Mathematical modelling of ultrasonic waves in heterogeneous materials using stochastic differential equations


Ultrasonic transducer arrays are having an increasing impact on the NDT associated with the imaging of flaws and, more recently, in the construction of spatial maps of the localised pre-crack structures and textures within solid materials. Armed with the huge datasets that arise from this advance in technology (circa 100 megabytes per scan), the challenge is now for mathematicians to develop fast, real-time algorithms that maximise the use of this data in flaw detection and characterisation. One key application is so-called ‘difficult’ materials, where the spatial distribution of the material’s properties change in a random fashion (heterogeneous) and, in addition, this variation is commensurate with the wavelength of the ultrasonic wave. So, the transmitted wave undergoes a high degree of scattering and this leads to the received wave being severely attenuated and having an extended support in the time domain with a stretched coherent wave and a long incoherent coda wave. In order to develop better algorithms, it is important to develop a fundamental understanding of the manner in which waves propagate in these materials. This can be achieved by building up, from basic physical principles, mathematical models of these materials and wave propagation within them. This paper sets out a mathematical approach to the physical modelling of ultrasound wave propagation in difficult materials based on stochastic differential equations. The models will provide means of gaining analytical insight into how the statistical distribution of the material’s properties affects the form of the received waves, in particular, the amplitude and spread of the received coherent wave and the statistical properties of the incoherent received wave. This knowledge can then be used to analytically assess the resolution performance of imaging algorithms and in the use of ultrasound arrays in design-phase NDT.