THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

140 7. Pipes, Waveguides, and Resonatorsdirection of its propagation vector ˆk which has projections ±k xl ′, ±k ym ′, ±k zn ′ onthe coordinate axes. The standing wave solution results from the superpositionof eight traveling waves (one into each quadrant) whose directions of travel areconstrained by the boundary conditions. The methodology involving separation ofvariables can also be used to treat standing waves in other simple cavities, suchas cylindrical and spherical cavities, with the eigenfunctions entailing Bessel andLegendre functions.7.8 A Waveguide with Constant Cross SectionIn Figure 7.3 a waveguide of rectangular cross section is assumed to have rigidside walls and a source of acoustic energy located at its boundary z = 0. Thereis no other boundary on the z-axis, which permits energy to propagate down thewaveguide. This results in a situation where the wave pattern consists of standingwaves in the transverse directions x and y and a traveling wave in the z-direction.The mathematical solution which would contain applicable eigenfunctionsisp lmn = A lmn cos k xl x cos k ym ye i(ωt−k zz)(7.38)Upon substituting Equation (7.38) into the wave equation (7.31) we obtain thefollowing relationship:ω 2c 2 = k2 = k 2 xl + k2 ym + k2 z (7.39)Figure 7.3. A waveguide having a rectangular cross section. The travel is along the z-axis.