Elastic Waves at High Frequencies

Techniques for Radiation and Diffraction of Elastic and Surface Waves

Elastic Waves at High Frequencies

John G. Harris intended to explain in this book the special techniques required to model the radiation and diffraction of elastic and surface waves. He sadly died before he could fulfil this ambition, but his plan has been brought to fruition by a team of his distinguished collaborators. The book begins with the basic underlying equations for wave motion and then builds upon this foundation by solving a number of fundamental scattering problems. The remaining chapters provide a thorough introduction to modern techniques that have proven essential to understanding radiation and diffraction at high frequencies. Graduate students, researchers and professionals in applied mathematics, physics and engineering will find that the chapters increase in complexity, beginning with plane-wave propagation and spectral analyses. Other topics include elastic wave theory, the Wiener–Hopf technique, the effects of viscosity on acoustic diffraction, and the phenomenon of channelling of wave energy along guided structures.


 Reviews:

"It is written in an elegant, engaging style, reflects well the research interests of the expert team, and as such, is an invaluable new source for graduate students, researchers, and professionals working in the fields of applied mathematics, physics and engineering who are faced with similar modelling tasks."
Larissa Ju. Fradkin, Mathematical Reviews

'it is written in an elegant engaging style, reflects well the research interests of the expert team, and as such is an invaluable new source for graduate students, researchers, and professionals working in the fields of applied mathematics, physics and engineering who are faced with similar modelling tasks.' Mathematical Reviews

