exerted on the surface moreover, surface or Rayleigh waves are of particular interest since their amplitude is significantly higher than the amplitude of bulk waves (longitudinal and shear) [16]. This is in line with the theoretical energy partition among wave modes in a semi-infinite medium excited by a harmonic point-load, as described in [17], where they found that 67% of the wave energy created by a sinusoidal force on the surface is carried by surface waves. These waves are useful to estimate graded material properties [18] and have some remarkable properties, studied both theoretically and experimentally. A first well-studied feature of spherical surface waves is their dispersive behavior due to the curvature of the surface [1, 19]. This has been experimentally validated in [20]. Second, surface waves favour certain frequencies due to the cyclic symmetry of the wave guide and stress April 11, 2014
Page 2 of 7
The setup of the experiments consists of a 100 mm-diameter stainless steel sphere (AISI 316), resting on a rotating table with a 0.005◦ rotation angle precision (Fig. 1). The turntable contact zone is much smaller than any wavelength of surface waves and it also coincides with the location with lowest amplitude of surface waves. Piezo actuators are used as an acoustic source. For the low frequency ultrasound range (50-150 kHz), a small piezostack is used (Piezomechanik 2 × 3 × 5 mm3 ). Higher frequencies (up to 350 kHz) can be achieved by a piezoelectric transducer accommodated in a steel casing with a 6 mm diameter (DAKEL midi). The transducers are driven by a Tektronix AFG 3022C arbitrary waveform generator. The source signals are short sinusoidal bursts (1-3 periods, depending on the frequency), modulated by a hamming window to limit the frequency content. The generated waves are measured by a Polytec OFV534 laser vibrometer and recorded with a Tektronix DPO 3012 digital oscilloscope. The source and the laser are aligned so that a great circle through the source and its opposite pole can be scanned by rotating the sphere. The advantage of this arrangement over previously described experiments is the absence of additional mass altering boundary conditions at the antipode of the source. No modifications of the surface, like a thin film [9, 20] or a measuring transducer [24] are required. Moreover, the investigated surface needs no special treatment. However, multiple surface-wave roundtrips are disturbed by piezoelectric transducers used for excitation. These may induce mode conversion at the excitation location, reducing amplitude, but still allow an analysis of the surface-wave frequency content.
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boundary-conditions which lead to a discrete spectrum. [21, 10, 19, 22]. Finally, surface waves travelling across the source antipode undergo a π/2 phase shift [23], and their experiments show a signal that is much more complex in the poles than on any other point of the surface. Most of the experimental work carried out so far has focused on determining the wave (phase or group) velocity and the amplitude variation away from poles, usually by using a short excitation source generated by a pulsed laser [15, 20, 24]. This approach however is well suited for high-frequency spectra given the pulse duration is in the ns range, while the aim of our investigations is the behavior of surface waves in the kHz range. In previous studies furthermore, the area near the source antipode has mostly been ignored because of the complexity of the local wavefield. Seismological studies have shown however that a small area around the antipode provides significant information about the various wave types and reflections in a sphere due to geometric focusing [2, 25]. This paper presents an experimental technique to investigate all of the aforementioned properties of spherical surface waves. The proposed experiments provide flexibility regarding amplitude and frequency of the wave source and both spatial and temporal-measurement accuracy. The use of windowed, sinusoidal bursts allows for the investigation of frequency dependence of these phenomena. The measured dispersion, frequency content and frequency-dependent focusing are compared to theoretical models predicting the dynamic response for the precise excitation signals we used. The focusing of surface waves at the antipode is of particular interest to investigate the complex wave behavior in that area.
3. Results and Discussion A typical response signal resulting from a short sinusoidal burst is shown in Fig. 2 at the antipode (a) and away from the antipode (b). The highamplitude pulses are surface waves arriving with regular intervals defined by their wave speed and the circumference of the sphere. For a 100 mmdiameter stainless-steel sphere and a pulse with central frequency 206 kHz, this interval is measured to be 107 µs. The measured surface waves can be visualized as a ring with the pole-to-pole line as an axis, with a radius and a particle amplitude that vary as the wave travels over the sphere. Due to conservation of energy, the amplitude is lowest at
2. Material and methods
Transducer
Laser vibrometer
Turntable Figure 1: Setup of the experiments. A sphere is positioned on a precision turntable to allow the laser vibrometer to scan a line along a great circle centered around the rotation axis.
