Rayleigh Waves

12 Key excerpts on "Rayleigh Waves"

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  eBook - ePub

  Fundamentals and Applications of Ultrasonic Waves

  • J. David N. Cheeke(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  8 Rayleigh Waves

  8.1 Introduction

  Like much of acoustics, surface acoustic waves (SAWs) go back to Lord Rayleigh, and because of this, SAWs and Rayleigh Waves are usually used synonymously. Rayleigh’s interest in the problem was brought about by his intuitive feeling that they could be a dominant acoustic signal triggered by earthquakes. His 1885 paper on the subject [1 ] concluded with the well-known remark, “… It is not improbable that the surface waves here investigated play an important part in earthquakes, and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great distance from the source a continually increasing preponderance.” This was indeed found to be the case and Rayleigh’s pioneering work stimulated a great deal of further study of other acoustic modes that could propagate in the layered structure of the earth’s crust.

  Rayleigh Waves are now standard fare not only in seismology but also in many areas of modern technology. With the introduction of IDTs in the 1960s, they have, as it were, been integrated into modern microelectronics in the form of filters, delay lines, and many other acoustoelectronic functions. They are ubiquitous in all of the applications of ultrasonics described in this book, and so it is incumbent upon us to have a good understanding of their propagation characteristics.

  Rayleigh Waves are the simplest cases of guided waves that we will examine. They are confined to within a wavelength or so of the surface along which they propagate. They are distinct from longitudinal and shear bulk acoustic wave (BAW) modes, which propagate independently at different velocities. In Rayleigh Waves, the longitudinal and shear motions are intimately coupled together, and they travel at a common velocity. In this chapter, we start with a detailed description of these waves on the surface of an isotropic solid in vacuum. In Section 8.3

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  Physical Acoustics V7

  Principles and Methods

  • Warren P. Mason(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Rayleigh Waves have a certain resemblance to waves on fluid surfaces: In both cases the particle motion occurs in elliptical orbits in a plane perpendicular to the surface and parallel to the direction of propagation. The restoring forces are, of course, very different: elastic forces in solids, gravity and surface tension in fluids (see, e.g., L a n d a u and Lifshitz 1959). Originally, Rayleigh Waves were of specific interest only to seismologists: As surface waves propagate into only two dimensions, their energy decays more slowly with increasing distance from their point of origin than that of bulk waves, namely, proportional to 1/r as compared with 1/r 2 for bulk waves. Therefore, at large distances from the epicenter of an earthquake, surface shocks are caused mainly by surface waves. (The seismological aspects of elastic surface waves are discussed in detail, e.g., by Ewing et al., 1957.) Rayleigh Waves became interesting for technical applications when Firestone and Frederick (1946) had succeeded in their piezoelectric generation. A t frequencies of about 5 MHz, the wavelength and hence the penetration depth is of the order of 1 mm, and this fact was utilized for materials testing: I t became possible to detect very tiny surface cracks, since Rayleigh Waves are reflected b y any discontinuities in the surface. I t was only in recent years that transduction methods for Rayleigh Waves have been sufficiently improved for their excitation up to frequencies in the gigahertz range.

