The existence of a circulation of sound energy in an enclosure of square cross section was demonstrated on theoretical grounds by Preston Smith in 1963. Recent work has confirmed experimentally the flow of sound energy in a vortex pattern in various sound fields. Here it is shown that for any rectangular enclosure (e.g., a room or duct) having reflecting walls with two dimensions in the ratio of any two integers, there exists an infinite set of vortex modes, whose frequencies form a harmonic series. Also, for an enclosure in the form of a rigid circular cylinder, there exists infinite sets of vortex modes whose frequencies depend on the extreae of the Bessel functionsJm. Energy streamlines are given for some of the above modes, and their practical significance is discussed.