ESDU 01015 introduces computer programs for a theoretical method for the prediction of the axisymmetric far-field sound radiation pattern resulting from the radiating end of an unflanged@ hard-walled circular section duct. The effect of flow in whch the duct may be immersed is also included. The method is applicable to aircraft jet engines and to large exhaust stacks and mufflers. The two programs are provided as source code and as executable versions@ one calculating the directivity in the far-field at chosen angles to the duct axis forwards or rearwards and the other assisting the user in creation of the required input file. The directivity may be calculated for a single mode or for multimode radiation. For multimode radiation the user may choose a uniform distribution of either dipoles or monopoles@ equal power in each mode@ or may specify the modes and amplitudes. Two worked examples illustrate the use of the program. No comparison is made with experimental data but it is indicated that the literature shows the method to be in good agreement with results obtained with a turbo-fan engine. The theoretical method is summarised and its assumptions and limitations are briefly indicated.