Mapped wave-envelope elements of variable radial order are presented for the computation of time-harmonic, unbounded, three-dimensional acoustical fields. Their application to transient problems is described in a companion article (Part II). Accuracy is assessed by a comparison of computed and analytic solutions for multi-pole fields generated by a vibrating sphere. Solutions are also presented for plane wave scattering. Elements of radial orderm+lare shown to be capable of modeling multi-pole components of orderm,although the provision of adequate transverse resolution is shown to be a stringent requirement, particularly at high frequencies. Ill-conditioning of the coefficient matrix limits the practical implementation of the method to elements of radial order eleven or less. The utility of the method for more general geometries is demonstrated by the presentation of computed solutions for the sound field generated by the vibration of a cylindrical piston in a plane baffle and of an idealised engine casing under anechoic conditions. The computed results are shown to be in close agreement with the analytic solution in the case of the cylindrical piston, and with a boundary element solution in the case of the engine casing.