Abrahams I. D. (2000). The application of Padé approximants to Wiener–Hopf factorization. IMA J. Appl. Math., 65: 257–281.
Achenbach J. D. (1973). Wave Propagation in Elastic Solids. North-Holland, Amsterdam.
Achenbach J. D. (2003). Reciprocity in Elastodynamics. Cambridge University Press, New York.
Achenbach J. D. , Gautesen A. K. and McMaken H. (1982). Ray Methods for Waves in Elastic Solids. Pitman, Boston.
Aki K. and Richards P. G. (2002). Quantitative Seismology. University Science Books, Sausalito, CA, second edition.
Atkin R. J. and Fox N. (1980). An Introduction to the Theory of Elasticity, volume 1. Longman, London, second edition.
Auld B. A. (1979). General electromechanical reciprocity relations applied to the calculation of elastic wave scattering coefficients. Wave Motion, 1: 3–10.
Auld B. A. (1990a). Acoustic Fields and Waves in Solids, volume 1. Krieger, Malabar, FL, second edition.
Auld B. A. (1990b). Acoustic Fields and Waves in Solids, volume 2. Krieger, Malabar, FL, second edition.
Babic V. M. and Buldyrev V. S. (1991). Short-Wavelength Diffraction Theory. Springer-Verlag, Berlin.
Baker B. B. and Copson E. T. (1987). The Mathematical Theory of Huygens' Principle. Chelsea, New York, third edition.
Batchelor G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, U.K.
Bell J. C. (1979). Stresses from arbitrary loads on a circular crack. Int. J. Fracture, 15: 85–104.
Besserer H. and Malischewsky P. G. (2004). Mode series expansions at vertical boundaries in elastic waveguides. Wave Motion, 39: 41–59.
Biryukov S. V. , Gulyaev Yu. V. , Krylov V. V. and Plesky V. P. (1995). Surface Acoustic Waves in Inhomogeneous Media. Springer-Verlag, Berlin.
Bleistein N. and Handelsman R. A. (1975). Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York.
Block G. , Harris J. G. and Hayat T. (2000). Measurement models for ultrasonic nondestructive evaluation. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 47: 604–611.
Born M. and Wolf E. (1999). Principles of Optics. Cambridge University Press, Cambridge, U.K., seventh (expanded) edition.
Borovikov V. A. (1994). Uniform Stationary Phase Method. The Institution of Electrical Engineers, London.
Borovikov V. A. and Kinber B. Ye. (1994). Geometrical Theory of Diffraction. The Institution of Electrical Engineers, London.
Boström A. (2003). Review of hypersingular integral equation method for crack scattering and application to modeling of ultrasonic nondestructive evaluation. Appl. Mech. Rev., 56: 383–405.
Boström A. and Olsson P. (1987). Scattering of elastic waves by non-planar cracks. Wave Motion, 9: 61–76.
Bouwkamp C. J. (1946). A contribution to the theory of acoustic radiation. Philips Res. Rep., 1: 251–277.
Bouwkamp C. J. (1954). Diffraction theory. Rep. Prog. Phys., 17: 35–100.
Brekhovskikh L. M. and Godin O. A. (1999). Acoustics of Layered Media, volume 2. Springer-Verlag, Berlin, second edition.
Brekhovskikh L. M. and Goncharov V. (1985). Mechanics of Continua and Wave Dynamics. Springer-Verlag, Berlin.
Briggs A. (1992). Acoustic Microscopy. Oxford University Press, New York.
Chako N. (1965). Asymptotic expansions of double and multiple integrals occurring in diffraction theory. J. Inst. Math. Appl., 1: 372–422.
Clemmow P. C. (1966). The Plane Wave Spectrum Representation of Electromagnetic Waves. Pergamon, Oxford.
Collin R. E. (1991). Field Theory of Guided Waves. IEEE Press, New York, second edition.
Comninou M. and Dundurs J. (1977). Reflexion and refraction of elastic waves in the presence of separation. Proc. R. Soc. A, 356: 509–528.
Copson E. T. (1971). Asymptotic Expansions. Cambridge University Press, Cambridge, U.K.
Davis A. M. J. and Nagem R. J. (2002). Acoustic diffraction by a half-plane in a viscous fluid medium. J. Acoust. Soc. Am., 112: 1288–1296.
Davis A. M. J. and Nagem R. J. (2006). Curle's equation and acoustic scattering by a sphere. J. Acoust. Soc. Am., 119: 2018–2026.
de Bruijn N. G. (1970). Asymptotic Methods in Analysis. North-Holland, Amsterdam.
Doetsch G. (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer-Verlag, New York. Translated from the German by W. Nader.
Eringen A. C. and Şuhubi E. S. (1975). Elastodynamics, volume 2. Academic Press, New York.
Felsen L. B. and Marcuvitz N. (1994). Radiation and Scattering of Waves. IEEE and Oxford University Press, New York.
Folguera A. and Harris J. G. (1998). Propagation in a slowly varying waveguide. In J. A. DeSanto , editor, Mathematical and Numerical Aspects of Wave Propagation, pages 434–436. SIAM, Philadelphia.
Friedman B. (1956). Principles and Techniques of Applied Mathematics. Wiley, New York.
Glushkov Y. V. and Glushkova N. V. (1996). Diffraction of elastic waves by three-dimensional cracks of arbitrary shape in a plane. J. Appl. Math. Mech. (PMM), 60: 277–283.
