Tracing lined cylindrical duct modes ~ eiωt-ikx from hard wall to hard wall, by varying X in Z = R + i·X

R = 1.4, ω = 5, m = 0, M = 0

R = 0.4, ω = 5, m = 0, M = 0

R = 0.09, ω = 5, m = 0, M = 0

R = 1.4, ω = 5, m = 0, M = 0.5

R = 0.4, ω = 5, m = 0, M = 0.5

R = 0.001, ω = 5, m = 0, M = 0.5

(see A Classification of Duct Modes Based on Surface Waves, Wave Motion, 37, p.119-135, 2003.)