ABSTRACT
In this chapter wave propagation in 2D isotropic waveguides is presented. Additional dimensions introduce an additional horizontal wavenumber. The governing Navier's equations is first subjected to Helmholtz decomposition to split the original Partial differential equation into two different partial differential equations in terms of scalar and vector potentials. Such a splitting give rise to two different waves propagating in 2D isotropic waveguides, namely the P-waves and S-waves, respectively. S-waves, propagating in an infinite media can further be split into SV-waves and SH-Waves. The solution of these equations requires two transforms, one in time and other in one of the spatial dimensions. Here, instead of the second transform in the spatial dimension, Fourier series is used. Firstly, the propagation of P-waves, S-waves and SH-waves in an infinite media is discussed, and their propagation characteristics are evaluated by plotting their wavenumber and speed variation as a function of horizontal wavenumber. Next, wave propagation in a semi-infinite media is discussed, where one edge is bounded. Here, the propagation characteristics for different boundary condition at the bounded edge is discussed and their respective wavenumbers are obtained. These boundary conditions couple the P and S-waves and generate an entirely new set of waves. One such important boundary condition that has great significance is the stress-free boundary condition at the bounded edge, which generates what is well known as is the Rayleigh wave. This case is discussed in detail, and it is shown that the obtained wavenumbers are Poisson's ratio-dependent and using these wavenumbers and the solutions for the governing equations, Rayleigh wave responses are plotted to bring out the wave propagation characteristics. Next, wave propagation in a doubly bounded media is presented. Here the 2 edges of the 2D waveguides are bounded. Again, here different boundary conditions at the top and bottom edges are considered, and their respective wave propagation characteristics are evaluated. Two important boundary conditions that lead to complex transcendental equation for solution of wavenumbers and phase speeds are the fix-fixed boundary conditions and the stress-free boundary conditions. The latter is normally referred to as Lamb wave propagation. Two cases arise in the case of wave propagation in doubly bounded media, namely the symmetric and antisymmetric loading cases. Although there are several methods available in the literature to solve these transcendental equations, here bracketing technique is implemented to obtain the wavenumbers and their corresponding speeds. Here, first, the method of solution of these transcendental equations using bracketing technique is discussed followed by the presentation of real, imaginary and complex wavenumbers as well as their corresponding phase speeds for both the loading cases. The last part of this chapter deal with wave propagation in plates, wherein first, the spectral analysis is performed on the governing equation of plates to obtain its spectrum and dispersion relations. This analysis is followed by the presentation of wave propagation in a plate edge, wherein the plate is subjected to oblique incidence. The chapter ends with a summary, the details of the MATLAB Codes provided and some exercise problems.