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Phase6velocity6Ekm/sx
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Figure 2: Time histories of signals measured in the antipode of an acoustic source generating a short sinusoidal burst (a) and at 25◦ away from the antipode (b). The source signal is a 3-period 206 kHz hanning-windowed sinusoidal burst. The periodic arrivals of Rayleigh wave pulses are clearly visible.
ed
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Time6Emsx
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Theory Experiment Rayleigh6wave6speed
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the equator and highest at the poles, as the ring focuses theoretically in a single point [19]. Although viscous damping and dispersion of the pulses make the amplitude decay, up to 50 roundtrips can be measured. The amplitude is much lower for signals away from the antipode, as is shown in Fig. 2 (b). At 25◦ away from the pole, the measured velocity is almost reduced by a factor 10. Each high amplitude peak now consists of two minor peaks. The first is the signal still converging towards the pole, whereas the second already traveled over the pole and is diverging [20]. The authors first verified the dispersion relation and the surface wave frequencies of the considered sphere by measuring the time history of surface signals along a line with a 0.04◦ step over a total angle of 120◦ . The sampled time history of the surface velocity at each location is assembled into a matrix s (xk , tl ), where xk in the location of the kth station and tl is the lth time step. A two-dimensional, discrete Fourier transform Mr,m =
0.1
0
(c) 3.6 Group6velocity6Ekm/sx
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T206
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S6 Eω,t)
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(b) Velocity (m/s)
6
(a)
ip t
(a)
0
100
200 Frequency6EkHzx
300
Figure 3: Measured phase velocity dispersion (a) and group velocity dispersion (c) of surface acoustic waves on the surface of a 100 mm-diameter stainless steel sphere. The white (blue) line shows the theoretical values. The green dotted line is the theoretical Rayleigh wave speed on a plane surface. Two measurements have been overlapped to cover a larger frequency range, hence the difference in color in part (a). The convolution signal used to determine the group velocity is shown for 50 kHz and 206 kHz (b).
Eq. (1) is implemented numerically in Matlab via the function fft2. The magnitude of complex coefficients Mr,m provides the amplitude for each wave number-frequency combination of the measured signals. The highest amplitude from these data is used to evaluate the phase-velocity (cph = f λ) dispersion relation which is shown in Fig. 3(a). The experiment was carried out twice, with excitations at 60 kHz and 200-kHz-central frequency in order to span a wide frequency range. The agreement between the theoretical prediction [1] (white line) and the measured wave speeds is remarkable, especially for the higher frequencies. Discrete frequencies with high energies are well-guided, Rayleighwave frequencies as predicted from theory [1]. The corresponding wavelengths are determined by symmetry of the waveguide leading to discreteness. An
s(xk , tl )e−i2πk/K/λm e−i2πfr l/L ,
k=0 l=0
(1) where i = −1, r, m are integers fk and λm are the frequency and wavelength respectively; K = 3000 and L = 16384. The bandwidth of signals obtained with the laser vibrometer extends to 3 MHz. √
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focal area is clearly visible, and the opening angle of the focus diminishes with frequency. For the 61 kHz pulse, the full focus angle at half height of the focus is approximately 20◦ , whereas it is only 7◦ for the excitation at 206 kHz. The time of arrival of the maximum amplitude is comparable for both frequencies at approximately 50-60 µs, half of the time it takes a surface wave to travel around the entire circumference. The low-frequency wave arrives slightly earlier due to dispersion. The focusing effect is particularly visible in the 206 kHz plot Fig.4(c): low-amplitude waves converge from both sides of the antipode and diverge afterwards, resulting in a X-shaped pattern. Before and after the high-amplitude focus in time, a low-amplitude ripple is visible consisting of the various bulk waves. Just like surface waves, bulk waves reach their highest amplitudes at the poles due to reflections. An interesting observation is the presence of spatial side lobes in the focus area for both frequencies. These can be explained by interference of surface waves. Since surface waves consist of a train of several periods of a sine, the early part of the signal is already diverging when the last part of the signal is still traveling towards the antipode. The diverging and converging parts of the waves interfere, thereby