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  Physical Acoustics V10

  Principles and Methods

  • Warren P. Mason(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  They propagate with speeds equal to the bulk velocities of the media, and hence represent simply bulk waves in the medium on either side of the interface, which may propagate along the latter due to the discontinuity which it represents. These waves are known as lateral waves (see, e.g., Brekhovskikh, 1960). Waves of all these types have been studied extensively on plane interfaces, both theoretically and experimentally. In particular, Rayleigh Waves have received considerable attention in view of their usefulness for the detection of surface faults in ultrasonic nondestructive testing of materials. In addition, they have been employed in the construction of electromechanical delay lines, used for radar and communications systems. Several review papers have appeared in the recent literature emphasizing one or the other of these technological aspects of Rayleigh Waves (Becker and Richardson, 1970; Dransfeld and Salzmann, 1970; Farnell, 1970; Nickerson, 1970; Kallard, 1971); furthermore, the more fundamental properties of surface waves have also been treated (Brekhovskikh, 1959; Viktorov, 1967). Analogous surface waves exist also on surfaces with simple or arbitrary curvature. In this case the situation is considerably more complex and much less well understood. Investigations are less numerous, although a certain amount of progress has been made in the last few years. In the present article we intend to survey the essential results obtained concerning acoustic surface waves on curved surfaces, and to establish their relationships with the corresponding plane surface waves. In view of the limited generality of the existing literature, the excitation mechanisms considered here for the generation of such surface waves have mostly been taken as those provided by a plane incident acoustic wave; more general mechanisms are briefly mentioned in Section V I .

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  Fundamental Physics of Ultrasound

  • Shutilov, Vladimir Alexandrovich Shutilov, Yelena Vladimirovna Tcharnaya(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  1

  We recall that, according to the meaning of the solution obtained (X.65), the velocity c appearing in it through the parameter χ is different for different components of the displacement: the component u l corresponds to the velocity c l and the shear component u τ corresponds to the velocity

  cτ

  . These components can propagate independently in the volume of a solid body with corresponding velocities, i.e., volume waves can be purely longitudinal as well as purely shear waves. In a surface wave, however, due to the presence of a free boundary the displacement u is always mixed: it includes different components which, generally speaking, are no longer “longitudinal” or “transverse.” The corresponding calculation, using the boundary conditions, shows that the displacement trajectory of particles in the surface wave is an ellipse whose long axis is perpendicular to the surface and whose short axis is parallel to the surface and oriented along the direction of propagation of the surface wave, i.e., along the y -axis in this case. The ration of the axes depends on the ratio of the velocities c l /c τ , i.e., on Poisson’s ratio, and for v 0 = 0.3, ≃ 1.5 for particles on the surface (x = 0). The velocity of propagation of a Rayleigh surface wave c R = ω /k R aslo depends on

  cl

  /c τ , i.e., on v 0 , and does not depend on the frequency ω .

  It is interesting to note that these results can be formally obtained based on the relations derived in the preceding section by interpreting a Rayleigh wave as a degenerate case of reflection of plane waves, in which the coefficient of reflection of the incident wave from the free boundary becomes infinite. Since the physical origin of the reflection and refraction of waves at boundaries lies in the radiation by the oscillating boundary, the indicated condition (

  ρA

  = ∞) corresponds to a wave process propagating along the boundary without the incident wave, i.e., a free surface wave. Its propagation velocity c R can be determined from the velocity of the track of the reflected wave with an infinite reflection coefficient. For example, for a reflected shear wave

  c R

  =

  c τ

  / sin 

  θ τ ∞

  at

  ρAτ

  = ∞. Setting z liq = 0 in Eq. (X.51) for the reflection coefficient of the shear wave and equating its denominator to zero, we obtain the equation (

  cl

  /cos θ l ) cos2

  θτ

  + (c τ /cos θ τ ) sin2 2θτ = 0, whence, taking into account the relation between the angles

  θl

  and

  θτ

  (sin

  θτ

  /sin

  θl

  = c τ /

  cl

  ), it is not difficult to calculate the values of

  θ τ ∞

  that determine the velocity of the Rayleigh wave as a function of the ratio

  cτ /cl

  for a given medium, i.e., as a function of Poisson’s ratio, since according to (X.16)

  cτ

  /

  cl

  = {(1 − 2v 0 )/[2(1 − V 0 )] }1/2 . The results of such a calculation are presented in Fig. 68 , whence it is evident that as v 0 varies between the two limiting values 0 and 1/2, the velocity of the Rayleigh wave for different media varies from 0.874

  cτ

  to 0.955

  cτ

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  Guided Waves in Structures for SHM

  The Time - domain Spectral Element Method

  • Wieslaw Ostachowicz, Pawel Kudela, Marek Krawczuk, Arkadiusz Zak(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  y , starting from the wave crest. Rayleigh Waves propagate along surfaces of elastic bodies of thickness many times exceeding the wave height. Sea waves are a natural example of Rayleigh Waves.

