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Fourier series multiscale method, a concise and efficient analytical approachfor multiscale computation, will be developed out of this series of papers. Inthe sixth paper, exact analysis of the wave propagation in a beam withrectangular cross section is extended to a thorough multiscale analysis for asystem of completely coupled second order linear differential equations formodal functions, where general boundary conditions are prescribed. For thispurpose, the modal function each is expressed as a linear combination of thecorner function, the two boundary functions and the internal function, toensure the series expressions obtained uniformly convergent and termwisedifferentiable up to second order. Meanwhile, the sum of the corner functionand the internal function corresponds to the particular solution, and the twoboundary functions correspond to the general solutions which satisfy thehomogeneous form of the equations. Since the general solutions haveappropriately interpreted the meaning of the differential equations, thespatial characteristics of the solution of the equations are expected to bebetter captured in separate directions. With the corner function, the twoboundary functions and the internal function selected specifically aspolynomials, one-dimensional full-range Fourier series along the x2 (orx1)-direction, and two-dimensional full-range Fourier series, the Fourierseries multiscale solution of the wave propagation in a beam with rectangularcross section is derived. For the beam with various boundary conditions,computation and analysis are performed, and the propagation characteristics ofelastic waves in the beam are presented. The newly proposed accurate wave modelhas laid a solid foundation for simultaneous control of coupled waves in thebeam and establishment of guided wave NDE techniques.

Fourier series (based) multiscale method for computational analysis in  science and engineering: VI. Fourier series multiscale solution for wave  propagation in a beam with rectangular cross section