Gniadek K. and Petykiewicz J. (1971). Applications of optical methods in diffraction theory of elastic waves. In E. Wolf , editor, Progress in Optics, volume 9, pages 281–310. North Holland, Amsterdam.
Graff K. F. (1991). Wave Motion in Elastic Solids. Dover, New York (reprint; first published in 1975).
Greenspan M. (1979). Piston radiator: some extensions of the theory. J. Acoust. Soc. Am., 65: 608–621.
Gridin D. , Craster R. V. and Adamou A. T. I. (2005). Trapped modes in curved elastic plates. Proc. R. Soc. A, 461: 1181–1197.
Hansen R. C. , editor. (1981). Geometrical Theory of Diffraction, IEEE Press Selected Reprint Series. IEEE Press, New York.
Harris J. G. (1987). Edge diffraction of a compressional beam. J. Acoust. Soc. Am., 82: 635–646.
Harris J. G. (1997). Modeling scanning acoustic imaging of defects at solid interfaces. In G. Chavent , G. Papanicolaou , P. Sacks , and W. W. Symes , editors, Inverse Problems in Wave Propagation, pages 237–257. Springer-Verlag, New York.
Harris J. G. (2001). Linear Elastic Waves. Cambridge University Press, New York. A list of errors is maintained at http://www.diffractedwave.com.
Harris J. G. and Block G. (2005). The coupling of elastic, surface-wave modes by a slow, interfacial inclusion. Proc. R. Soc. A, 461: 3765–3783.
Herrera I. and Spence D. A. (1981). Framework of biorthogonal series. Proc. Nat. Acad. Sci. Am., 78: 7240–7244.
Hudson J. A. (1980). The Excitation and Propagation of Elastic Waves. Cambridge University Press, Cambridge, U.K.
James G. L. (1980). Geometrical Theory of Diffraction for Electromagnetic Waves. Peter Peregrinus, London, second edition.
Jull E. V. (1981). Aperture Antennas and Diffraction Theory. IEE and Peter Peregrinus, London.
Kaplunov J. D. , Rogerson G. A. and Tovstik P. E. (2005). Localized vibration in elastic structures with slowly varying thickness. Quart. J. Mech. Appl. Math., 58: 645–664.
Keller J. B. (1957). Diffraction by an aperture. J. Appl. Phys., 28: 426–444.
Keller J. B. (1958). A geometrical theory of diffraction. In L. M. Graves , editor, Calculus of Variations and its Applications, pages 27–52. AMS, Providence, RI.
King L. V. (1934). On the acoustic radiation field of the piezoelectric oscillator and the effect of viscosity on the transmission. Can. J. Res., 11: 135–155.
Kirrmann P. (1995). On the completeness of Lamb modes. J. Elast., 37: 39–69.
Krenk S. (1979). A circular crack under asymmetric loads and some related integral equations. J. Appl. Mech., 46: 821–826.
Kreyszig E. (1975). Introduction to Differential Geometry and Riemannian Geometry. University of Toronto Press, Toronto.
L. B. Felsen , and G. A. Deschamps editors. (1974). Special Issue on Rays and Beams, volume 62 of Proc. IEEE.
Lamb H. (1906). On Sommerfeld's diffraction problem; and on reflection by a parabolic mirror. Proc. Lond. Math. Soc., 2: 190–203.
Levine H. (1978). Unidirectional Wave Motions. North-Holland, Amsterdam.
Liang K. K. , Kino G. S. and Khuri-Yakub B. T. (1985). Material characterization by the inversion of v(z). IEEE Trans. Sonics Ultrason., SU 32: 213–224.
Lighthill M. J. (1965). Group velocity. J. Inst. Math. Appl., 1: 1–28.
Lighthill M. J. (1978a). Fourier Analysis and Generalized Functions. Cambridge University Press, Cambridge, U.K.
Lighthill M. J. (1978b). Waves in Fluids. Cambridge University Press, Cambridge, U.K.
Magnus W. , Oberhettinger F. and Soni R. P. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, New York, third (enlarged) edition.
Malischewsky P. (1987). Surface Waves and Discontinuities. Akademie-Verlag, Berlin.
Martin P. A. (1996). Mapping flat cracks onto penny-shaped cracks, with applications to somewhat circular tensile cracks. Quart. Appl. Math., 54: 663–675.
Martin P. A. (1998). On potential flow past wrinkled discs. Proc. R. Soc. A, 454: 2209–2221.
Martin P. A. (2001). On wrinkled penny-shaped cracks. J. Mech. Phys. Solids, 49: 1481–1495.
Martin P. A. (2006). Multiple Scattering: Interaction of Time-Harmonic Waves with N Scatterers. Cambridge University Press, Cambridge, U.K.
Maupin V. (1988). Surface waves across 2-d structures: a method based on coupled local modes. Geophys. J., 93: 173–185.
Mikhas'kiv V. V. and Butrak I. O. (2006). Stress concentration around a spheroidal crack caused by a harmonic wave incident at an arbitrary angle. Int. Appl. Mech., 42: 61–66.
Miklowitz J. (1978). Elastic Waves and Waveguides. North-Holland, Amsterdam.
Miles J. W. (1971). Scattering by a spherical cap. J. Acoust. Soc. Am., 50: 892–903.
Morse P. M. and Ingard K. U. (1968). Theoretical Acoustics. McGraw-Hill, New York.