  Figure 1.3 Distribution of displacements for the Rayleigh wave

  1.1.4 Love Waves

  Love waves (Figure 1.4 ) are characterised by particle oscillations involving alternating transverse movements. The direction of medium particle oscillations is horizontal (in the xz plane) and perpendicular to the direction of propagation. As in the case of Rayleigh Waves, wave amplitude decreases with depth.

  Figure 1.4 Distribution of displacements for the Love wave

  1.1.5 Lamb Waves

  These waves were named after their discoverer, Horace Lamb, who developed the theory of their propagation in 1917 [1]. Curiously, Lamb was not able to physically generate the waves he discovered. This was achieved by Worlton [2], who also noticed their potential usefulness for damage detection. Lamb waves propagate in infinite media bounded by two surfaces and arise as a result of superposition of multiple reflections of longitudinal P waves and shear SV waves from the bounding surfaces. In the case of these waves medium particle oscillations are very complex in character. Depending on the distribution of displacements on the top and bottom bounding surface, two forms of Lamb waves appear: symmetric, denoted as S 0 , S 1 , S 2 , … , and antisymmetric, denoted as A 0 , A 1 , A 2 , … . It should be noted that numbers of these forms are infinite. Displacement fields of medium points for the fundamental symmetric mode S 0 and fundamental antisymmetric mode A 0 of Lamb waves are illustrated in Figures 1.5 and 1.6

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  Modern Global Seismology

  • Thorne Lay, Terry C. Wallace(Authors)
  • 1995(Publication Date)
  • Academic Press

    (Publisher)

  International Geophysics , Vol. 58, Suppl. (C), 1995

  ISSN: 0074-6142

  doi: 10.1016/S0074-6142(05)80004-X

  Chapter 3 Body Waves and Ray Theory

  In the last chapter we derived the existence of P and S waves, the only transient solutions to a stress imbalance suddenly introduced to a homogeneous elastic space. P and S waves are known as body waves because they travel along paths throughout the continuum. The solutions for P and S waves, like those given in Eqs. (2.85) and (2.90) , give the locations of wavefronts , which are loci of points that undergo the same motion at a given instant in time. Rays are defined as the normals to the wavefront and thus point in the direction of propagation. In the case of a plane wave, the rays are a family of parallel straight lines; in the case of a spherical wave, the rays are spokes radiating out from the seismic source. Rays provide a convenient means of tracking an expanding wavefront, and they provide an intuitive framework for extending elastic-wave solutions from homogeneous to inhomogeneous materials. If the inhomogeneities in velocity are not excessively chaotic, then the rays corresponding to P or S waves behave very much as light does in traveling through materials of varying indices of refraction. This leads to many parallels with optics: rays bend, focus, and defocus depending on the velocity distribution. Strictly speaking, we will have to approximate our displacement solutions to extract the ray behavior, for it cannot describe all wave phenomena. These approximations are collectively known as geometric ray theory and are the standard basis for seismic body-wave interpretation.

  In classical optics, the geometry of a wave surface is governed by Huygens’ principle, which states that every point on a wavefront can be considered the source of a small secondary wavelet that travels outward in every forward direction with the velocity of the medium at that point. The wavefront at a later instant in time is found by drawing a tangent to the secondary wavelets, as shown in Figure 3.1 . Thus, given the location of a wavefront at a certain instant in time, we can predict future positions of the wavefront. Portions of the wavefront which are located in relatively high-velocity material produce wavelets that travel farther in a given time interval than those produced by points in relatively low-velocity material. This causes a temporal dependence in the shape of the wavefront. Because rays are the normals to the wavefront, the rays will also change with time. Fermat’s principle governs the geometry of raypaths. This usually means that the ray will follow a minimum-time path , which is the path that will allow the wavefront to move from point A to point B

