The propagation of transient waves in a homogeneous, isotropic, linearly elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half-space and for all load speeds. Wave front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb's point load problem. INTRODUCTIONGround motions excited by moving surface forces arise, for example, from nuclear blasts and from shock waves generated by supersonic aircraft, and they interact with structures causing extensive damage. A mathematical problem of fundamental importance in these applications is that of an elastic half-space whose surface is excited by a normal point load which is suddenly applied and which subsequently moves rectilinearly at a constant speed.In recent years several solutions to this probl em have been given for the surface of the half-space. t Numbers in brackets designate References at end of paper. shown to form the steady-state displacement field when the load speed exceeds both of the body wave speeds. FORMULATION OF THE PROBLEMThe subject half-space problem is depicted in Fig. 1 The constants cd and cs, defined by c! = (X. + 2µ)/v and c; = µ/v, represent the dilatational and equivoluminal body wave speeds, respectively, where X. and µ are the Lam~ constants and v is the material density. The stresses T .. are related to the displacements by lJwhere tensor notation is employed.The boundary conditions at z = 0 take the formwhere o is the Dirac delta function. To represent quiescence at t = 0, the initial conditions appear asFinally, the potentials cf> and ljJ, and the space derivatives of the patentials, are required to vanish at infinity. FORMAL SOL UT IONA solution of the wave equations (1) that satisfies the initial conditions (6) and the boundedness condition at infinity can be computed Soos-n z+i(kx+vy) (1 7a)(1 7b)(1 7c)(1 7d) In view of the symmetry properties u (r,9,z,p) =u (r,-9,z,p) xa xa u (r,9,z,p) = -u (r,-9, z ,p)u (r,9,z,p) =u (r,-9,z,p), za za u. , and hence u., are only inverted for 0 :S 9 <'TT. Since u. has different Ja J Ja forms depending on the speed of the load relative to the body wave speeds, the inversion of each u. is separated into three cases. In particular, S(X)s(X)_ _E...c (mdz-iqr)where ~ is the position vector. uzd is converted into the Laplace transform of a known function by mapping (1 /cd)(mdz-iqr) into t through a contour integration in a complex q-plane. To this end, the singularities of the integrand of U:zd are branch points at -9-for t > twd, where Equation (26) defines one branch of a hyperbola with vertex 2 .!.2 r/p and asymptotes arg q = ±r/z. As shown in Fig. 2 (27) + -by a solid line labeled with qd and qd, this hyperbola is parametrically described by t as t varies from twd ...

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A class of elastic half-space problems involving axisymmetric, normally applied, surface loads is investigated. Each load is assumed to suddenly emanate from a point on the surface and expand radially at a constant rate. The cases of loads shaped like a ring and a disk are considered in detail. Exact solutions are derived for the displacements at every point in the half space in terms of single integrals. Each integral is identified as a specific wave. The integrals are evaluated analytically and numerically for different depths in the half space, for loads expanding at superseismic, transeismic, and subseismic rates, and for different values of Poisson’s ratio. Moreover, the interaction of the loads and the Rayleigh wave is described. Then solutions are obtained for loads of other shapes by convoluting the ring and disk load solutions.

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