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Basic Equations and Solutions for Normal Modes of
Chapter k Approximations of the Ray Integrals
9 other sections not shown
A3 H H acoustic emission amplitude approximation arrival Backscattering Bessel functions boundary conditions branch points calculated Chapter chord rays circular cylinder contact the inner creep waves creeping ray curvature cylindrical cavity defined elastic waves expressions Fourier transform H|OJ Hankel functions head wave Hollow Cylinder hoop stress inner boundary inner surface integral transforms Integration Contour inverse transformation Laplace transform line source mapping matrix medium Mid-field Receiver normal modes normal modes solution obtained from Eq odd function P-ray P-source in Hollow P-source with Step Pekeris phase function Poisson's problem propagation pSpS radius ray expansion ray integrals ray parameter ray path ray-integrals receiver function reflected rays reflection coefficients S-wave shadow zone shown in Fig Side-scattering slip stream ray source function sS ray stationary point Step Time Function t-plane Table thesis tion transient waves u-plane vanish variable vector Velocity wave number