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  Seismic Resistant Design and Technology

  • Dentcho Ivanov(Author)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  If, however, the third and fourth harmonic waves are damped simultaneously, the maximum amplitude of initial Rayleigh wave can be amplified via parametric amplification up to 10 times. Nonlinear interactions between surface and bulk waves have been studied both experimentally and theoretically. The results can of mixing nonlinear waves be summarized as follow: i) a surface wave and a bulk wave generate a surface wave ii) two surface waves generate a bulk wave iii) a surface wave and a bulk wave generate a bulk wave iv) two bulk waves generate a surface wave It is interesting to discuss how nonlinearity and dispersion affect the propagation of bulk and surface seismic waves. During distant earthquakes often two distinct stages have been observed— the first characterized by a preliminary weak motion followed by the second main shock characterized by a much stronger tremor. First Rayleigh (Rayleigh 1885) suggested that surface seismic waves play an important role in earthquakes. Later Oldham (Oldham 1900) recorded two phases in the preliminary weak motions that he identified by their travel times as the direct bulk P- and S-waves traveling at different but almost constant velocities. The main shock Oldham attributed to Rayleigh surface elastic Surface Elastic Wave Propagation 147 wave traveling also at almost constant speed on the Earth’s surface but much slower than the bulk P- and S-waves. Lamb (Lamb 1904) confirmed Oldham’s observations on the surface seismic waves during the main shock, but couldn’t explain why in some cases the ground was moving vertically which shows a typical Rayleigh wave and in other it moved sidewinding in the horizontal ground plane. The sidewinding surface elastic waves were later were explained by Love (Love 1911) and called Love waves. As we have discussed already, instead of considering homogeneous half-space Love considered a layer on the top of the ground half space as a boundary between the ground half space and the air.

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  Principles of Seismology

  • Agustín Udías, Elisa Buforn(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The amplitudes of these waves decrease exponentially with depth. In a certain way, these waves may be considered to be generated by energy brought to the surface by incident P and S waves that produce no re fl ections. 12.1.1 Displacements of Rayleigh Waves Displacements of Rayleigh Waves are obtained by substituting expressions ( 12.13 ) and ( 12.14 ) for the potentials ϕ and ψ into ( 12.1 ) and ( 12.3 ). For the particular case of σ ¼ 1 4 , we have c R = 0.9194 β , r = 0.85 i , and s = 0.39 i . By fi rst substituting these values into ( 12.22 ), we obtain Β = − 1.47 iA . Finally, taking only the real part, the displacements are given by u 1 ¼ Ak ð e 0 : 85 kx 3 0 : 58 e 0 : 39 kx 3 Þ sin ½ k ð x 1 c R t ފ ; ð 12 : 29 Þ u 3 ¼ Ak ð 0 : 85 e 0 : 85 kx 3 þ 1 : 47 e 0 : 39 kx 3 Þ cos ½ k ð x 1 c R t ފ : ð 12 : 30 Þ 0.25 0.5 0.5 1 q 0 ξ Fig. 12.2 Variation of q ( β / α ) and ξ ( c / β ) with Poisson ’ s ratio. 263 12.1 Rayleigh Waves in a half-space We must remember that A is the potential amplitude (in units of m 2 ) and Ak is the displacement amplitude (in units of meters). For points at the free surface ( x 3 = 0), letting a = -Ak , we fi nd that u 1 ¼ 0 : 42 a sin ½ k ð x 1 c R t ފ ; ð 12 : 31 Þ u 3 ¼ 0 : 62 a cos ½ k ð x 1 c R t ފ : ð 12 : 32 Þ Since Rayleigh Waves have no transverse component, they are polarized in the vertical plane. The horizontal and vertical components are shifted in phase by π /2 and thus the motion is elliptical. If we substitute into ( 12.31 ) and ( 12.32 ) the values of t during a complete cycle (0 to Τ , where Τ = 2 π / ω is the period), we obtain for the particle ’ s motion an ellipse with a vertical major axis and retrograde motion (opposite to that of wave propagation) ( Fig. 12.3 ). The dependence of the displacement components u 1 and u 3 on depth ( -x 3 ) is given by ( 12.29 ) and ( 12.30 ). There is a value of x 3 for which u 1 is null, x 3 = -0.19 λ ( λ = 2 π / k is the wave length), whereas u 3 is never null.

