The method of multiple scales is employed to analyze the wave propagation in a rectangular hard-walled duct whose walls have weak periodic undulations. Interacting modes traveling in the same direction propagate unattenuated. The energy is continuously exchanged between the two modes. The modes that travel in opposite directions are attenuated and, there, may be cut off. This cutoff will depend upon the geometry of the cross section as well as the phase-angle differences between the undulations of the opposite walls.
Starting with magnetohydrodynamic (MHD) equilibria for plasma field, and under certain conditions these MHD equations were reduced to a single nonlinear elliptic equation for magnetic potential ũ , known as Grad-Shafranov equation. By specifying the arbitrary functions in this equation, the Bullough-Dodd equation was obtained. Travelling wave solutions were given which turned out to be a very unimportant special case. In this paper, we give exact solutions using rational sech-csch functions.
The method of straightforward expansion is applied to acoustic waves in a half-space with sinusoidal boundary. It is found that the expansion is uniform. Propagating waves show their dependence on the undulations of the surface in a proportional manner to the order of the terms. Moreover, it is found that for certain values of wavenumbers the propagating waves have terms that are body waves.
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