A WAVE SPLITTING/INVARIANT EMBEDDING APPROACH TO AN ULTRASONIC INVERSE PROBLEM Paul Stucky Electrical and Comyuter Engineering Iowa State Universlty Ames, Iowa 50011 William Lord Electrical and Computer Engineering Iowa State University Ames, Iowa 50011 INTRODUCTION Material property measurement is an important area of basic and applied research and can be defined as an inverse problern in which knowleiige of the system input and output can be used to determine material properties. A tool for studying the forward and inverse problems, denoted here as wave splitting and invariant embedding (WSIE), has been developed and offers unique physical insights [1] [2] [3]. Wave splitting and invariant embedding (WSIE) is a theory which relates material properties to system tmpulse response functions in precise ways. WSIE theory provides a clear way for solving the inverse problern if the response functions are known. In turn 1mpulse response functions relate system inputs to the outputs. If response functions can be recovered from measured system inputs and outputs then WSIE has potential application for determining material properties. Recent experiments in an electromagnetic coaxialline system utilizing wave splitting and invariant embedding signal processing teclmiques have demonstrated the proof of principle of material property reconstruction for linear, isotropic and spatially inhomogeneous dielectric material [4] as weil as for LHI dispersive material [5]. The mechanical analog to electromagnetic LHI dispersive media is LHI viscoelastic media. Most polymeric solids and liquids can be described by linear viscoelasticity theory under conditions of infinitesimal stress, strain and displacement [6]. Considering the success of these two experiments employing electromagnetic waves 1 the question is posed: 'can the viscoelastic moduli of an LHI viscoelast1c medium be reconstructea from knowledge of the incident 1 reflected and transmitted ultrasonic waves using wave splitting and invariant embedding tecnniques?' WAVE SPLITTINGANDINVARIANT EMBEDDING Ultrasonic wave propagation through LHI viscoelastic has been previously discussed in the context of time-domain finite element modeling [7] [8] [9]. For one dimensional wave propagation the appropriate wave equations for longitudinal displacement in terms of a stress relaxation modulus or, equivalently, a creep compliance are, respectively, [10]: 1 .. ( ) z, t t) t) + m * t) 1 .. ( t) 1. ""() z, + C2n * 'Uz z, t (1) (2) where the normalized longitudinal stress relaxation modulus, m(t), creep compliance, n(t), and the wave speed, c, are defined by m(t) n(t) c M(t) M(O) N(t) N(O) [ii(O) __ 1 V p- .jpN(O). ReVIew g{ Pro.J!ress m Quantztanve Nondestrucnve Evaluatw11. Yal. Jfi Edtted by 0.0. Thompson and D.E. Chtmentt, Plenum Press, New York, 1997 (3) (4) (5) 75