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  Waves in Layered Media

  • Leonid Brekhovskikh(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Apart from the attenuating system of waves considered above, which approach Rayleigh Waves as the liquid density approaches zero, an undamped surface wave having a somewhat different nature can propagate along the boundary between a solid and a liquid. Its velocity is less t h a n the velocity c, and therefore, in the liquid, it will have the form of the usual inhomogeneous wave, decreasing in amplitude with distance from the boundary. The intensity in both media will be directed parallel to the boundary of separation. I n fact, when p/pi^I, r = ( 6 1 / c ) 2 > l we have the approximate solution of Eq. 4.53 which is easily found by representing the right hand side of Eq. 4.53 in the form 44 P L A N E W A V E S I N L A Y E R S where λ is the wavelength of the sound wave in the liquid. I n the solid, the decrease of the amplitude of the longitudinal and transverse waves will take place according to a law, an approximate expression of which has the form exp [ — (2π/λ) ζ |]. Thus, since pc 2 /p 1 6f is small, the amplitude in the liquid decreases very slowly with distance from the boundary, while in the solid, the entire wave process is concentrated in a layer of thickness of the order of λ. This type of surface wave is nothing other than a sound wave, incident on the solid boundary at grazing incidence. I t was shown in § 3 t h a t when a plane sound wave strikes a boundary of separation between two liquids at grazing incidence, the reflected wave completely cancels the incident wave, as a result of which the total field is zero, i.e. no wave can exist. I n the case under consideration, we see t h a t a sound wave gliding along the boundary can exist, b u t it must then be slightly inhomogeneous, i.e. its amplitude must decrease, at least slowly, with distance from the boundary. Because of the compliance of the solid, the velocity of the wave will be somewhat lower t h a n t h a t of a sound wave in the unbounded liquid.

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  Statistical Methods Of Geophysical Data Processing

  • Vladimir Troyan, Yurii Kiselev(Authors)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 5 Ray theory of wave field propagation The exact solutions of problems of sounding signals propagation are constructed only for limited number of media models. As a rule, this set of media includes uniform medium, layered homogeneous medium, uniform medium with the inclu-sions of high symmetry. For the interpretation of real geophysical fields (seismic, acoustic, electromagnetic) it is necessary to construct the approximate solutions for the wave propagation in non-uniform media. So, it exists in the Earth not only the interfaces, on which one the elastic properties vary by jump, but also areas, inside which there is a smoothly varying variation of elastic properties. From the physical point of view the ray theory is interpreted as follows: the waves propagate with local velocities along ray pathways and arrive in the observation point with ampli-tudes described by a geometrical spreading of rays from a source to the receiver point. At an enunciating of this chapter we shall follow the description introduced in (Ryzhikov and Troyan, 1994). 5.1 Basis of the Ray Theory One of the most common method of the solution of the equations of wave propa-gation is the method of geometrical optics (Babic and Buldyrev, 1991; Babic et al. , 1999; Kravtsov, 2005; Bleistein et al. , 2000). This method is a shortwave asymp-totic of a field in weak non-uniform, slow non-stationary and weak-conservative media: the sizes of the inhomogeneity are much greater than the wavelength and time intervals of the non-stationarity much more than the period of oscillation. The shortwave asymptotic allows to consider the medium locally as homogeneous and stationary and is based on the assumption of a wave field in a form ( ϕ ) a “quick” phase and a “slow” amplitude multiplier factors. Let us consider a formal scheme of the space-time ray method.

