A brief review of the modern state-of-the art and tendencies of further development of various methods of solution of wave dispersion problems in heterogeneous functionally graded elastic waveguides is presented. Main types of functionally graded materials and structures, including gradient thon-walled structures, and their main engineering applications is discussed. The main difficulties of modelling of the stress-strain state of functionally graded shells and plates are pointed, as well as the possible ways to overcome such difficulties. The main theoretical bases of definition of effective constitutive constants of functionally graded materials and their possible estimates used in the practice are considered. Main dependencies of the effective constitutive constants of a functionally graded material on coordinates used for the mathematical modelling of the dynamics are also shown. The statement of the dynamics problem for a functionally graded waveguide and the appropriate statement of the normal wave dispersion problem are pointed. The presented Part I of the review consider some analytical methods of solution of dispersion problems, mainly the matrix ones based on the formulation of the steady dynamics problem in the image space as a first-order ordinary differential equations system. The state vectors corresponding to the useful Cauchy and Stroh formalisms are introduced, and the appropriate governing equations and the boundary conditions on waveguide’s faces are presented. Classical methods for solving the steady dynamics problem for a laminated waveguide are briefly described, which could be a basis for the further approximation of a functionally graded material by a system of layers with constant properties, i.e. the transfer matrix method, its main modifications developed to ensure the stability of calculations, and the global matrix method. Then, the intensively developed last 15 years reverberation matrix method, stiffness matrix method, and the Peano series method are discussed. Some key solutions of the wave dispersion problems for heterogeneous layers are presented; such solutions improve the efficiency of approximation of a functionally graded structure by a laminated one. The implicit solution of the general problem of steady dynamics for a waveguide with arbitrary gradation law is shown. The key features of the discussed matrix methods are pointed briefly as well as their main drawbacks. In the Part II, the main attention will be paid to methods of semi-analytical solution of dispersion problems based on the approximation of a waveguide by an equivalent system with a finite number of degrees of freedom: power series, generalized Fourier series, semi-analytical finite elements. spectral elements, as well as methods based on various theories of plates and shells.

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