6 - Guided Waves and Dispersion

Summary

Synopsis

Chapter 6 discusses guided waves and the dispersion they experience. Only the antiplane shear problem is treated. The guided waves are constructed by using partial waves and their dispersion calculated by using the transverse resonance principle. Both harmonic and transient excitations of a closed waveguide are studied by using an expansion of modes. The harmonic excitation of an open waveguide by a line source is also studied, though in this case by using both ray and mode representations. As a last example, we examine propagation in a closed waveguide with a slowly varying thickness, using an asymptotic expansion that combines features of both rays and modes. We close by examining how information and energy propagate at the group velocity.

Harmonic Waves in a Closed Waveguide

We consider a layer of infinite extent in the x₁ direction and of finite thickness in the x₂ direction. Within the layer, the coordinate x₂ ∈ (–h, h) and the plane x₂ = 0 is a plane of reflection symmetry. This structure is a waveguide or guide because the waves are forced to propagate in the x₁ direction and the guide is closed because waves are completely trapped within the structure. We are interested in learning what kinds of antiplane waves propagate in the guide without, at present, seeking to know how they are excited. Accordingly, we seek possible solutions to the following antiplane problem.