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  Low and High Frequency Asymptotics

  Acoustic, Electromagnetic and Elastic Wave Scattering

  • V.K. Varadan(Author)
  • 2013(Publication Date)
  • North Holland

    (Publisher)

  C H A P T E R 1 Rayleigh Scattering R.E. KLEINMAN Department of Mathematical Sciences University of Delaware Newark, Delaware, USA and T.B.A. SENIOR Department of Electrical and Computer Engineering University of Michigan Ann Arbor, Michigan, USA Low and High Frequency Asymptotics Edited by V.K. Varadan and V.V. Varadan © Elsevier Science Publishers B.V., 1986 2 R.E. Kleinman, Τ.Β.Λ. Senior 1. Introduction The scattering of acoustic and electromagnetic waves from bounded objects whose dimensions are small compared with the length of the incident wave has been the subject of considerable study for more than a century. Lord Rayleigh found this problem of continuing interest and his contributions in this area (e.g., Rayleigh, 1881, 1897) provide the foundation on which almost all subsequent work is based. It is only fitting that his name should grace the subject. Some idea of the history of this area of scientific inquiry and its use in light scattering may be gained from Twersky (1964) and the books of van de Hülst (1957) and Kerker (1969); and much relevant material, especially concerning spherical scatterers, is contained in the fascinating study of Logan (1965). Despite this long history, it was not until relatively recently that a rigorous mathematical definition of Rayleigh scattering was attempted (Kleinman, 1965, 1978). Although there is not necessarily universal agreement, it is generally accepted that in three-dimensional problems Rayleigh scattering concerns the determination of the first non-vanishing term in a series expansion in powers of wavenumber of a relevant quantity of interest such as the scattered field or far-field coefficient. This is the definition used here and made precise in Section 2. The present study is concerned with the determination of the first or Rayleigh term in the far-field coefficient for a variety of scattering problems.

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  Landmark Writings in Western Mathematics 1640-1940

  • Ivor Grattan-Guinness(Author)
  • 2005(Publication Date)
  • Elsevier Science

    (Publisher)

  This experiment was first reported in Philosophical magazine, published in June 1877. After strengthening the intensities of the sound by the addition of resonators to tuning forks, he detected the points of silence at places where the theory indicated, which confirmed his own expectation. In Chapter 15, Rayleigh put forth his original theory on the secondary waves that are produced when the plane waves impinge on different media. He deduced that the amplitude of the secondary waves varies inversely to the distance through the medium and to the square value of the wavelength, and that while a region in which the compressibility varies acts like a simple source, a region at which the density varies acts like a double source. 596 J.H. Ku He illustrated these theoretical arguments by means of harmonic echoes, on which he had written a paper in 1873. In Chapter 16, on the theory of resonators, Rayleigh presented the results of mathe- matical and experimental investigations on resonators he had performed. A considerable portion of the chapter came from his paper ‘On the theory of resonance’ of 1870. Chapter 17 included discussion on sound waves that are generated as a reaction to the vibration of a rigid body and propagate in the air. Referring to George Green (§30.5) and S.D. Poisson as predecessors, Rayleigh developed his own analytical theory. He solved Laplace’s equations in various situations, employing such functions as spherical harmon- ics, Legendre’s functions, and Bessel’s functions. He proceeded from this to the case of disturbance confined to a small portion of a spherical surface. By using the reciprocal the- orem, he transformed this problem into that of a sound wave which arrived at any point on the spherical obstacle from an external source, and he could therefore discuss the ob- structive effect of a head in the path of the transmission of sound in the air. Then, Rayleigh discussed the application of the general equations when there was no sound source.

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