Naze Tjøtta J. and Tjøtta S. (1980). An analytical model for the nearfield of a baffled piston transducer. J. Acoust. Soc. Am., 68: 334–339.
Nieto-Vesperinas M. (1991). Scattering and Diffraction in Physical Optics. Wiley-Interscience, New York.
NIST , (2008). Digital library of mathematical functions. http://dlmf.nist.gov, accessed August. This is an ongoing project.
Noble B. (1988). Methods Based on the Wiener-Hopf Technique. Chelsea, New York, second edition.
Norris A. N. (2008). Faxen relations in solids: a generalized approach to particle motion in elasticity and viscoelasticity. J. Acoust. Soc. Am., 123: 99–108.
Oestreicher H. L. (1951). Field and impedance of an oscillating sphere in a viscoelastic medium with an application to biophysics. J. Acoust. Soc. Am., 23: 707–714.
Osipov A. V. and Norris A. N. (1999). The Malyuzhinets theory for scattering from wedge boundaries: a review. Wave Motion, 29: 313–340.
Oughstun K. E. editor (1991). Selected Papers on Scalar Wave Diffraction, volume 51. SPIE, Bellingham, WA.
Phillips H. B. (1933). Vector Analysis. Wiley, New York.
Pierce A. D. (1981). Acoustics. McGraw-Hill, New York.
Porter D. and Stirling D. S. G. (1990). Integral Equations. Cambridge University Press, Cambridge, U.K.
Poruchikov V. B. (1993). Methods of the Classical Theory of Elastodynamics. Springer-Verlag, Berlin. Translated from the Russian by V. A. Khokhryakov and G. P. Groshev .
Pott J. and Harris J. G. (1984). Scattering of an acoustic Gaussian beam from a fluid-solid interface. J. Acoust. Soc. Am., 76: 1829–1838.
Rebinsky D. A. (1991). Asymptotic description of the acoustic microscopy of a surface-breaking crack. PhD thesis, University of Illinois, Urbana-Champaign.
Rebinsky D. A. and Harris J. G. (1992a). The acoustic signature for a surface-breaking crack produced by a point focus microscope. Proc. R. Soc. A, 438: 47–65.
Rebinsky D. A. and Harris J. G. (1992b). An asymptotic calculation of the acoustic signature of a cracked surface for the line focus scanning acoustic microscope. Proc. R. Soc. A, 436: 251–265.
Rose J. L. (1999). Ultrasonic Waves in Solids. Cambridge University Press, Cambridge, U.K.
Schmerr Jr. L. W. and Song S. J. (2007). Ultrasonic Nondestructive Evaluation Systems: Models and Measurements. Springer-Verlag, New York.
Skudrzyk E. (1971). The Foundations of Acoustics. Springer-Verlag, Vienna.
Sneddon I. N. (1951). Fourier Transforms. McGraw-Hill, New York.
Somekh M. G. , Bertoni H. L. , Briggs G. A. D. and Burton N. J. (1985). A two-dimensional imaging theory of surface discontinuities with the scanning acoustic microscope. Proc. R. Soc. A, 401: 29–51.
Sommerfeld A. (1967). Optics: Lectures on Theoretical Physics, Vol. 4. Academic Press, New York. Translated from the German by O. Laporte and P. A. Moldauer .
Sommerfeld A. (2004). Mathematical Theory of Diffraction. Birkhäuser, Boston. Translated from the German by R. J. Nagem , M. Zampolli , and G. Sandri . An introduction and translators' notes are provided.
Spies M. (1994). Elastic waves in homogeneous and layered transversely isotropic media: plane waves and Gaussian wave packets. A general approach. J. Acoust. Soc. Am., 95: 1748–1760.
Spies M. (1999). Transducer field modeling in anisotropic media by superposition of Gaussian base functions. J. Acoust. Soc. Am., 105: 633–638.
Stamnes J. J. (1986). Waves in Focal Regions. Adam Hilger, Bristol, U.K.
Tada T. , Fukuyama E. and Madariaga R. (2000). Non-hypersingular boundary integral equations for 3-D non-planar crack dynamics. Comput. Mech., 25: 613–626.
Thomas D. P. (1963). Diffraction by a spherical cap. Proc. Camb. Phil. Soc., 59: 197–209.
Thurston R. N. (1974). Waves in solids. In C. Truesdell , editor, Mechanics of Solids, volume 4, pages 109–308. Springer-Verlag, New York.
Ti B. W. , O'Brien W. D. and Harris J. G. (1997). Measurement of coupled wave propagation in an elastic plate. J. Acoust. Soc. Am., 102: 1528–1531.
Titchmarsh E. C. (1939). The Theory of Functions. Clarendon Press, Oxford, second edition.
Titchmarsh E. C. (1948). Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford, second edition.
Ufimtsev P. Ya. (1989). Theory of acoustical edge waves. J. Acoust. Soc. Am., 86: 463–474.
Visscher W. M. (1983). Theory of scattering of elastic waves from flat cracks of arbitrary shape. Wave Motion, 5: 15–32.
Weinstein L. A. (1969). The Theory of Diffraction and the Factorization Method. The Golem Press, Boulder, CO. Translated from the Russian by P. Beckmann .
Wen J. J. and Breazeale M. A. (1988). A diffraction beam field expressed as the superposition of Gaussian beams. J. Acoust. Soc. Am., 83: 1752–1756.
Zhang Ch. and Gross D. (1998). On Wave Propagation in Elastic Solids with Cracks. Computational Mechanics Publications, Southampton, U.K.