resulting in a typical interference pattern with clear amplitude minima. These minima occur when the converging and diverging waves are out of phase, i.e. when the difference in the traveled path is a multiple of the half wavelength. A similar phenomenon occurs when waves in a medium are scattered by a rigid spherical obstruction [26, 27]. Although the interference of different wave types is of large interest for seismic research [2, 28], the interactions of waves of the same type have not been studied extensively. The interference behavior can be modeled by geometric considerations. The converging and diverging parts of the wave encounter destructive interference if their travelled path differs by a multiple of half the wavelength. At an angle θ away from the source antipode, the converging wave has travelled over a distance
an
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energy source and an energy sink at antipodes are employed to find physical solutions of the equation of motion [19]. Solutions to the characteristic equation resulting from symmetry conditions indicate the lowest possible frequency that can travel over a sphere: the cut-off frequency in the case of a 10 cm steel sphere is 27 kHz. The wave speeds evaluated from experiments are not very accurate for low frequencies. Since the maximum distance over which the time signals can be measured is limited to half a circumference of the sphere, the wave number cannot be determined accurately with the method of Eq. (1). To resolve this experimental challenge, a different approach was used to calculate the group velocity of each frequency component. In a first step, the measured time signal s(t) is convolved with a windowed sine burst fω (t) with the frequency of interest ω: Z S(ω, t) = fω (τ − t)s(τ )dτ . (2)
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The number of periods in fω (t) is taken equal to the number of periods in the source signal. This results in a signal S(ω, t) still showing the repeated arrival of the pulses at the location of the measurement, but the convolution acts as a narrow-band filter (Fig. 3(b)). In a second step, the time lag Tω between the maximum of two consecutive pulses is used to calculate the group velocity of the pulse containing the selected frequency only, since the traveled distance is known to be one circumference. A comparison of theoretical and experimental group velocities is shown in Fig. 3(c). Errorbars are determined by the standard deviation on the mean time interval between two maxima of the signal. The authors’ second point of interest is the dynamic behavior of the antipode. In order to visualize the influence of the frequency on the wavefocusing area in the antipode, a line scan was performed measuring the normal surface velocity over a 60◦ angle centered around the pole. Surface waves around the antipode can be easily distinguished. Due to multiple reflections, bulk waves, which travel faster than surface waves, arrive at the antipode as a long train of low amplitude disturbances. Surface-wave pulses on the other hand, are guided and they reach the antipode almost undisturbed. Consequent arrivals of the latter can therefore be easily distinguished. The maximum velocity in each point is shown in Fig. 4(a,c) for a 61 kHz and a 206 kHz sinusoidal burst, before and after the first arrival of the surface wave. In both cases, the
d1 = R(π − θ),
(3)
and the wave that already passed the antipode has gone a distance R(π + θ). In [23], it is shown that a passage over the antipode additionally results in a phase shift of the wave, inducing a reduction of the travelled wave path by a quarter of the wavelength 4
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61 kHz
(b) interference model measurement
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Figure 4: Surface velocity around the source antipode, before and after the first surface wave arrival (a,c). Results for two frequencies reveal that the focusing area of higher frequencies is smaller than for low frequencies. Interference pattern near the antipode of a point source on a sphere for two frequencies (b,d). Measurements and analytic results are shown. The minima are reached at the angles predicted by geometric considerations, shown as red dashed lines.
the material, geometry and colatitudinal angle. To achieve reasonable fidelity of the wave response to a windowed sine burst, the first 50 modes are taken into account including their overtones up to 1 MHz. The calculations show that even a point source results in spatial side lobes around the source antipode, confirming that the size of the source does not affect the interference. By tuning the calculated displacements to the measurements, the stress applied by the piezo transducers is estimated to be approximately 2 kPa. The measured and calculated interference patterns are shown in Fig. 4(b,d).
