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Wave Motion 36 (2002) 311333 On the application of the WienerHopf technique

Summary: Wave Motion 36 (2002) 311­333
On the application of the Wiener­Hopf technique
to problems in dynamic elasticity
I. David Abrahams
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Received 9 January 2002; received in revised form 3 March 2002; accepted 3 March 2002
Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener­Hopf functional
equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between
shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener­Hopf kernels
are of matrix form. The key step in the solution of a Wiener­Hopf equation, which is to decompose the kernel into a product
of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from
special matrices which yield commutative factorizations, no procedure has yet been devised to factorize exactly general matrix
This paper shall demonstrate, by way of example, that the Wiener­Hopf approximant matrix (WHAM) procedure for
obtaining approximate factors of matrix kernels (recently introduced by the author in [SIAM J. Appl. Math. 57 (2) (1997)
541]) is applicable to the class of matrix kernels found in elasticity, and in particular to problems in QNDE. First, as a
motivating example, the kernel arising in the model of diffraction of skew incident elastic waves on a semi-infinite crack
in an isotropic elastic space is studied. This was first examined in a seminal work by Achenbach and Gautesen [J. Acoust.
Soc. Am. 61 (2) (1977) 413] and here three methods are offered for deriving distinct non-commutative factorizations of the


Source: Abrahams, I. David - Department of Mathematics, University of Manchester


Collections: Mathematics