The real roots of the fluid-loaded plate

The vibration of an infinite, thin plate, immersed in a fluid and excited by a time-harmonic line-force, is a classical problem in acoustics. The classical solution to the fluid-loaded plate argues that one pair of real roots, equal in magnitude and opposite in sign, exists to the equation defining the free wavenumber. However, recent papers by Stuart [J. Acoust. Soc. Am. 59, 1160–1169; 1170–1174 (1976)] and by Pierucci and Graham [J. Acoust. Soc. Am. 62, S84(A) (1977)] argue that, for certain plate-fluid combinations, three such pairs of real roots exist: the classical pair plus two additional pairs. This paper reexamines this problem to clarify the existence and physical significance of these multiple pairs of real roots. It is shown that only the classical pair of real roots (1) can exist when the radiation condition is satisfied, (2) characterizes the waves in a free plate, and (3) characterizes significant wavenumber contributions to the displacement of the line-force excited plate at large distances from the line excitation. The other two paris of real roots result only if the radiation condition is ignored, and therefore cannot characterize free waves in the plate. Further, these wavenumbers do not characterize any nondecaying contributions to the displacement of the line-force excited plate.