Theoretical investigation on end correction problems. IV Transient flow analysis by multi-mode wave equation

An exact description of the transitional wave flow in the 'transient' end correction problem is presented. The propagation of a transient 'nonplane' wave in a tube generated by a step change of its cross-section is examined. The multimode wave equation is applied to the analysis together with the Laplace transformation, yielding series expansions by Bessel functions as solutions. After the transient properties of an individual higher-mode wave are made clear, some intriguing behaviors of the entire nonplane wave in a tube are revealed. For an infinitely augmented tube radius, these results turn out to be useful for describing an expanding wave flow around an aperture in an infinite plane wall. Finally, the pressure and velocities in the flow are determined in terms of definite functions through reduction of the infinite series.