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

7.8 A Waveguide with Constant Cross Section 141with permitted values of k xl and k ym , resulting from the boundary conditions ofrigidity, these beingk xl = lπ ,L xl = 0, 1, 2,...k ym = mπ ,L ym = 0, 1, 2,... (7.40)We rearrange Equation (7.39) to find k z√ω2k z =c − 2 k2 xl − k2 ym (7.41)Because ω may have any value, Equation (7.38) comprises a solution for all valuesof ω, in contrast to the totally enclosed cavity which allows for only quantizedfrequencies. Setting√k lm = kxl 2 + k2 ym (7.42)we can shorten Equation (7.41) to√ω2k z =c − 2 k2 lm(7.43)The value k z is real when ω/c > k lm . We then obtain a propagating mode,asthewave moves in the +z-direction. The cutoff frequency occurs when ω/c = k lm ,sodefining the limit for which k z remains real; is given byω lm = ck lm (7.44)for the (l, m) mode. A frequency below the threshold value of ω lm results in apurely imaginary value of k z ,√k z ≡±i klm 2 − ω2c 2 (7.45)We need to include the negative sign in Equation (7.45) so that p → 0asz →∞and the eigenfunctions assume the form(√p lm = A lm cos k xl x cos k ym ye − klm )z 2 − ω2c 2 eiωt(7.46)Equation (7.46) represents a standing wave that decays exponentially with z. Thisform of eigenfunction is termed an evanescent mode—i.e., no energy is propagatedalong the waveguide. If the frequency exciting the waveguide is just below thecutoff value of some particular mode, then this and higher modes are evanescentand are of little consequence at appreciable distances from the source. The lowermodes may propagate energy and can be detected at large distances from thesource.Only plane waves can propagate in a rigid-walled waveguide if the frequencyof the sound is sufficiently low. This frequency is less than c/(2L), where L is thelarger dimension of the rectangular cross section.