M
λ. The total travel distance of the diverging wave is thus d2 = R(π + θ) − λ/4. (4)
3λ 8R
(5)
pt
θmin =
ed
The difference of both wave paths |d1 − d2 | is half a wavelength for the first interference minimum, which results in an angle
Ac ce
away from the source antipode. In the case of the 206 kHz wave, the first two minima are expected at 6.3◦ and 12.5◦ which agrees well with the experiment. In the case of the 61 kHz wave the minimum is expected at 20◦ . The measurements however show a smaller angle of 17◦ due to a very strong frequency component at 80-90 kHz as shown in Fig. 3, which agrees well with this first minimum. The diameter of the source transducer might introduce a wave-path difference as well, and so supplies an alternative explanation to the presence of the interference pattern. In order to check if the side lobes are indeed due to the interaction of the converging and diverging part of the wave pulse, an analytical calculation of the radial surface motion is carried out. The wave propagation in a sphere is a superposition of the spectral components of the source signal and of the possible vibration modes of the sphere under consideration [1, 16, 22].The amplitude of each spectral component depends on
4. Conclusions In this letter, a non-contact experimental technique for the investigation of surface-wave coalescence over a solid sphere is presented. Theoretically-predicted dispersion of surface waves and the characteristics of their focusing at the pole opposite to the excitation location are validated experimentally with a laser vibrometer. Measurements of bandwidth-limited signals indicate a significant frequency-dependence for the focusing of surface waves. A model to predict interference patterns based on geometric notions and the antipode phase-shift of surface waves on a sphere is proposed. The knowledge of surface waves in spheriodal bodies is important for the dynamics of granular solids, 5
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[18] S. Chirit¸a ˘, Thermoelastic surface waves on an exponentially graded half-space, Mechanics Research Communications 49 (0) (2013) 27 – 35. [19] I. A. Viktorov, Rayleigh and Lamb waves: physical theory and applications, Vol. 147, Plenum press New York, 1967. [20] P. Otsuka, O. Matsuda, M. Tomoda, O. Wright, Interferometric imaging of surface acoustic waves on a glass sphere, Journal of Applied Physics 108 (12) (2010) 123508–123508. [21] L. Rayleigh, Cxii. the problem of the whispering gallery, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 20 (120) (1910) 1001– 1004. [22] E. R. Lapwood, T. Usami, Free oscillations of the Earth, Cambridge University Press, Cambridge, 1981. [23] J. Brune, J. Nafe, L. Alsop, The polar phase shift of surface waves on a sphere, Bulletin of the Seismological Society of America 51 (2) (1961) 247–257. [24] Y. Tsukahara, N. Nakaso, H. Cho, K. Yamanaka, Observation of diffraction-free propagation of surface acoustic waves around a homogeneous isotropic solid sphere, Applied Physics Letters 77 (2000) 2926. [25] E. P. Chael, D. L. Anderson, Global q estimates from antipodal rayleigh waves, Journal of Geophysical Research: Solid Earth (1978–2012) 87 (B4) (1982) 2840– 2850. [26] L. Rayleigh, A. Lodge, On the acoustic shadow of a sphere. with an appendix, giving the values of legendre’s functions from p {0} to p {20} at intervals of 5 degrees, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203 (359-371) (1904) 87–110. [27] J. Foxwell, The diffraction of sound round a rigid sphere from a point source on the sphere, Journal of Sound and Vibration 11 (1) (1970) 103–113. [28] H. Kawase, The cause of the damage belt in kobe:the basin-edge effect, constructive interference of the direct s-wave with the basin-induced diffracted/rayleigh waves, Seismological Research Letters 67 (5) (1996) 25– 34.
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for applications such as acoustophoresis, gas sensors, optomechanics, and nano-printing. Understanding the convergence of surface waves into narrow, high amplitude waveforms sheds new light on energy transmission between contacting particles, but can also be used to enhance performance of devices exploiting surface waves.
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