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

142 7. Pipes, Waveguides, and ResonatorsFrom Equation (7.38), the phase speed for mode lm is not c butc p = ω/k z = √1 −c(klmk)= √21 −c( ωlmω) 2(7.47)A physically meaningful solution can be written in the complex exponential formp lm = 1 4 A ∑lm e i(ωt±k xlx±k ym y−k z z)(7.48)±The absence of the boundary at a location on the +z-axis necessitates only anegative sign for k z , since there will be no reflected waves propagating in the −zdirection.The propagation vector k for each of the four traveling waves forms anangle θ with the z-axis, according tocos θ = k z√k = 1 − ( ω lm)(7.49)2ωand the corresponding phase speed c p is˘c p =c(7.50)cos θThe surfaces of constant phase for two component waves representing the (0,1)mode are shown in Figure 7.4 for a rigid waveguide. The waves cancel eachother precisely for y = 1/2L y , with the result there exists a nodal plane halfwayFigure 7.4. The surfaces of constant phase for two-component waves in the (0,1) mode.