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

7.9 A Waveguide with Constant Cross Section 143between the walls. The waves remain in phase at the upper and the lower walls,so the pressure amplitude is maximized at these rigid boundaries. The apparentwavelength λ z in the z-direction is given byλ z =λ(7.51)cos θIn the lowest mode (0,0), k z = k and the four component waves form a singleplane wave that travels down the axis of the waveguide with speed c. For allthe other modes, the propagation vectors of the component waves generally formangles with the z-axis, with one aimed into each of the four forward quadrants.According to Equation (7.47), at frequencies much greater than the cutoff of the(l, m) mode, i.e., ω ≫ ω lm , the angle θ approaches zero and the waves are travelingfairly straight down the waveguide. As the input frequency approaches the cutoffvalue, θ increases with the result that the component waves move in increasinglytransverse directions. In fact, when the frequency reaches the stage that ω = ω lm ,the component waves are traveling transversely to the axis of the waveguide.Each component wave carries energy along the waveguide through the process ofcontinual reflection from the rigid walls (in the same manner a signal is transmittedthrough a fiberoptic line, bouncing from one wall back to the opposite wall down˘along the line). With the energy of the wave propagating at a speed c in the directionk, the corresponding speed c p (the group speed) of the energy in the z-direction isgiven by the component of the plane-wave velocity c along the waveguide axis:√( ωlm) 2c g = c cos θ = c 1 −(7.52)ωGiven the driving frequency ω, each normal mode, in which ω lm