Going beyond the ideal open end condition, we may assume that only plane waves propagate in the duct but that a non-zero pressure arises at the end due to the radiation of sound from the end. This is called the piston approximation. In the low frequency limit, the result is that the pressure node is moved a fraction of a tube radius down the axis of the duct from the actual tube end. This is known as length correction and is discussed in pp.180-181 of Fletcher and Rossing . The full expression for the piston approximation radiation impedance is available for an unflanged cylindrical duct due to Levine and Schwinger  and in the case of a flanged cylindrical duct due to Rayleigh . Both are graphed in Fletcher and Rossing  pp.181-182. In rectangular geometry the radiation impedance of a rectangular piston (which is the equivalent to the impedance for the plane velocity and plane pressure modes in a rectangular duct) mounted in an infinite baffle has been treated [47,48,49,50,51].