Convolution Formulation for Hi~h-Frequency Leak Wave Scatterin Enhancements for Solids and Shells with Truncations: d i valuation of the urface Integral and Experimental and Computational Tests Philip L. Marston, Karen Gipson, and Scot F. Morse Department of Physics, Washington State UniversiQ, Pullman, WA 99164-2814 Abstract: A convolution formulation has been developed for leaky wave scattering enhancements. Progress is summarized in the testing of this formulation for several situations in which exact solutions are (1) not known (e.g. solid and hollow finite tilted cylinders and retroreflective flat tilted surfaces with comers) or (2) known (e.g. tilted infinite cylinders). Some of the examples considered are for leaky Rayleigh waves on steel objects in water, INTRODUCTION AND MOTIVATION The high frequency backscattering from finite targets with truncations can be significantly enhanced at appropriate aspect angles over a wide range of frequencies. The observed contributions from meridional and helical rays reflected from the end of a cylindrical shell as shown in Fig. 1 are examples [1]. The enhancements are typically associated with backscattered wavefronts having a vanishing gaussian curvature which produce a ftileld caustic. A formulation has been developed which first approximates the pressure amplitude pi radiated by the leaky wave at the surface of the scatterer by convolving the local amplitude pi of the incident wave on the illuminated portion of the surface with a two-dimensional response function [2,3,4]. The resulting two-dimensional integral for pf at a surface point S may be expressed in the form PJ(S) = ~pj (S )[KHr)(kps)~ , D (1) where Hjl ) is the Hankel function having an argument proportional to the geodesic distance s between S and an illuminated point at S’ where the incident wave complex amplitude is pi(S’) and dfi’ is the differential area of the contributing surface patch. The domain D is restricted depending on the directional properties of the waves of interest such that the integral may be expressed as a convolution of pi with a response function of limited angular support [2,3]. The leaky wavenumber for the lth class of leaky wave is kp = k! + ia where kl = kclcl and k is the wavenumber in water, K= -tXk/eXp(iQb/), and qbf is a background phase. For typical applications of interest, the leaky wave damping rate tx is sufficiently large that leaky wave reverberations may be neglected so that global resonances are unimportant. The farfield amplitude is then calculated through the evaluation of a Rayleigh propagation integral. We summarize below the status of computational and experimental tests of this formulation. TILTED INFINITE CIRCULAR CYLINDERS: HELICAL AND MERIDIONAL RAYS When applied to helical rays on circular cylinders [2], the integral, Eq. (1), was confirmed to recover results from others derived specifically for leaky waves on thin shells. When applied to leaky wave scattering into the meridional plane numerical tests for Rayleigh waves at high frequencies support the use of the approximation [3]. Recently one of us (SFM) confirmed that the partial wave series for thick and thin infinite shells supports the meridional result. MERIDIONAL RAY END-REFLECTION BACKSCATTERING ENHANCEMENT The farfield caustic due to the reflected meridional ray in Fig. 1 makes this contribution important when the cylinder tilt y is close to the leaky wave trace velocity matching angle 81 = sin- 1(c/c/). The surface pressure is approximated using Eq. (1) by introducing a leaky wave amplitude reflection coefficient B at the end of the cylinder and introducing other approximations [3]. The resulting amplitude is geometrically propagated to a plane tangent to the cylinder and a Rayleigh integral is evaluated to give the farfield amplitude. The original result [3] for the case y = el has recently been extended to describe the degradation of the amplitude when y is shifted away from 0/. For scattering by finite cylinders, it is convenient to relate the incident and ftileld scattered pressures, p. and pls, by a dimensionless form function fl through the relationship pls = (pof~a/2R) exp(ikR) where a is the radius of the cylinder and R is the distance from a reference point on the cylinder. The analysis based on Eq. (1) shows that for leaky waves of interest when ka is large and the tilt y = 8[, typical values of If/lare greater than unity. It follows that this elastic contribution to the backscattering amplitude from an appropriately tilted cylinder can be greater than 583