It is impossible to predict theoretically the time evolution of the back scattered signals even though the individual interaction mechanisms of the acoustic wave with microstructure is known mathematically [2] because of the presence of intrinsic microscopic randomness in the microstructure. Again the interaction of the acoustic wave with the individual microscopic inhomogeneity is not governed by its former interaction process so the mechanism turns out to be Non-Markovian.

Since an exact theoretical analysis of the entire back scattered signal is practically impossible, a statistical approach is very essential. Mandelbrot [3] introduced the fractal and the concept of the fractal dimension as a means of describing and quantifying these random signals. The concept of fractal dimension has been introduced to characterize discrete time domain back scattered ultrasonic signals from polycrystalline copper (Cu) and single crystal potassium chloride (KCl).

For the analysis of the back scattered ultrasonic signals, a broad band ultrasonic probe with central frequency of 5 MHz and of crystal diameter of 12.5 mm for Cu and a broadband probe of central frequency of 10 MHz with crystal diameter 6 mm for KCl was used. The probe was excited by a pulser receiver unit to generate a short duration ultrasonic pulse in the plane parallel specimens of Cu (50 mm x 50 mm x 10 mm) and KCl (50 mm dia, 7.56 mm thickness). Scattered ultrasonic signals starting from the first back wall echo from these materials were captured by the same probe and were sampled at a sampling rate of 10 ns for 512 data points. The position of the probe on the sample was changed at random and the backscattered signals of 512 data points from each location were captured. These eight individual scattered signals (each being a representative of the microstructure of a particular location along its thickness) were then joined together to form the entire signal of 4096 data points. The analysis of this signal will certainly give the global information of the microstructure. The fractal dimension for these scattered discrete ultrasonic signals from Cu and KCl were evaluated by the box counting method [4]. The values of the fractal dimension for Cu and KCl are found to be 1.5297 and 1.5458 respectively. These are two unique distinct values representing the respective microstructure. The observed variation of the fractal dimension is discussed in light of acoustic scattering mechanism in this paper. REFERENCES

1. E. P. Papadakis, "Ultrasonic Attenuation Caused by Scattering in Polycrystalline Media", in Physical Acoustics: Principles and Methods, Vol. IV, Part-B (Edited by W. P. Mason), New York: Academic Press, 1968
2. R. Truell, C. Elbaum and B. B. Chick, "Ultrasonic Methods in Solid State Physics", New York and London: lAcademic Press, 1969
3. B. B. Mandelbrot, "The Fractal Geometry of Nature", New York: Freeman, 1982
4. H. O. Peitgen, H. Jurgens and D. Saupe, "Chaos and Fractals: New Frontiers of Science", New York: Springer-Verlag, 1992